chapter 11 mastering physics
TRANSCRIPT
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Chapter 11 Homework Assignment
Due 1100pm on Saturday October 10 2015
You will receive no credit for items you complete after the assignment is due Grading Policy
Video Tutor Balancing a Meter Stick
First launch the video below You will be asked to use your knowledge of physics to predict the outcome of an
experiment Then close the video window and answer the question at right You can watch the video again at any
point
Part A
Suppose we replace the mass in the video with one that is four times heavier How far from the free end must we
place the pivot to keep the meter stick in balance
Hint 1 How to approach the problem
For the meter stick to be in equilibrium the net torque on it must be zero Torques about the fulcrum may
be exerted by the mass hanging from the end of the stick and by the stickrsquos own weight
Use the condition that the net torque must be equal to zero to obtain a relationship involving 1) the distance
between the left end of the stick and the fulcrum and 2) the distance between the center of mass of the
stick and the fulcrum These two distances must add up to a constant You should get two equations that
you can solve for the location of the fulcrum
ANSWER
Correct
90 cm (10 cm from the weight)
75 cm (25 cm from the weight)
10 cm
50 cm (in the middle)
25 cm
Typesetting math 20
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A Bar Suspended by Two Wires
A nonuniform horizontal bar of mass is supported by two massless wires against gravity The left wire makes an
angle with the horizontal and the right wire makes an angle The bar has length
Part A
Find the position of the center of mass of the bar measured from the bars left end
Express the center of mass in terms of and
Hint 1 Nature of the problem
This is a statics problem There is no net force or torque acting on the bar
Hint 2 Torques about left end of bar
The net torque is zero about any point you select Here we ask you to find the net torque of the system
about the left end of the bar Label the tension in the left wire and label the other wires tension The
weight of the bar is Note that the vector sum of and is zero Using the sign
convention shown in the picture express the sum of the torques about the left end of the bar
Answer in terms of andor Note that not all of these quantities will appear
in your answer
ANSWER
Hint 3 Forces x components
Assume that the tensions in the left and right wires are and respectively What is the sum of the x
components of the forces Because this is a statics problem these forces will sum to zero
Use the sign convention indicated in the figure and express your answer in terms of
andor Note that not all of these quantities will appear in your answer
983149
ϕ
1
ϕ
2
L
983160
L ϕ
1
ϕ
2
T
1
T
2
W =
983149 983143 T
1
T
2
W
L 983160 W T
2
T
1
ϕ
2
ϕ
1
=sum = 0 τ
l e f t
L s i n ( ) minus W 983160 T
2
ϕ
2
T
1
T
2
Σ F
983160
L 983160 W T
2
T
1
ϕ
2
ϕ
1
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ANSWER
Hint 4 Forces y components
Assuming that the tensions in the left and right wires are and respectively what is the sum of the y
components of the forces Because this is a statics problem these forces will sum to zero
Use the sign convention indicated in the figure and express your answer in terms of
andor Note that not all of these quantities will appear in your answer
ANSWER
Hint 5 Eliminate weight from your equations
You should have found three equations by now It is possible to eliminate two variables and solve for interms of the others As an intermediate step solve your torque equation for in terms of texttipT_rm
2T_2 texttipLL etc and then solve your y -component force equation for texttipWW and substitute
back into your expression for texttipxx In other words find an expression for texttipxx
Answer in terms of texttipT_rm 1T_1 texttipT_rm 2T_2 texttipphi _rm 1phi_1
texttipphi _rm 2phi_2 and texttipLL
ANSWER
Hint 6 A useful trig identity
The dimensions for the expression you just found for texttipxx are correct since the units of the
tensions cancel out leaving the units of length in the numerator If you now solve the x -component force
equation for texttipT_rm 1T_1 in terms of texttipT_rm 2T_2 and substitute into your equation for
texttipxx you should find the following trig identity useful
sin(a+b) = sin(a)cos(b)+cos(a)sin(b)
Alternatively you could express your answer in terms of tan(phi_1) and tan(phi_2)
ANSWER
Correct
Precarious Lunch
=sum = 0 F
983160
minus c o s ( ) + c o s ( ) T
1
ϕ
1
T
2
ϕ
2
T
1
T
2
Σ F
983161
L 983160 W T
2
T
1
ϕ
2
ϕ
1
=sum = 0 F
983161
s i n ( ) + s i n ( ) minus W T
1
ϕ
1
T
2
ϕ
2
983160
983160 W
texttipxx
=
largefracT_2 sinleft(phi_2right) LT_1 sinleft(phi_1right)+T_2
sinleft(phi_2right)
texttipxx = largeL fractanleft(phi_2right)tanleft(phi_2right)+tanleft(phi_1right)
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A uniform steel beam of length texttipLL and mass texttipm_rm 1m_1 is attached via a hinge to the side of a
building The beam is supported by a steel cable attached to the end of the beam at an angle texttiptheta theta a
shown Through the hinge the wall exerts an unknown force texttipFF on the beam A workman of mass
texttipm_rm 2m_2 sits eating lunch a distance texttipdd from the building
Part A
Find texttipTT the tension in the cable Remember to account for all the forces in the problem
Express your answer in terms of texttipm_rm 1m_1 texttipm_rm 2m_2 texttipLL texttipd
d texttiptheta theta and texttipgg the magnitude of the acceleration due to gravity
Hint 1 Pick the best origin
This is a statics problem so the sum of torques about any axis a will be zero In order to solve for texttipT
T you want to pick the axis such that texttipTT will give a torque but as few as possible other
unknown forces will enter the equations So where should you place the origin for the purpose of calculating
torques
ANSWER
Hint 2 Calculate the sum torques
Now find the sum of the torques about center of the hinge Remember that a positive torque will tend to
rotate objects counterclockwise around the origin
Answer in terms of texttipTT texttipLL texttipdd texttipm_rm 1m_1
texttipm_rm 2m_2 texttiptheta theta and texttipgg
At the center of the bar
At the hinge
At the connection of the cable and the bar
Where the man is eating lunch
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ANSWER
ANSWER
Correct
Part B
Find texttipF_mit xF_x the texttipxx-component of the force exerted by the wall on the beam ( texttipF
F) using the axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT and other given quantities
Hint 1 Find the sign of the force
The beam is not accelerating in the texttipxx-direction so the sum of the forces in the texttipxx-
direction is zero Using the given coordinate system is texttipF_mit xF_x going to have to be positive
or negative
ANSWER
Correct
Part C
Find texttipF_mit yF_y the y-component of force that the wall exerts on the beam ( texttipFF) using the
axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT texttiptheta theta texttipm_rm 1m_1
texttipm_rm 2m_2 and texttipgg
ANSWER
Sigmatau_a = 0 = largeleft(fracm_1 L2+m_2 dright) g-T L sinleft(thetaright)
texttipTT = largefracgleft(m_1 fracL2+m_2 dright)L sinleft(thetaright)
texttipF_mit xF_x = -T cosleft(thetaright)
texttipF_mit yF_y = gleft(m_1+m_2right)-T sinleft(thetaright)
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Correct
If you use your result from part (A) in your expression for part (C) youll notice that the result simplifies
somewhat The simplified result should show that the further the luncher moves out on the beam the lower
the magnitude of the upward force the wall exerts on the beam Does this agree with your intuition
Three-Legged Table
The top view of a table with weight texttipW_rm tW_t is shown in the figure The table has lost the leg at
(texttipL_mit xL_x texttipL_mit yL_y) in the upper
right corner of the diagram and is in danger of tipping over
Company is about to arrive so the host tries to stabilize the
table by placing a heavy vase (represented by the green
circle) of weight textt ipW_rm vW_v at ( texttipXX
texttipYY) Denote the magnitudes of the upward forces
on the table due to the legs at (0 0) (texttipL_mit xL_x
0) and (0 texttipL_mit yL_y) as texttipF_rm 0F_0
texttipF_mit xF_x and texttipF_mit yF_y
respectively
Part A
Find texttipF_mit xF_x the magnitude of the upward force on the table due to the leg at (texttipL_mit x
L_x 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipXX texttipYY
texttipL_mit xL_x andor texttipL_mit yL_y Note that not all of these quantities may appear in
the answer
Hint 1 Find the useful vector relations
This is a statics problem so about any point texttippp we have sum vectau_p = 0 and at texttipp
p sum vecF = 0 Each is a vector equation yielding algebraic expressions for each of the three
components x y and z Using the coordinate system shown in the figure indicate for each component of these equations whether it gives a useful relationship among the various forces
Check all that apply
Hint 1 Direction of applied forces
All of the forces are in the z direction Figure out what nonzero components of the torque these could
possibly generate
ANSWER
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Hint 2 Find the y component of the torque
What is sum tau_y the y component of the torque equation
Express the y component of the torque in terms of texttipW_rm vW_v texttipW_rm tW_t
texttipF_mit xF_x texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y
texttipXX andor texttipYY Note that not all terms may appear in the answer
Hint 1 Choice of origin for torques
Take torques around the origin of the coordinate system (ie this becomes point texttippp in the
torque equations) as this will result in an equation with one and only one unknown force
Hint 2 Sign convention
If you do not get the sign from the cross-product expression tau_y = z F_x - x F_z remember that
the y torque is positive for clockwise rotation about the y axis when looking in the positive y
direction This is a subtle point--normally you look down on the x -y axes from above that is towardnegative z and so positive torque (about the z axis) is counterclockwise
ANSWER
ANSWER
x component of the forces
y component of the forces
z component of the forces
x component of the torques
y component of the torques
z component of the torques
sum tau_y=0 = large-W_v X+left(F_x-fracW_t2right) L_x
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Correct
Part B
Find texttipF_mit yF_y the magnitude of the upward force on the table due to the leg at (0 texttipL_mit y
L_y)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipYY texttipXX
texttipL_mit yL_y andor texttipL_mit xL_x Note that not all of these quantities may appear in
the final answer
Hint 1 Find the x component of torque
What is sum tau_x the x component of the torque equation
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y texttipXX andor
texttipYY Note that not all terms may appear in the answer Remember to pay attention to the
signs of the torque
ANSWER
ANSWER
Correct
Part C
Find texttipF_rm 0F_0 the magnitude of the upward force on the table due to the leg at (0 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipL_mit xL_x texttipL_mit yL_y texttipXX andor texttipYY Note that not all terms may appear in the
answer
Hint 1 Summing the total forces
What is sum F_z the total vertical force on the table
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y andor texttipF_rm 0F_0 Remember to pay attention to signs (with a
positive force being upward on the table)
texttipF_mit xF_x = largefracW_v XL_x+fracW_t2
sum tau_x=0 = large-W_v Y-left(fracW_t2-F_yright) L_y
texttipF_mit yF_y = largefracW_v YL_y+fracW_t2
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ANSWER
ANSWER
Correct
While the host is greeting the guests the cat (of weight texttipW_rm cW_c) gets on the table and walks until her
position is (xy)=(L_xL_y)
Part D
Find the maximum weight texttipW_rm maxW_max of the cat such that the table does not tip over and brea
the vase
Express the cats weight in terms of texttipW_rm vW_v texttipXX texttipYY texttipL_mit
xL_x and texttipL_mit yL_y
Hint 1 Force when table begins to tip
When the table is just beginning to tip over what is the force texttipF_rm critF_crit supplied by the leg
at the origin
sum F_z= 0 = F_0+F_x+F_y-W_v-W_t
texttipF_rm 0F_0 = largeW_v left(1-fracXL_x-fracYL_yright)
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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A Bar Suspended by Two Wires
A nonuniform horizontal bar of mass is supported by two massless wires against gravity The left wire makes an
angle with the horizontal and the right wire makes an angle The bar has length
Part A
Find the position of the center of mass of the bar measured from the bars left end
Express the center of mass in terms of and
Hint 1 Nature of the problem
This is a statics problem There is no net force or torque acting on the bar
Hint 2 Torques about left end of bar
The net torque is zero about any point you select Here we ask you to find the net torque of the system
about the left end of the bar Label the tension in the left wire and label the other wires tension The
weight of the bar is Note that the vector sum of and is zero Using the sign
convention shown in the picture express the sum of the torques about the left end of the bar
Answer in terms of andor Note that not all of these quantities will appear
in your answer
ANSWER
Hint 3 Forces x components
Assume that the tensions in the left and right wires are and respectively What is the sum of the x
components of the forces Because this is a statics problem these forces will sum to zero
Use the sign convention indicated in the figure and express your answer in terms of
andor Note that not all of these quantities will appear in your answer
983149
ϕ
1
ϕ
2
L
983160
L ϕ
1
ϕ
2
T
1
T
2
W =
983149 983143 T
1
T
2
W
L 983160 W T
2
T
1
ϕ
2
ϕ
1
=sum = 0 τ
l e f t
L s i n ( ) minus W 983160 T
2
ϕ
2
T
1
T
2
Σ F
983160
L 983160 W T
2
T
1
ϕ
2
ϕ
1
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ANSWER
Hint 4 Forces y components
Assuming that the tensions in the left and right wires are and respectively what is the sum of the y
components of the forces Because this is a statics problem these forces will sum to zero
Use the sign convention indicated in the figure and express your answer in terms of
andor Note that not all of these quantities will appear in your answer
ANSWER
Hint 5 Eliminate weight from your equations
You should have found three equations by now It is possible to eliminate two variables and solve for interms of the others As an intermediate step solve your torque equation for in terms of texttipT_rm
2T_2 texttipLL etc and then solve your y -component force equation for texttipWW and substitute
back into your expression for texttipxx In other words find an expression for texttipxx
Answer in terms of texttipT_rm 1T_1 texttipT_rm 2T_2 texttipphi _rm 1phi_1
texttipphi _rm 2phi_2 and texttipLL
ANSWER
Hint 6 A useful trig identity
The dimensions for the expression you just found for texttipxx are correct since the units of the
tensions cancel out leaving the units of length in the numerator If you now solve the x -component force
equation for texttipT_rm 1T_1 in terms of texttipT_rm 2T_2 and substitute into your equation for
texttipxx you should find the following trig identity useful
sin(a+b) = sin(a)cos(b)+cos(a)sin(b)
Alternatively you could express your answer in terms of tan(phi_1) and tan(phi_2)
ANSWER
Correct
Precarious Lunch
=sum = 0 F
983160
minus c o s ( ) + c o s ( ) T
1
ϕ
1
T
2
ϕ
2
T
1
T
2
Σ F
983161
L 983160 W T
2
T
1
ϕ
2
ϕ
1
=sum = 0 F
983161
s i n ( ) + s i n ( ) minus W T
1
ϕ
1
T
2
ϕ
2
983160
983160 W
texttipxx
=
largefracT_2 sinleft(phi_2right) LT_1 sinleft(phi_1right)+T_2
sinleft(phi_2right)
texttipxx = largeL fractanleft(phi_2right)tanleft(phi_2right)+tanleft(phi_1right)
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A uniform steel beam of length texttipLL and mass texttipm_rm 1m_1 is attached via a hinge to the side of a
building The beam is supported by a steel cable attached to the end of the beam at an angle texttiptheta theta a
shown Through the hinge the wall exerts an unknown force texttipFF on the beam A workman of mass
texttipm_rm 2m_2 sits eating lunch a distance texttipdd from the building
Part A
Find texttipTT the tension in the cable Remember to account for all the forces in the problem
Express your answer in terms of texttipm_rm 1m_1 texttipm_rm 2m_2 texttipLL texttipd
d texttiptheta theta and texttipgg the magnitude of the acceleration due to gravity
Hint 1 Pick the best origin
This is a statics problem so the sum of torques about any axis a will be zero In order to solve for texttipT
T you want to pick the axis such that texttipTT will give a torque but as few as possible other
unknown forces will enter the equations So where should you place the origin for the purpose of calculating
torques
ANSWER
Hint 2 Calculate the sum torques
Now find the sum of the torques about center of the hinge Remember that a positive torque will tend to
rotate objects counterclockwise around the origin
Answer in terms of texttipTT texttipLL texttipdd texttipm_rm 1m_1
texttipm_rm 2m_2 texttiptheta theta and texttipgg
At the center of the bar
At the hinge
At the connection of the cable and the bar
Where the man is eating lunch
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ANSWER
ANSWER
Correct
Part B
Find texttipF_mit xF_x the texttipxx-component of the force exerted by the wall on the beam ( texttipF
F) using the axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT and other given quantities
Hint 1 Find the sign of the force
The beam is not accelerating in the texttipxx-direction so the sum of the forces in the texttipxx-
direction is zero Using the given coordinate system is texttipF_mit xF_x going to have to be positive
or negative
ANSWER
Correct
Part C
Find texttipF_mit yF_y the y-component of force that the wall exerts on the beam ( texttipFF) using the
axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT texttiptheta theta texttipm_rm 1m_1
texttipm_rm 2m_2 and texttipgg
ANSWER
Sigmatau_a = 0 = largeleft(fracm_1 L2+m_2 dright) g-T L sinleft(thetaright)
texttipTT = largefracgleft(m_1 fracL2+m_2 dright)L sinleft(thetaright)
texttipF_mit xF_x = -T cosleft(thetaright)
texttipF_mit yF_y = gleft(m_1+m_2right)-T sinleft(thetaright)
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Correct
If you use your result from part (A) in your expression for part (C) youll notice that the result simplifies
somewhat The simplified result should show that the further the luncher moves out on the beam the lower
the magnitude of the upward force the wall exerts on the beam Does this agree with your intuition
Three-Legged Table
The top view of a table with weight texttipW_rm tW_t is shown in the figure The table has lost the leg at
(texttipL_mit xL_x texttipL_mit yL_y) in the upper
right corner of the diagram and is in danger of tipping over
Company is about to arrive so the host tries to stabilize the
table by placing a heavy vase (represented by the green
circle) of weight textt ipW_rm vW_v at ( texttipXX
texttipYY) Denote the magnitudes of the upward forces
on the table due to the legs at (0 0) (texttipL_mit xL_x
0) and (0 texttipL_mit yL_y) as texttipF_rm 0F_0
texttipF_mit xF_x and texttipF_mit yF_y
respectively
Part A
Find texttipF_mit xF_x the magnitude of the upward force on the table due to the leg at (texttipL_mit x
L_x 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipXX texttipYY
texttipL_mit xL_x andor texttipL_mit yL_y Note that not all of these quantities may appear in
the answer
Hint 1 Find the useful vector relations
This is a statics problem so about any point texttippp we have sum vectau_p = 0 and at texttipp
p sum vecF = 0 Each is a vector equation yielding algebraic expressions for each of the three
components x y and z Using the coordinate system shown in the figure indicate for each component of these equations whether it gives a useful relationship among the various forces
Check all that apply
Hint 1 Direction of applied forces
All of the forces are in the z direction Figure out what nonzero components of the torque these could
possibly generate
ANSWER
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Hint 2 Find the y component of the torque
What is sum tau_y the y component of the torque equation
Express the y component of the torque in terms of texttipW_rm vW_v texttipW_rm tW_t
texttipF_mit xF_x texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y
texttipXX andor texttipYY Note that not all terms may appear in the answer
Hint 1 Choice of origin for torques
Take torques around the origin of the coordinate system (ie this becomes point texttippp in the
torque equations) as this will result in an equation with one and only one unknown force
Hint 2 Sign convention
If you do not get the sign from the cross-product expression tau_y = z F_x - x F_z remember that
the y torque is positive for clockwise rotation about the y axis when looking in the positive y
direction This is a subtle point--normally you look down on the x -y axes from above that is towardnegative z and so positive torque (about the z axis) is counterclockwise
ANSWER
ANSWER
x component of the forces
y component of the forces
z component of the forces
x component of the torques
y component of the torques
z component of the torques
sum tau_y=0 = large-W_v X+left(F_x-fracW_t2right) L_x
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Correct
Part B
Find texttipF_mit yF_y the magnitude of the upward force on the table due to the leg at (0 texttipL_mit y
L_y)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipYY texttipXX
texttipL_mit yL_y andor texttipL_mit xL_x Note that not all of these quantities may appear in
the final answer
Hint 1 Find the x component of torque
What is sum tau_x the x component of the torque equation
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y texttipXX andor
texttipYY Note that not all terms may appear in the answer Remember to pay attention to the
signs of the torque
ANSWER
ANSWER
Correct
Part C
Find texttipF_rm 0F_0 the magnitude of the upward force on the table due to the leg at (0 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipL_mit xL_x texttipL_mit yL_y texttipXX andor texttipYY Note that not all terms may appear in the
answer
Hint 1 Summing the total forces
What is sum F_z the total vertical force on the table
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y andor texttipF_rm 0F_0 Remember to pay attention to signs (with a
positive force being upward on the table)
texttipF_mit xF_x = largefracW_v XL_x+fracW_t2
sum tau_x=0 = large-W_v Y-left(fracW_t2-F_yright) L_y
texttipF_mit yF_y = largefracW_v YL_y+fracW_t2
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ANSWER
ANSWER
Correct
While the host is greeting the guests the cat (of weight texttipW_rm cW_c) gets on the table and walks until her
position is (xy)=(L_xL_y)
Part D
Find the maximum weight texttipW_rm maxW_max of the cat such that the table does not tip over and brea
the vase
Express the cats weight in terms of texttipW_rm vW_v texttipXX texttipYY texttipL_mit
xL_x and texttipL_mit yL_y
Hint 1 Force when table begins to tip
When the table is just beginning to tip over what is the force texttipF_rm critF_crit supplied by the leg
at the origin
sum F_z= 0 = F_0+F_x+F_y-W_v-W_t
texttipF_rm 0F_0 = largeW_v left(1-fracXL_x-fracYL_yright)
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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ANSWER
Hint 4 Forces y components
Assuming that the tensions in the left and right wires are and respectively what is the sum of the y
components of the forces Because this is a statics problem these forces will sum to zero
Use the sign convention indicated in the figure and express your answer in terms of
andor Note that not all of these quantities will appear in your answer
ANSWER
Hint 5 Eliminate weight from your equations
You should have found three equations by now It is possible to eliminate two variables and solve for interms of the others As an intermediate step solve your torque equation for in terms of texttipT_rm
2T_2 texttipLL etc and then solve your y -component force equation for texttipWW and substitute
back into your expression for texttipxx In other words find an expression for texttipxx
Answer in terms of texttipT_rm 1T_1 texttipT_rm 2T_2 texttipphi _rm 1phi_1
texttipphi _rm 2phi_2 and texttipLL
ANSWER
Hint 6 A useful trig identity
The dimensions for the expression you just found for texttipxx are correct since the units of the
tensions cancel out leaving the units of length in the numerator If you now solve the x -component force
equation for texttipT_rm 1T_1 in terms of texttipT_rm 2T_2 and substitute into your equation for
texttipxx you should find the following trig identity useful
sin(a+b) = sin(a)cos(b)+cos(a)sin(b)
Alternatively you could express your answer in terms of tan(phi_1) and tan(phi_2)
ANSWER
Correct
Precarious Lunch
=sum = 0 F
983160
minus c o s ( ) + c o s ( ) T
1
ϕ
1
T
2
ϕ
2
T
1
T
2
Σ F
983161
L 983160 W T
2
T
1
ϕ
2
ϕ
1
=sum = 0 F
983161
s i n ( ) + s i n ( ) minus W T
1
ϕ
1
T
2
ϕ
2
983160
983160 W
texttipxx
=
largefracT_2 sinleft(phi_2right) LT_1 sinleft(phi_1right)+T_2
sinleft(phi_2right)
texttipxx = largeL fractanleft(phi_2right)tanleft(phi_2right)+tanleft(phi_1right)
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A uniform steel beam of length texttipLL and mass texttipm_rm 1m_1 is attached via a hinge to the side of a
building The beam is supported by a steel cable attached to the end of the beam at an angle texttiptheta theta a
shown Through the hinge the wall exerts an unknown force texttipFF on the beam A workman of mass
texttipm_rm 2m_2 sits eating lunch a distance texttipdd from the building
Part A
Find texttipTT the tension in the cable Remember to account for all the forces in the problem
Express your answer in terms of texttipm_rm 1m_1 texttipm_rm 2m_2 texttipLL texttipd
d texttiptheta theta and texttipgg the magnitude of the acceleration due to gravity
Hint 1 Pick the best origin
This is a statics problem so the sum of torques about any axis a will be zero In order to solve for texttipT
T you want to pick the axis such that texttipTT will give a torque but as few as possible other
unknown forces will enter the equations So where should you place the origin for the purpose of calculating
torques
ANSWER
Hint 2 Calculate the sum torques
Now find the sum of the torques about center of the hinge Remember that a positive torque will tend to
rotate objects counterclockwise around the origin
Answer in terms of texttipTT texttipLL texttipdd texttipm_rm 1m_1
texttipm_rm 2m_2 texttiptheta theta and texttipgg
At the center of the bar
At the hinge
At the connection of the cable and the bar
Where the man is eating lunch
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ANSWER
ANSWER
Correct
Part B
Find texttipF_mit xF_x the texttipxx-component of the force exerted by the wall on the beam ( texttipF
F) using the axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT and other given quantities
Hint 1 Find the sign of the force
The beam is not accelerating in the texttipxx-direction so the sum of the forces in the texttipxx-
direction is zero Using the given coordinate system is texttipF_mit xF_x going to have to be positive
or negative
ANSWER
Correct
Part C
Find texttipF_mit yF_y the y-component of force that the wall exerts on the beam ( texttipFF) using the
axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT texttiptheta theta texttipm_rm 1m_1
texttipm_rm 2m_2 and texttipgg
ANSWER
Sigmatau_a = 0 = largeleft(fracm_1 L2+m_2 dright) g-T L sinleft(thetaright)
texttipTT = largefracgleft(m_1 fracL2+m_2 dright)L sinleft(thetaright)
texttipF_mit xF_x = -T cosleft(thetaright)
texttipF_mit yF_y = gleft(m_1+m_2right)-T sinleft(thetaright)
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Correct
If you use your result from part (A) in your expression for part (C) youll notice that the result simplifies
somewhat The simplified result should show that the further the luncher moves out on the beam the lower
the magnitude of the upward force the wall exerts on the beam Does this agree with your intuition
Three-Legged Table
The top view of a table with weight texttipW_rm tW_t is shown in the figure The table has lost the leg at
(texttipL_mit xL_x texttipL_mit yL_y) in the upper
right corner of the diagram and is in danger of tipping over
Company is about to arrive so the host tries to stabilize the
table by placing a heavy vase (represented by the green
circle) of weight textt ipW_rm vW_v at ( texttipXX
texttipYY) Denote the magnitudes of the upward forces
on the table due to the legs at (0 0) (texttipL_mit xL_x
0) and (0 texttipL_mit yL_y) as texttipF_rm 0F_0
texttipF_mit xF_x and texttipF_mit yF_y
respectively
Part A
Find texttipF_mit xF_x the magnitude of the upward force on the table due to the leg at (texttipL_mit x
L_x 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipXX texttipYY
texttipL_mit xL_x andor texttipL_mit yL_y Note that not all of these quantities may appear in
the answer
Hint 1 Find the useful vector relations
This is a statics problem so about any point texttippp we have sum vectau_p = 0 and at texttipp
p sum vecF = 0 Each is a vector equation yielding algebraic expressions for each of the three
components x y and z Using the coordinate system shown in the figure indicate for each component of these equations whether it gives a useful relationship among the various forces
Check all that apply
Hint 1 Direction of applied forces
All of the forces are in the z direction Figure out what nonzero components of the torque these could
possibly generate
ANSWER
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Hint 2 Find the y component of the torque
What is sum tau_y the y component of the torque equation
Express the y component of the torque in terms of texttipW_rm vW_v texttipW_rm tW_t
texttipF_mit xF_x texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y
texttipXX andor texttipYY Note that not all terms may appear in the answer
Hint 1 Choice of origin for torques
Take torques around the origin of the coordinate system (ie this becomes point texttippp in the
torque equations) as this will result in an equation with one and only one unknown force
Hint 2 Sign convention
If you do not get the sign from the cross-product expression tau_y = z F_x - x F_z remember that
the y torque is positive for clockwise rotation about the y axis when looking in the positive y
direction This is a subtle point--normally you look down on the x -y axes from above that is towardnegative z and so positive torque (about the z axis) is counterclockwise
ANSWER
ANSWER
x component of the forces
y component of the forces
z component of the forces
x component of the torques
y component of the torques
z component of the torques
sum tau_y=0 = large-W_v X+left(F_x-fracW_t2right) L_x
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Correct
Part B
Find texttipF_mit yF_y the magnitude of the upward force on the table due to the leg at (0 texttipL_mit y
L_y)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipYY texttipXX
texttipL_mit yL_y andor texttipL_mit xL_x Note that not all of these quantities may appear in
the final answer
Hint 1 Find the x component of torque
What is sum tau_x the x component of the torque equation
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y texttipXX andor
texttipYY Note that not all terms may appear in the answer Remember to pay attention to the
signs of the torque
ANSWER
ANSWER
Correct
Part C
Find texttipF_rm 0F_0 the magnitude of the upward force on the table due to the leg at (0 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipL_mit xL_x texttipL_mit yL_y texttipXX andor texttipYY Note that not all terms may appear in the
answer
Hint 1 Summing the total forces
What is sum F_z the total vertical force on the table
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y andor texttipF_rm 0F_0 Remember to pay attention to signs (with a
positive force being upward on the table)
texttipF_mit xF_x = largefracW_v XL_x+fracW_t2
sum tau_x=0 = large-W_v Y-left(fracW_t2-F_yright) L_y
texttipF_mit yF_y = largefracW_v YL_y+fracW_t2
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ANSWER
ANSWER
Correct
While the host is greeting the guests the cat (of weight texttipW_rm cW_c) gets on the table and walks until her
position is (xy)=(L_xL_y)
Part D
Find the maximum weight texttipW_rm maxW_max of the cat such that the table does not tip over and brea
the vase
Express the cats weight in terms of texttipW_rm vW_v texttipXX texttipYY texttipL_mit
xL_x and texttipL_mit yL_y
Hint 1 Force when table begins to tip
When the table is just beginning to tip over what is the force texttipF_rm critF_crit supplied by the leg
at the origin
sum F_z= 0 = F_0+F_x+F_y-W_v-W_t
texttipF_rm 0F_0 = largeW_v left(1-fracXL_x-fracYL_yright)
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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A uniform steel beam of length texttipLL and mass texttipm_rm 1m_1 is attached via a hinge to the side of a
building The beam is supported by a steel cable attached to the end of the beam at an angle texttiptheta theta a
shown Through the hinge the wall exerts an unknown force texttipFF on the beam A workman of mass
texttipm_rm 2m_2 sits eating lunch a distance texttipdd from the building
Part A
Find texttipTT the tension in the cable Remember to account for all the forces in the problem
Express your answer in terms of texttipm_rm 1m_1 texttipm_rm 2m_2 texttipLL texttipd
d texttiptheta theta and texttipgg the magnitude of the acceleration due to gravity
Hint 1 Pick the best origin
This is a statics problem so the sum of torques about any axis a will be zero In order to solve for texttipT
T you want to pick the axis such that texttipTT will give a torque but as few as possible other
unknown forces will enter the equations So where should you place the origin for the purpose of calculating
torques
ANSWER
Hint 2 Calculate the sum torques
Now find the sum of the torques about center of the hinge Remember that a positive torque will tend to
rotate objects counterclockwise around the origin
Answer in terms of texttipTT texttipLL texttipdd texttipm_rm 1m_1
texttipm_rm 2m_2 texttiptheta theta and texttipgg
At the center of the bar
At the hinge
At the connection of the cable and the bar
Where the man is eating lunch
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ANSWER
ANSWER
Correct
Part B
Find texttipF_mit xF_x the texttipxx-component of the force exerted by the wall on the beam ( texttipF
F) using the axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT and other given quantities
Hint 1 Find the sign of the force
The beam is not accelerating in the texttipxx-direction so the sum of the forces in the texttipxx-
direction is zero Using the given coordinate system is texttipF_mit xF_x going to have to be positive
or negative
ANSWER
Correct
Part C
Find texttipF_mit yF_y the y-component of force that the wall exerts on the beam ( texttipFF) using the
axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT texttiptheta theta texttipm_rm 1m_1
texttipm_rm 2m_2 and texttipgg
ANSWER
Sigmatau_a = 0 = largeleft(fracm_1 L2+m_2 dright) g-T L sinleft(thetaright)
texttipTT = largefracgleft(m_1 fracL2+m_2 dright)L sinleft(thetaright)
texttipF_mit xF_x = -T cosleft(thetaright)
texttipF_mit yF_y = gleft(m_1+m_2right)-T sinleft(thetaright)
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Correct
If you use your result from part (A) in your expression for part (C) youll notice that the result simplifies
somewhat The simplified result should show that the further the luncher moves out on the beam the lower
the magnitude of the upward force the wall exerts on the beam Does this agree with your intuition
Three-Legged Table
The top view of a table with weight texttipW_rm tW_t is shown in the figure The table has lost the leg at
(texttipL_mit xL_x texttipL_mit yL_y) in the upper
right corner of the diagram and is in danger of tipping over
Company is about to arrive so the host tries to stabilize the
table by placing a heavy vase (represented by the green
circle) of weight textt ipW_rm vW_v at ( texttipXX
texttipYY) Denote the magnitudes of the upward forces
on the table due to the legs at (0 0) (texttipL_mit xL_x
0) and (0 texttipL_mit yL_y) as texttipF_rm 0F_0
texttipF_mit xF_x and texttipF_mit yF_y
respectively
Part A
Find texttipF_mit xF_x the magnitude of the upward force on the table due to the leg at (texttipL_mit x
L_x 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipXX texttipYY
texttipL_mit xL_x andor texttipL_mit yL_y Note that not all of these quantities may appear in
the answer
Hint 1 Find the useful vector relations
This is a statics problem so about any point texttippp we have sum vectau_p = 0 and at texttipp
p sum vecF = 0 Each is a vector equation yielding algebraic expressions for each of the three
components x y and z Using the coordinate system shown in the figure indicate for each component of these equations whether it gives a useful relationship among the various forces
Check all that apply
Hint 1 Direction of applied forces
All of the forces are in the z direction Figure out what nonzero components of the torque these could
possibly generate
ANSWER
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Hint 2 Find the y component of the torque
What is sum tau_y the y component of the torque equation
Express the y component of the torque in terms of texttipW_rm vW_v texttipW_rm tW_t
texttipF_mit xF_x texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y
texttipXX andor texttipYY Note that not all terms may appear in the answer
Hint 1 Choice of origin for torques
Take torques around the origin of the coordinate system (ie this becomes point texttippp in the
torque equations) as this will result in an equation with one and only one unknown force
Hint 2 Sign convention
If you do not get the sign from the cross-product expression tau_y = z F_x - x F_z remember that
the y torque is positive for clockwise rotation about the y axis when looking in the positive y
direction This is a subtle point--normally you look down on the x -y axes from above that is towardnegative z and so positive torque (about the z axis) is counterclockwise
ANSWER
ANSWER
x component of the forces
y component of the forces
z component of the forces
x component of the torques
y component of the torques
z component of the torques
sum tau_y=0 = large-W_v X+left(F_x-fracW_t2right) L_x
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Correct
Part B
Find texttipF_mit yF_y the magnitude of the upward force on the table due to the leg at (0 texttipL_mit y
L_y)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipYY texttipXX
texttipL_mit yL_y andor texttipL_mit xL_x Note that not all of these quantities may appear in
the final answer
Hint 1 Find the x component of torque
What is sum tau_x the x component of the torque equation
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y texttipXX andor
texttipYY Note that not all terms may appear in the answer Remember to pay attention to the
signs of the torque
ANSWER
ANSWER
Correct
Part C
Find texttipF_rm 0F_0 the magnitude of the upward force on the table due to the leg at (0 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipL_mit xL_x texttipL_mit yL_y texttipXX andor texttipYY Note that not all terms may appear in the
answer
Hint 1 Summing the total forces
What is sum F_z the total vertical force on the table
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y andor texttipF_rm 0F_0 Remember to pay attention to signs (with a
positive force being upward on the table)
texttipF_mit xF_x = largefracW_v XL_x+fracW_t2
sum tau_x=0 = large-W_v Y-left(fracW_t2-F_yright) L_y
texttipF_mit yF_y = largefracW_v YL_y+fracW_t2
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ANSWER
ANSWER
Correct
While the host is greeting the guests the cat (of weight texttipW_rm cW_c) gets on the table and walks until her
position is (xy)=(L_xL_y)
Part D
Find the maximum weight texttipW_rm maxW_max of the cat such that the table does not tip over and brea
the vase
Express the cats weight in terms of texttipW_rm vW_v texttipXX texttipYY texttipL_mit
xL_x and texttipL_mit yL_y
Hint 1 Force when table begins to tip
When the table is just beginning to tip over what is the force texttipF_rm critF_crit supplied by the leg
at the origin
sum F_z= 0 = F_0+F_x+F_y-W_v-W_t
texttipF_rm 0F_0 = largeW_v left(1-fracXL_x-fracYL_yright)
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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ANSWER
ANSWER
Correct
Part B
Find texttipF_mit xF_x the texttipxx-component of the force exerted by the wall on the beam ( texttipF
F) using the axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT and other given quantities
Hint 1 Find the sign of the force
The beam is not accelerating in the texttipxx-direction so the sum of the forces in the texttipxx-
direction is zero Using the given coordinate system is texttipF_mit xF_x going to have to be positive
or negative
ANSWER
Correct
Part C
Find texttipF_mit yF_y the y-component of force that the wall exerts on the beam ( texttipFF) using the
axis shown Remember to pay attention to the direction that the wall exerts the force
Express your answer in terms of texttipTT texttiptheta theta texttipm_rm 1m_1
texttipm_rm 2m_2 and texttipgg
ANSWER
Sigmatau_a = 0 = largeleft(fracm_1 L2+m_2 dright) g-T L sinleft(thetaright)
texttipTT = largefracgleft(m_1 fracL2+m_2 dright)L sinleft(thetaright)
texttipF_mit xF_x = -T cosleft(thetaright)
texttipF_mit yF_y = gleft(m_1+m_2right)-T sinleft(thetaright)
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Correct
If you use your result from part (A) in your expression for part (C) youll notice that the result simplifies
somewhat The simplified result should show that the further the luncher moves out on the beam the lower
the magnitude of the upward force the wall exerts on the beam Does this agree with your intuition
Three-Legged Table
The top view of a table with weight texttipW_rm tW_t is shown in the figure The table has lost the leg at
(texttipL_mit xL_x texttipL_mit yL_y) in the upper
right corner of the diagram and is in danger of tipping over
Company is about to arrive so the host tries to stabilize the
table by placing a heavy vase (represented by the green
circle) of weight textt ipW_rm vW_v at ( texttipXX
texttipYY) Denote the magnitudes of the upward forces
on the table due to the legs at (0 0) (texttipL_mit xL_x
0) and (0 texttipL_mit yL_y) as texttipF_rm 0F_0
texttipF_mit xF_x and texttipF_mit yF_y
respectively
Part A
Find texttipF_mit xF_x the magnitude of the upward force on the table due to the leg at (texttipL_mit x
L_x 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipXX texttipYY
texttipL_mit xL_x andor texttipL_mit yL_y Note that not all of these quantities may appear in
the answer
Hint 1 Find the useful vector relations
This is a statics problem so about any point texttippp we have sum vectau_p = 0 and at texttipp
p sum vecF = 0 Each is a vector equation yielding algebraic expressions for each of the three
components x y and z Using the coordinate system shown in the figure indicate for each component of these equations whether it gives a useful relationship among the various forces
Check all that apply
Hint 1 Direction of applied forces
All of the forces are in the z direction Figure out what nonzero components of the torque these could
possibly generate
ANSWER
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Hint 2 Find the y component of the torque
What is sum tau_y the y component of the torque equation
Express the y component of the torque in terms of texttipW_rm vW_v texttipW_rm tW_t
texttipF_mit xF_x texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y
texttipXX andor texttipYY Note that not all terms may appear in the answer
Hint 1 Choice of origin for torques
Take torques around the origin of the coordinate system (ie this becomes point texttippp in the
torque equations) as this will result in an equation with one and only one unknown force
Hint 2 Sign convention
If you do not get the sign from the cross-product expression tau_y = z F_x - x F_z remember that
the y torque is positive for clockwise rotation about the y axis when looking in the positive y
direction This is a subtle point--normally you look down on the x -y axes from above that is towardnegative z and so positive torque (about the z axis) is counterclockwise
ANSWER
ANSWER
x component of the forces
y component of the forces
z component of the forces
x component of the torques
y component of the torques
z component of the torques
sum tau_y=0 = large-W_v X+left(F_x-fracW_t2right) L_x
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Correct
Part B
Find texttipF_mit yF_y the magnitude of the upward force on the table due to the leg at (0 texttipL_mit y
L_y)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipYY texttipXX
texttipL_mit yL_y andor texttipL_mit xL_x Note that not all of these quantities may appear in
the final answer
Hint 1 Find the x component of torque
What is sum tau_x the x component of the torque equation
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y texttipXX andor
texttipYY Note that not all terms may appear in the answer Remember to pay attention to the
signs of the torque
ANSWER
ANSWER
Correct
Part C
Find texttipF_rm 0F_0 the magnitude of the upward force on the table due to the leg at (0 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipL_mit xL_x texttipL_mit yL_y texttipXX andor texttipYY Note that not all terms may appear in the
answer
Hint 1 Summing the total forces
What is sum F_z the total vertical force on the table
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y andor texttipF_rm 0F_0 Remember to pay attention to signs (with a
positive force being upward on the table)
texttipF_mit xF_x = largefracW_v XL_x+fracW_t2
sum tau_x=0 = large-W_v Y-left(fracW_t2-F_yright) L_y
texttipF_mit yF_y = largefracW_v YL_y+fracW_t2
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ANSWER
ANSWER
Correct
While the host is greeting the guests the cat (of weight texttipW_rm cW_c) gets on the table and walks until her
position is (xy)=(L_xL_y)
Part D
Find the maximum weight texttipW_rm maxW_max of the cat such that the table does not tip over and brea
the vase
Express the cats weight in terms of texttipW_rm vW_v texttipXX texttipYY texttipL_mit
xL_x and texttipL_mit yL_y
Hint 1 Force when table begins to tip
When the table is just beginning to tip over what is the force texttipF_rm critF_crit supplied by the leg
at the origin
sum F_z= 0 = F_0+F_x+F_y-W_v-W_t
texttipF_rm 0F_0 = largeW_v left(1-fracXL_x-fracYL_yright)
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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Correct
If you use your result from part (A) in your expression for part (C) youll notice that the result simplifies
somewhat The simplified result should show that the further the luncher moves out on the beam the lower
the magnitude of the upward force the wall exerts on the beam Does this agree with your intuition
Three-Legged Table
The top view of a table with weight texttipW_rm tW_t is shown in the figure The table has lost the leg at
(texttipL_mit xL_x texttipL_mit yL_y) in the upper
right corner of the diagram and is in danger of tipping over
Company is about to arrive so the host tries to stabilize the
table by placing a heavy vase (represented by the green
circle) of weight textt ipW_rm vW_v at ( texttipXX
texttipYY) Denote the magnitudes of the upward forces
on the table due to the legs at (0 0) (texttipL_mit xL_x
0) and (0 texttipL_mit yL_y) as texttipF_rm 0F_0
texttipF_mit xF_x and texttipF_mit yF_y
respectively
Part A
Find texttipF_mit xF_x the magnitude of the upward force on the table due to the leg at (texttipL_mit x
L_x 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipXX texttipYY
texttipL_mit xL_x andor texttipL_mit yL_y Note that not all of these quantities may appear in
the answer
Hint 1 Find the useful vector relations
This is a statics problem so about any point texttippp we have sum vectau_p = 0 and at texttipp
p sum vecF = 0 Each is a vector equation yielding algebraic expressions for each of the three
components x y and z Using the coordinate system shown in the figure indicate for each component of these equations whether it gives a useful relationship among the various forces
Check all that apply
Hint 1 Direction of applied forces
All of the forces are in the z direction Figure out what nonzero components of the torque these could
possibly generate
ANSWER
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Hint 2 Find the y component of the torque
What is sum tau_y the y component of the torque equation
Express the y component of the torque in terms of texttipW_rm vW_v texttipW_rm tW_t
texttipF_mit xF_x texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y
texttipXX andor texttipYY Note that not all terms may appear in the answer
Hint 1 Choice of origin for torques
Take torques around the origin of the coordinate system (ie this becomes point texttippp in the
torque equations) as this will result in an equation with one and only one unknown force
Hint 2 Sign convention
If you do not get the sign from the cross-product expression tau_y = z F_x - x F_z remember that
the y torque is positive for clockwise rotation about the y axis when looking in the positive y
direction This is a subtle point--normally you look down on the x -y axes from above that is towardnegative z and so positive torque (about the z axis) is counterclockwise
ANSWER
ANSWER
x component of the forces
y component of the forces
z component of the forces
x component of the torques
y component of the torques
z component of the torques
sum tau_y=0 = large-W_v X+left(F_x-fracW_t2right) L_x
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Correct
Part B
Find texttipF_mit yF_y the magnitude of the upward force on the table due to the leg at (0 texttipL_mit y
L_y)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipYY texttipXX
texttipL_mit yL_y andor texttipL_mit xL_x Note that not all of these quantities may appear in
the final answer
Hint 1 Find the x component of torque
What is sum tau_x the x component of the torque equation
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y texttipXX andor
texttipYY Note that not all terms may appear in the answer Remember to pay attention to the
signs of the torque
ANSWER
ANSWER
Correct
Part C
Find texttipF_rm 0F_0 the magnitude of the upward force on the table due to the leg at (0 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipL_mit xL_x texttipL_mit yL_y texttipXX andor texttipYY Note that not all terms may appear in the
answer
Hint 1 Summing the total forces
What is sum F_z the total vertical force on the table
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y andor texttipF_rm 0F_0 Remember to pay attention to signs (with a
positive force being upward on the table)
texttipF_mit xF_x = largefracW_v XL_x+fracW_t2
sum tau_x=0 = large-W_v Y-left(fracW_t2-F_yright) L_y
texttipF_mit yF_y = largefracW_v YL_y+fracW_t2
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ANSWER
ANSWER
Correct
While the host is greeting the guests the cat (of weight texttipW_rm cW_c) gets on the table and walks until her
position is (xy)=(L_xL_y)
Part D
Find the maximum weight texttipW_rm maxW_max of the cat such that the table does not tip over and brea
the vase
Express the cats weight in terms of texttipW_rm vW_v texttipXX texttipYY texttipL_mit
xL_x and texttipL_mit yL_y
Hint 1 Force when table begins to tip
When the table is just beginning to tip over what is the force texttipF_rm critF_crit supplied by the leg
at the origin
sum F_z= 0 = F_0+F_x+F_y-W_v-W_t
texttipF_rm 0F_0 = largeW_v left(1-fracXL_x-fracYL_yright)
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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Hint 2 Find the y component of the torque
What is sum tau_y the y component of the torque equation
Express the y component of the torque in terms of texttipW_rm vW_v texttipW_rm tW_t
texttipF_mit xF_x texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y
texttipXX andor texttipYY Note that not all terms may appear in the answer
Hint 1 Choice of origin for torques
Take torques around the origin of the coordinate system (ie this becomes point texttippp in the
torque equations) as this will result in an equation with one and only one unknown force
Hint 2 Sign convention
If you do not get the sign from the cross-product expression tau_y = z F_x - x F_z remember that
the y torque is positive for clockwise rotation about the y axis when looking in the positive y
direction This is a subtle point--normally you look down on the x -y axes from above that is towardnegative z and so positive torque (about the z axis) is counterclockwise
ANSWER
ANSWER
x component of the forces
y component of the forces
z component of the forces
x component of the torques
y component of the torques
z component of the torques
sum tau_y=0 = large-W_v X+left(F_x-fracW_t2right) L_x
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Correct
Part B
Find texttipF_mit yF_y the magnitude of the upward force on the table due to the leg at (0 texttipL_mit y
L_y)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipYY texttipXX
texttipL_mit yL_y andor texttipL_mit xL_x Note that not all of these quantities may appear in
the final answer
Hint 1 Find the x component of torque
What is sum tau_x the x component of the torque equation
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y texttipXX andor
texttipYY Note that not all terms may appear in the answer Remember to pay attention to the
signs of the torque
ANSWER
ANSWER
Correct
Part C
Find texttipF_rm 0F_0 the magnitude of the upward force on the table due to the leg at (0 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipL_mit xL_x texttipL_mit yL_y texttipXX andor texttipYY Note that not all terms may appear in the
answer
Hint 1 Summing the total forces
What is sum F_z the total vertical force on the table
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y andor texttipF_rm 0F_0 Remember to pay attention to signs (with a
positive force being upward on the table)
texttipF_mit xF_x = largefracW_v XL_x+fracW_t2
sum tau_x=0 = large-W_v Y-left(fracW_t2-F_yright) L_y
texttipF_mit yF_y = largefracW_v YL_y+fracW_t2
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ANSWER
ANSWER
Correct
While the host is greeting the guests the cat (of weight texttipW_rm cW_c) gets on the table and walks until her
position is (xy)=(L_xL_y)
Part D
Find the maximum weight texttipW_rm maxW_max of the cat such that the table does not tip over and brea
the vase
Express the cats weight in terms of texttipW_rm vW_v texttipXX texttipYY texttipL_mit
xL_x and texttipL_mit yL_y
Hint 1 Force when table begins to tip
When the table is just beginning to tip over what is the force texttipF_rm critF_crit supplied by the leg
at the origin
sum F_z= 0 = F_0+F_x+F_y-W_v-W_t
texttipF_rm 0F_0 = largeW_v left(1-fracXL_x-fracYL_yright)
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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Correct
Part B
Find texttipF_mit yF_y the magnitude of the upward force on the table due to the leg at (0 texttipL_mit y
L_y)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipYY texttipXX
texttipL_mit yL_y andor texttipL_mit xL_x Note that not all of these quantities may appear in
the final answer
Hint 1 Find the x component of torque
What is sum tau_x the x component of the torque equation
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y texttipL_mit xL_x texttipL_mit yL_y texttipXX andor
texttipYY Note that not all terms may appear in the answer Remember to pay attention to the
signs of the torque
ANSWER
ANSWER
Correct
Part C
Find texttipF_rm 0F_0 the magnitude of the upward force on the table due to the leg at (0 0)
Express the force in terms of texttipW_rm vW_v texttipW_rm tW_t texttipL_mit xL_x texttipL_mit yL_y texttipXX andor texttipYY Note that not all terms may appear in the
answer
Hint 1 Summing the total forces
What is sum F_z the total vertical force on the table
Answer in terms of texttipW_rm vW_v texttipW_rm tW_t texttipF_mit xF_x
texttipF_mit yF_y andor texttipF_rm 0F_0 Remember to pay attention to signs (with a
positive force being upward on the table)
texttipF_mit xF_x = largefracW_v XL_x+fracW_t2
sum tau_x=0 = large-W_v Y-left(fracW_t2-F_yright) L_y
texttipF_mit yF_y = largefracW_v YL_y+fracW_t2
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ANSWER
ANSWER
Correct
While the host is greeting the guests the cat (of weight texttipW_rm cW_c) gets on the table and walks until her
position is (xy)=(L_xL_y)
Part D
Find the maximum weight texttipW_rm maxW_max of the cat such that the table does not tip over and brea
the vase
Express the cats weight in terms of texttipW_rm vW_v texttipXX texttipYY texttipL_mit
xL_x and texttipL_mit yL_y
Hint 1 Force when table begins to tip
When the table is just beginning to tip over what is the force texttipF_rm critF_crit supplied by the leg
at the origin
sum F_z= 0 = F_0+F_x+F_y-W_v-W_t
texttipF_rm 0F_0 = largeW_v left(1-fracXL_x-fracYL_yright)
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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ANSWER
ANSWER
Correct
While the host is greeting the guests the cat (of weight texttipW_rm cW_c) gets on the table and walks until her
position is (xy)=(L_xL_y)
Part D
Find the maximum weight texttipW_rm maxW_max of the cat such that the table does not tip over and brea
the vase
Express the cats weight in terms of texttipW_rm vW_v texttipXX texttipYY texttipL_mit
xL_x and texttipL_mit yL_y
Hint 1 Force when table begins to tip
When the table is just beginning to tip over what is the force texttipF_rm critF_crit supplied by the leg
at the origin
sum F_z= 0 = F_0+F_x+F_y-W_v-W_t
texttipF_rm 0F_0 = largeW_v left(1-fracXL_x-fracYL_yright)
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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ANSWER
Hint 2 Putting it all together
Go back to your previous equations put in the weight andor torque from the weight of the cat
(texttipW_rm cW_c) to obtain a new expression for texttipF_rm 0F_0 (checking your signs
carefully) equate this to the value you just obtained for texttipF_rm critF_crit and solve for
textt ipW_rm cW_c (now texttipW_rm maxW_max)
ANSWER
Correct
Exercise 1122
You are doing exercises on a Nautilus machine in a gym to strengthen your deltoid (shoulder) muscles Your arms are
raised vertically and can pivot around the shoulder joint and you grasp the cable of the machine in your hand 640 rm
rm cm from your shoulder joint The deltoid muscle is attached to the humerus 150 rm rm cm from the shoulde
oint and makes a 120 rm ^circ angle with that bone (See the figure below )
texttipF_rm critF_crit = 0
texttipW_rm maxW_max = largeW_v left(1-fracXL_x-fracYL_yright)
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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Part A
If you have set the tension in the cable of the machine to 360 rm rm N on each arm what is the tension ineach deltoid muscle if you simply hold your outstretched arms in place
ANSWER
Correct
Exercise 1130
A vertical solid steel post of diameter texttipdd = 22 rm rm cm and length texttipLL = 240 rm rm m is
required to support a load of mass texttipmm = 7700 rm rm kg You can ignore the weight of the post Take
free fall acceleration to be g=98 rm ms^2
Part A
What is the stress in the post
Express your answer using two significant figures
ANSWER
Correct
Part B
T = 605 rm N
P = 20times106 rm Pa
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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What is the strain in the post
Express your answer using two significant figures
ANSWER
Correct
Part C
What is the change in the posts length when the load is applied
Express your answer using two significant figures
ANSWER
Correct
Spinning Mass on a Spring
An object of mass texttipMM is attached to a spring with spring constant texttipkk whose unstretched length is
texttipLL and whose far end is fixed to a shaft that is rotating with angular speed texttipomega omega Neglec
gravity and assume that the mass rotates with angular speed texttipomega omega as shown When solving this
problem use an inertial coordinate system as drawn here
Part A
largefrac Delta L L = 99times10minus6
Delta L = 24times10minus5 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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Given the angular speed texttipomega omega find the radius texttipRleft(omega right)R(omega) at whic
the mass rotates without moving toward or away from the origin
Express the radius in terms of texttipkk texttipLL texttipMM and texttipomega omega
Hint 1 Acceleration at a certain point
Find texttipaleft(omega right)a(omega) the x component of the acceleration of mass texttipMM at
the instant pictured in the figure
Answer in terms of texttipRR and texttipomega omega
Hint 1 Expression for centripetal acceleration
Recall that the magnitude of centripetal acceleration for constant circular motion is a_rm
cent=v^2R This expression can be converted into texttipomega omega and texttipRR with
the standard equation for velocity of distancetime
ANSWER
Hint 2 Graphical description of forces
Which figure describes correctly the inertial force(s) acting on mass texttipMM
ANSWER
Hint 3 Spring force at a particular instant
texttipaleft(omega right)a(omega) = R omega^2
Figure 1
Figure 2
Figure 3
Figure 4
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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What is texttipF_rm springleft(Rright)F_spring(R) the x component of the force exerted by the spring
at the instant pictured in the figure
Answer in terms of texttipkk texttipRR and texttipLL
Hint 1 What determines the spring force
Recall that the amount of force a spring exerts is proportional to the distance it is stretched or
compressed with respect to its equilibrium length ( texttipLL in this case)
ANSWER
Hint 4 Newtons 2nd law
The mass is moving on a circular path and so there is a constant change in the direction of its velocity
vector and hence an inward radial acceleration The vector sum of all forces acting on the object
textt ipvecF_rm totalF_total_vec must act to keep the mass on the circular trajectory at constantspeed Hence the forces are radial and
F_rm total = ma_rm radial
Keep in mind that we are (as always) using an inertial reference system
ANSWER
Correct
Part B
Assume that at a certain angular s peed texttipomega _rm 2omega_2 the radius texttipRR becomes
twice texttipLL Find texttipomega _rm 2omega_2
Answer in terms of texttipkk and texttipMM
Hint 1 How to approach the problem
In Part A you have obtained the formula largeR(omega)=frackLk-Momega^2 Now use the additional
piece of information R(omega_2)=2L to solve for texttipomega _rm 2omega_2
ANSWER
texttipF_rm springleft(Rright)F_spring(R) = k left(R-Lright)
texttipRleft(omega right)R(omega) = largefrackLk-Momega^2
texttipomega _rm 2omega_2 = largesqrtfrack2M
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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Correct
Part C
You probably have noticed that as you increase texttipomega omega there will be a value texttipomega
_rm critomega_crit for which texttipRleft(omega right)R(omega) goes to infinity Find texttipomega
_rm critomega_crit
Answer in terms of texttipkk and texttipMM
Hint 1 What equation to use
Note that the denominator of your expression for texttipRleft(omega right)R(omega) is a function of
texttipomega omega The critical point(s) of texttipRleft(omega right)R(omega) will coincide with
the root(s) of the denominator Before answering analyze the physical validity of your answer
ANSWER
Correct
Part D
What is happening to the spring as the angular velocity approaches texttipomega _rm critomega_crit
Choose the best option
Hint 1 Graphical help
Any spring behaves as shown in the two graphs
textt ipomega _rm critomega_crit = largesqrtfrackM
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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ANSWER
Correct
plusmn Youngs Modulus
Learning Goal
To understand the meaning of Youngs modulus to perform some real-life calculations related to stretching steel a
common construction material and to introduce the concept of breaking stress
Hookes law states that for springs and other elastic objects
F=kDelta x
where texttipFF is the magnitude of the stretching force Delta x is the corresponding elongation of the spring from
equilibrium and texttipkk is a constant that depends on the geometry and the material of the spring If the
deformations are small enough most materials in fact behave like springs Their deformation is directly proportional
the external force Therefore it may be useful to operate with an expression that is similar to Hookes law but describe
the properties of various materials as opposed to objects such as springs Such an expression does exist Consider
for instance a bar of initial length texttipLL and cross-sectional area texttipAA stressed by a force of magnitud
texttipFF As a result the bar stretches by Delta L
Let us define two new terms
Tensile stress is the ratio of the stretching force to the
cross-sectional area
largerm stress=fracFA
Tensile strain is the ratio of the elongation of the rod to
the initial length of the bar
largerm strain=fracDelta LL
It turns out that the ratio of the tensile stress to the tensile
strain is a constant as long as the tensile stress is not too
large That constant which is an inherent property of a
material is called Youngs modulus and is given by
largeY=f racFADelta LL
Part A
What is the SI unit of Youngs modulus
Hint 1 Look at the dimensions
If you look at the dimensions of Youngs modulus you will see that they are equivalent to the dimension of
The spring streches linearly then breaks at omega = omega_rm crit
The value of texttipomega _rm critomega_crit is so large that the spring will behave linearly for any
practically attainable texttipomega omega
As texttipomega omega approaches texttipomega _rm critomega_crit the spring stops
behaving linearly and begins to act more like an unstretchable rod until it eventually breaks
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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pressure Use the SI unit of pressure
ANSWER
Correct
Part B
Consider a metal bar of initial length texttipLL and cross-sectional area texttipAA The Youngs modulus of
the material of the bar is texttipYY Find the spring constant texttipkk of such a bar for low values of
tensile strain
Express your answer in terms of texttipYY texttipLL and texttipAA
Hint 1 Use the definition of Youngs modulusConsider the equation defining texttipYY Then isolate texttipFF and compare the result with Hookes
law F = kDelta x
ANSWER
Correct
Part C
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
connected end to end so that the resultant wire has length 10L What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in the Part B
you can determine what happens to the spring constant when the length of the spring increases
ANSWER
Pa
texttipkk = largeY fracAL
01k
texttipkk
10k
100k
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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1082015 Chapter 11 Homework Assignment
Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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Correct
Part D
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The wires are
slightly twisted together so that the resultant wire has length texttipLL and its cross-sectional area is ten time
that of the individual wire What is the spring constant of the resulting wire
Hint 1 The spring constant
Use the expression for the spring constant determined in Part B From the expression derived in Part B you
can determine what happens to the spring constant when the area of the spring increases
ANSWER
Correct
Part E
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngsmodulus of each wire is texttipYY The wires are connected end to end so that the resultant wire has length
10L What is the Youngs modulus of the resulting wire
ANSWER
Correct
Part F
Ten identical steel wires have equal lengths texttipLL and equal spring constants texttipkk The Youngs
modulus of each wire is texttipYY The wires are slightly twisted together so that the resultant wire has length
texttipLL and is ten times as thick as the individual wire What is the Youngs modulus of the resulting wire
ANSWER
01k
texttipkk
10k
100k
01Y
texttipYY
10Y
100Y
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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1082015 Chapter 11 Homework Assignment
Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
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CorrectBy rearranging the wires we create a new object with new mechanical properties However Youngs modulus
depends on the material which remains unchanged To change the Youngs modulus one would have to
change the properties of the material itself for instance by heating or cooling it
Part G
Consider a steel guitar string of initial length L=100 meter and cross-sectional area A=0500 square millimeters
The Youngs modulus of the steel is Y=20 times 10^11 pascals How far ( texttipDelta LDelta L) would suc
a string stretch under a tension of 1500 newtons
Use two significant figures in your answer Express your answer in millimeters
ANSWER
Correct
Steel is a very strong material For these numeric values you may assume that Hookes law holds However
for greater values of tensile strain the material no longer behaves elastically If the strain and stress are large
enough the material deteriorates The final part of this problem illustrates this point and gives you a sense of the stretching limit of steel
Part H
Although human beings have been able to fly hundreds of thousands of miles into outer space getting inside the
earth has proven much more difficult The deepest mines ever drilled are only about 10 miles deep To illustrate t
difficulties associated with such drilling consider the following The density of steel is about 7900 kilograms per
cubic meter and its breaking stress defined as the maximum stress the material can bear without deteriorating
about 20 times 10^9 pascals What is the maximum length of a steel cable that can be lowered into a mine
Assume that the magnitude of the acceleration due to gravity remains constant at 98 meters per second per
second
Use two significant figures in your answer expressed in kilometers
Hint 1 Why does the cable break
The cable breaks because of the stress exerted on it by its own weight At the moment that the breaking
stress is reached the stress at the top of the cable reaches its maximum and the material begins to
deteriorate
Introduce an arbitrary cross-sectional area of the cable (which will cancel out of the final answer) The mass
of the cable below the top point can be found as the product of its volume and its density Use this to find
01Y
texttipYY
10Y
100Y
texttipDelta LDelta L = 15 rm mm
7232019 Chapter 11 Mastering Physics
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1082015 Chapter 11 Homework Assignment
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the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
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1082015 Chapter 11 Homework Assignment
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
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1082015 Chapter 11 Homework Assignment
Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
7232019 Chapter 11 Mastering Physics
httpslidepdfcomreaderfullchapter-11-mastering-physics 2023
1082015 Chapter 11 Homework Assignment
httpssessionmasteringphysicscommyctassignmentPrintViewassignmentID=3673484 20
the force at the top that will lead to the breaking stress
Hint 2 Find the stress in the cable
Assume that the cable has cross-sectional area texttipAA and length texttipLL The density is
texttiprho rho The maximum stress in the cable is at the very top where it has to support its own
weight What is this maximum stress
Express your answer in terms of texttiprho rho texttipLL and texttipgg the magnitude
of the acceleration due to gravity Recall that the stress is the force per unit area so the area will notappear in your expression
ANSWER
ANSWER
Correct
This is only about 16 miles and we have assumed that no extra load is attached By the way this length is
small enough to justify the assumption of virtually constant acceleration due to gravity When making such
assumptions one should always check their validity after obtaining a result
Problem 1176
You are trying to raise a bicycle wheel of mass texttipmm and radius texttipRR up over a curb of height
texttiphh To do this you apply a horizontal force texttipvecFF_vec
Part A
maximum stress = rho L g
26 rm km
7232019 Chapter 11 Mastering Physics
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1082015 Chapter 11 Homework Assignment
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What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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1082015 Chapter 11 Homework Assignment
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
7232019 Chapter 11 Mastering Physics
httpslidepdfcomreaderfullchapter-11-mastering-physics 2323
1082015 Chapter 11 Homework Assignment
Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
7232019 Chapter 11 Mastering Physics
httpslidepdfcomreaderfullchapter-11-mastering-physics 2123
1082015 Chapter 11 Homework Assignment
httpssessionmasteringphysicscommyctassignmentPrintViewassignmentID=3673484 2
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the center of the wheel
ANSWER
Correct
Part B
What is the least magnitude of the force texttipvecFF_vec that will succeed in raising the wheel onto the cu
when the force is applied at the top of the wheel
ANSWER
Correct
Part C
In which case is less force required
ANSWER
Correct
Problem 1194
Compressive strength of our bones is important in everyday life Youngs modulus for bone is about 14 times 10^10
rm Pa Bone can take only about a 10 change in its length before fracturing
Part A
What is the maximum force that can be applied to a bone whose minimum cross-sectional area is 30rm
cm^2 (This is approximately the cross-sectional area of a tibia or shin bone at its narrowest point)
Express your answer using two significant figures
ANSWER
F = largefracmg sqrt2Rh-h^2R-h
F = largemg sqrtfrach2R-h
case A
case B
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1082015 Chapter 11 Homework Assignment
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
7232019 Chapter 11 Mastering Physics
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1082015 Chapter 11 Homework Assignment
Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
7232019 Chapter 11 Mastering Physics
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1082015 Chapter 11 Homework Assignment
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Correct
Part B
Estimate the maximum height from which a 70rm -kg man could jump and not fracture the tibia Take the time
between when he first touches the floor and when he has stopped to be 0006 rm s and assume that the stress
is distributed equally between his legs
Express your answer using two significant figures
ANSWER
Correct
Problem 1183
A garage door is mounted on an overhead rail (the f igure )
The wheels at A and B have rusted so that they do not roll
but rather slide along the track The coefficient of kinetic
friction is 049 The distance between the wheels is 200 rm
m and each is 050 rm m from the vertical sides of the
door The door is uniform and weighs 945 rm rm N It is
pushed to the left at constant speed by a horizontal force
vec F
Part A
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel A by the
track
ANSWER
F_rm max = 42times104 rm N
h_rm max = 26 rm m
F_V = 130 rm N
7232019 Chapter 11 Mastering Physics
httpslidepdfcomreaderfullchapter-11-mastering-physics 2323
1082015 Chapter 11 Homework Assignment
Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m
7232019 Chapter 11 Mastering Physics
httpslidepdfcomreaderfullchapter-11-mastering-physics 2323
1082015 Chapter 11 Homework Assignment
Correct
Part B
If the distance h is 148 rm rm m what is the vertical component of the force exerted on the wheel B by the
track
ANSWER
Correct
Part C
Find the maximum value h can have without causing one wheel to leave the track
ANSWER
Correct
Score Summary
Your score on this assignment is 101
You received 1106 out of a possible total of 11 points
F_V = 815 rm N
h_rm max = 204 rm m