unit 3 - washingtonville central school district 3 work, energy & ... joules is a unit of...
TRANSCRIPT
2
Work
Work is equal to the force applied to an object that results in the object’s displacement.
Displacement is a vector! So again, direction and sign convention are important.
Eq: W = F x d W = Work (Joules OR N∙m)
Units: J = N x m F = Force (Newtons)
d = displacement (meters)
This J is a unit, it is not impulse!
↓
Joules
Joules is a unit of energy, therefore we must input energy into moving an object to do
work.
If I do work on an object, I have put energy into the object! I push a lawnmower, I put
energy into moving a lawnmower.
Can we derive Kinetic Energy equations from work and Newton’s 2nd
Law!
W = FNet∙d AND FNet = m∙a
W = m∙a∙d where v2 = 2a∙d from kinematics equations therefore v
2 / 2 = a∙d
W = m∙(v2 / 2) W = Joules = energy
KE = Joules = energy
KE = ½ m∙v2
a = ∆v/ t
v = d/t
Typically W = F∙d
Fapp. = 10N
d=10 m
Fapp.=10N
W = F∙d = 10N x 10 m = 100 N∙m = 100 J
3
Kinetic Energy of an Object
“Kinetic” means moving, therefore object has to be moving.
Does it have Kinetic Energy (KE)? Yes it moves, therefore it has KE.
KE = ½ mv2 KE = Kinetic Energy (Joules)
m = mass (kg)
v = Velocity (m/s) [has direction & magnitude]
↓
OR speed (m/s) [has only direction]
Ex. v = 10 m/s
m = 2kg
KE = ½ mv2 = ½ (2kg)(10m/s)
2 = 100 Joules
Different Forms of Energy
Gravitational Potential Energy (PEg)
When work is done against gravity
i.e. lifting a box, climbing stairs, climbing a mountain, etc.
∆PE = mg∆h ∆PE = change in potential energy (Joules)
m = mass (kg)
g = gravity (m/s2)
∆h = change in height OR elevation (m)
mg = weight = (N)
SKETCH
h g
F
m
4
Work Against Friction
Trick Question - Object moves at constant speed of 2 m/s for a distance of 5m. What
work is done on the object? [Ans: NONE b/c constant velocity = no accel.=No Force]
Sketch: FN
Ff Fapplied
Fg=m∙g
Elastic Potential Energy mg = Fg
Hooke’s Law = F = k∙x SKETCH
↓ ↓ x
(N)=(N/m)(m)
k = spring constant (N/m)
When we stretch a spring, we input stored energy (PEs) → Potential Energy of Spring
into the spring. We know this because when the spring is released, it moves and when it
moves it transforms PEs into KE.
Eq. for PEs
PEs = ½ k∙x2 Please note this is NOT F = K∙x
↓ ↓
This is energy This is force
k = spring constant (N/m) **If you use F=k∙x, you can first get k
and then k plug into PEs = ½ k∙x2**
x = displacement (m)
PEs = stored energy (J) or (N∙m)
5
Area Under Graphs and Work & Energy
F
d
Area of triangle = ½ b∙h
Units for energy Joules or N∙m
Slope = spring constant
UNITS ANALYSIS
Y – axis → N
X – axis → m
Therefore looking at units for the area under line
½ Base∙Height OR
½ Force∙distance = F∙d
Units Analysis = N∙ m = J which is energy!
7
Pendulum- Period
T=2Π√l/g Time to complete 1 full swing
No friction, pendulum will keep swinging
Period depends on length (l) and gravitational acceleration (g)
Period is independent of mass of the bob
v=0 here so MAX PEg!
h
Equilibrium point Vmax is here so MAX KE!
Period is the time taken by the bob to go from one end of its swing to the opposite end
and then return to the starting point
- Stick different masses on pendulum and determine T
- Set different length and determine T
Total Energy= ∆KE + ∆PE + Internal Energy
We will talk about forms of energy in motors, generators, photocell, battery.
Power
Power is the rate at which work is done, or the rate at which energy is used transferred.
The SI unit for power is the watt (W).
A power of 1W means that work is being done at the rate of 1 J/s.
Larger units for power are the kilowatt kW (1kW = 1000 W = 103
W) and
the megawatt MW (1 MW = 1000000 W = 106
W).
8
If work is being done by a machine moving at speed v against a constant force, or
resistance, F, then since work doe is force times distance, work done per second is Fv,
which is the same as power.
Example 1
A constant force of 2 kN pulls a crate along a level floor a distance of 10 m in 50s.
What is the power used?
Solution
Alternatively we could have calculated the speed first
and then calculated power
9
Example 2
A hoist operated by an electric motor has a mass of 500 kg. It raises a load of 300 kg
vertically at a steady speed of 0.2 m/s. Frictional resistance can be taken to be constant at
1200 N.
What is the power required?
Solution
From the Previous Power Equation
10
Name_______________________________ Date_________
Regents Physics Mr. Morgante
Energy Notesheet , Part I
Definitions:
1. Energy
2. Work
3. Joule
4. Power
5. Watt
6. Potential Energy
7. Gravitational Potential Energy
Equation Variables/ constants Units Can be used to find Vector/scalar
ΔPE
ΔPE = mgΔh m
g
Δh
W
W = Fd = ΔET F
d
ΔET
P
P = W = Fd =F v W
t t F
d
t
v
A
Ay
θ
Ax
Ay = A sin θ
Ax = A cos θ
11
+y
8. Work and gravitational potential energy examples +x
Case 1 Case 2 Case 3
F F F
no friction μ static μ kinetic
a = _______ a= ___0____ a = ___0____
v= ________ v = ________ v = ________
d = _______ d = ________ d = ________
Ff =_______ Ff= ________ Ff= ________
W = ______ W = ________ W = _______
Summary
______________________ ____________________ __________________
Case 4 Case 5 Case 6 (frictionless)
F
θ
h
μ kinetic
a = _______ h θ
v= ________
d = _______
Ff =_______
W = ______
Summary:
mass mass mass
12
Name_______________________________ Date___________
Regents Physics Mr. Morgante
Energy Notesheet, Part II
9. Force versus displacement graph
“The area under a Force versus displacement graph can be used to find _________”
Calculate the work done in each case:
F F F F
disp disp disp disp
Work =____________ ____________ __________ _________
10. Definitions: Forms of Energy/Devices for converting energy
Internal energy:
Nuclear energy:
Electromagnetic energy:
Photocell
Generator
Motor
Battery
13
Name_______________________________ Date___________
Regents Physics Mr. Morgante
Energy Notesheet, Part III, Elastic Potential Energy
Definitions:
1. Compression
2. Elastic Potential Energy
3. Elongation
4. Hooke’s Law
5. Spring constant
6. Equilibrium position
Equation Variables/ constants Units Can be used to find Vector/scalar
Fs Fs Fs
Fs = kx
k k k
x x x
PEs PEs PEs
PEs = ½ kx2
k k k
x x x
Hooke’s Law:
Sketch the graph of an elastic material that is being elongated according to Fs = kx.
a)What is the slope of this graph?__________
Fs b)What are the units of k?________________
c)What quantity can be computed for the area
under an F versus displacement graph?
x __________________
d)What is the equation for the area of a triangle?_______________________
e) Calculate the work done on a spring if the endpoints of the graph are (0m,0N) & (1.7m,20N)
14
Elastic Potential Energy:
a) Using PEs = ½ kx2 , and k = 15 N/m, complete the table below:
x value 0m 0.25m 0.5m 0.75m 1m 2m
PEs
b) Sketch the graph of an elastic material that is being elongated according to Fs = kx.
PEs
X
c) Using PEs = ½ kx2
Solve for k solve for x
____________________ ________________________
d) Prove that PEs is equivalent to Work is equivalent to ΔPE using units:
_______________ ________________ _______________
15
Name_______________________________ Date___________
Regents Physics Mr. Morgante
Energy Notesheet, Part IV, Kinetic Energy
Definitions:
1. Kinetic energy
2. Potential energy
3. Momentum
Equation Variables/ constants Units Can be used to find Vector/scalar
KE KE KE
KE = ½ mv2
m m m
v v v
ΔPE ΔPE ΔPE
ΔPE = mgΔh
m m m
g g g
Δh Δh Δh
p = mv p p p
m m m
v v v
Algebra review:
1. If KE = ½ mv2, solve for: m = ______________, v = ________________
2. If the velocity of an object is doubled, the KE is ______________________
3. If the velocity of an object is halved, the KE is ______________________
Graph review: Sketch the basic shapes of the graphs below(mass remains constant):
KE p
velocity velocity
16
Unit Review:
a)show how Work = KE with units
Practice Problem:
A 2-kg object falls from rest from a 490 m cliff. Ignore air resistance. Use g = 9.8 m/s2
Time (s) 0 1 2 3 4 5 6 7 8 9 10
Velocity
Displacement
PEg
KE
Momentum
(Not to scale)
+x
+y
17
Name_______________________________ Date_________
Regents Physics Mr. Morgante
Energy Notesheet, Part V, Conservation of Energy
Definitions:
4. Conservative force
5. Nonconservative force
6. Closed system
7. Law of conservation of energy:
8. Mechanical energy
9. Ideal mechanical system
10. Simple pendulum
11. Nonideal mechanical system
12. Total energy
Equation Variables/ constants Units Can be used to find Vector/scalar
ET ET ET
ET=PE+KE+Q
PE PE PE
KE KE KE
Q Q Q
ΔPE ΔPE ΔPE
ΔPE = mgΔh
m m m
g g g
Δh Δh Δh
KE = ½ mv2 KE KE KE
m m m
v v v
18
Algebra review:
1. Given: KE = ½ mv2, ΔPE = mgΔh, W = Fd, PEs = ½ kx
2
a) If ½ mv2
= mgΔh , solve for v: b) If ½ mv2
= mgΔh , solve for Δh:
a)_____________ b)_____________
c) If mgΔh = ½ kx2 , solve for k: d) If mgΔh = ½ kx
2, solve for x:
c)_______________ d)_____________
e) If mgΔh = ½ kx2 , solve for Δh: f ) If mgΔh = ½ kx
2 , solve for m:
e)_______________ f)_____________
g) If ½ mv2 = ½ kx
2 , solve for v: h ) If ½ mv
2 = ½ kx
2 , solve for m:
g)______________ h)____________
i) If ½ mv2 = ½ kx
2 , solve for k: j) If ½ mv
2 = ½ kx
2 , solve for x:
i)______________ j)_____________
19
Conservation of Energy Systems:
Case 1 Case 2
Object in free-fall above ground to h = 0 Object projected vertically (+y) from ground to
v i = 0, h top of arc
+y
+y
h = 0 h = 0
PE i = ________ KE i = ________ PE i = ________ KE i = _______
PE f = ________ KE f = ________ PE f = ________ KE f = _______
What Energy transformation occurs? What energy transformation occurs?
______________________________ ________________________________
Case 3 Case 4
Object on inclined plane Object launched vertically from spring
v i = 0 System:
h
h = 0
PE i = ________ KE i = ________ PE i = ________ KE i = _______
PE f = ________ KE f = ________ PE f = ________ KE f = _______
What Energy transformation occurs? What energy transformation occurs?
______________________________ ________________________________
20
Name:__________________ Date:__________
Regents Physics Mr. Morgante
Work Worksheet
1. What is the spring constant of a spring of negligible mass which gained 8 joules
of potential energy as a result of being compressed 0.4 meters?
(1 ) 100N/m (2 ) 50N/m
(3 ) 0.3N/m (4 ) 40N/m
2. Work is done when a force _______
(1 ) acts vertically on a cart that can only move horizontally
(2 ) exerted by one team in a tug of war when there is no movement
(3 ) is exerted while pulling a wagon up a hill
(4 ) of gravitational attraction acts on a person standing on the surface of the Earth
3. A spring of negligible mass with a spring constant of 200 newtons per meter is
stretched 0.2 meters. How much potential energy is stored in the spring?
(1 ) 40 J (2 ) 20 J
(3 ) 8 J (4 ) 4 J
4. An object gains 10 joules of potential energy as it is lifted vertically 2.0 meters. If
a second object with one-half the mass is lifted vertically 2.0 meters, the potential
evergy gained by the second object will be
(1 ) 10. J (2 ) 20. J
(3 ) 5.0 J (4 ) 2.5 J
5. A cart of mass M on a frictionless track starts from rest at the top of a hill having
height h1, as shown in the diagram below. What is the kinetic energy of the cart
when it reaches the top of the next hill, having height h2?
(1 ) mgh1 (2 ) mg(h1-h2)
(3 ) mg(h2-h3) (4 ) 0
21
6. A force is applied to a block, causing it to accelerate along a horizontal,
frictionless surface. The energy gained be the block is equal to the
(1 ) Work done on the block (2 ) power applied to the block
(3 ) impulse applied to the block (4 ) momentum given to the block
7. A 1.0 x103 -kilogram car is moving at a constant speed of 4.0 meters per second.
What is the kinetic energy of the car?
(1 ) 1.6 x 103 J (2 ) 2.0 x 10
4 J
(3 ) 8.0 x 103 J (4 ) 4.0 x 10
3 J
8. When a spring is stretched 0.200 meter from its equilibrium position, it possesses
a potential energy of 10.0 joules. What is the spring constant for this spring?
(1 ) 100. N/m (2 ) 125 N/m
(3 ) 250. N/m (4 ) 500. N/m
9. A constant force of 2.0 newtons is used to push a 3.0-kilogram mass 4.0 meters
across the floor. How much work is done on the mass?
(1 ) 6.0J (2 ) 8.0J
(3 ) 12J (4 ) 24J
10. A student rides a bicycle up a 30° hill at a constant speed of 6.00 meters per
second. The combined mass of the student and bicycle is 70.0 kilograms. What is
the kinetic energy of the student-bicycle system during this ride?
(1 ) 210. J (2 ) 420. J
(3 ) 1,260 J (4 ) 2,520 J
11. If the distance a spring is stretched is doubled, the potential energy is:
(1) four times as great (2) one fourth as great
(3) twice as great (4) the same
12. A 5-kg cart is rolling along on the ground when an additional 5-kg mass is placed
on the cart. The KE of the cart is now:
(1) four times as great (2) one fourth as great
(3) twice as great (4) the same
13. In raising an object vertically at a constant speed of 2.0 meters per second, 10
watts of power is developed. The weight of the object is
(1) 5.0 N (3) 40. N
(2) 20. N (4) 50. N 14. An object moving at a constant speed of 25 meters per second possesses 450 joules of
kinetic energy. What is the object’s mass?
(1) 0.72 kg (3) 18 kg
(2) 1.4 kg (4) 36 kg
22
Name:_________________ Date:__________ Regents Physics Mr. Morgante
Work and Power Worksheet Priscilla and Larry begin to climb the stairs at the end of the science wing in WHS to travel to physics class. The vertical rise of the stairs is 4.0 meters. Priscilla’s mass is 45.0 kg, while Larry has a mass of 60.0 kg. Priscilla makes the climb in 2.0 seconds, while Larry takes 30.0 seconds to climb the same distance. 1. Calculate Larry’s weight (metric) 2. Calculate the work done by Priscilla in climbing the stairs? 3. Calculate Priscilla’s power rating. Larry normally takes 30.0 seconds to climb the stairs, on a particular day he is the recipient of verbal help from a teacher. With the help he manages to climb the stairs in 2.0 seconds. 4. Compare the work he did climbing the stairs on a normal day to this special day. 5. Compare his usual power rating to his power rating on that special day.
Name:_______________ Date:__________
Regents Physics Mr. Morgante
23
Work, Power, KE, and PE Problems
Work and Power
A) What work is done by a girl who pushes a box along a floor with a force of 52.0 N
for a distance of 11.0 m?
B) A boy raises a 20.0 Kg rock 2.3 meters: -What is the force that the boy uses to raise the rock?
-Calculate the amount of work that he does?
C) A student is pulling on a wagon handle with a force of 40.0 N. The handle is at a 30-degree angle with the horizontal. The wagon moves 8 meters in 10 seconds. Find the
work done by the student and the power exerted by the student. Kinetic Energy
A) What is the kinetic energy of an object who’s mass is 5.0 Kg and is moving a 4.0
m/s? If the object was accelerated from rest for a distance of 10.0 m, what was the
force applied to it?
(OVER) B) A force of 10 N is applied to a body on a practically frictionless table over a distance
of 8.0 m, what is the Kinetic energy it imparts to the body? If the body starts at rest
24
and has a mass of 4.0 kg, what velocity does the force impart?
C) When the brake is applied to a car having a mass of 1000 Kg, its speed is reduced
from 30 m/s to 20 m/s. How much work does the brake do on the car? If the brake is applied for a distance of 25 m, what force does it exert on the car?
Potential Energy
A) A 5.0 Kg rock is located on a ledge 10 meters above the ground, calculate the
potential energy of the rock relative to the ground.
Name:_______________ Date:________
Regents Physics Mr. Morgante
25
Work and Energy Graph Worksheet
1. Calculate the work done on the cart by the force shown in the diagram below.
30N
60o
25 m
2. The force on an object varies as shown in the graph below.
Force (N) vs. Distance (m)
0
10
20
30
40
50
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Distance (m)
Fo
rce (
N)
Series1
What is the work done on the object in the first 1.1 meters?
3. How many joules of energy are produced by a 60 watt light bulb in 2 minutes?
(OVER)
26
4. The coefficient of friction between a 5.0 kg mass and a desktop is 0.20. An
unbalanced force of 50.0 N acts horizontally on the mass to move it along the
desktop as shown below.
5.0 kg
50 N
+X
a. What is the weight of the mass?
b. What is the size and direction of the frictional force acting on the mass?
c. What is the value and direction of the unbalanced force acting on the mass?
d. What is the acceleration experienced by the mass?
e. The unbalanced force is reduced to zero by lowering the outside push to
9.81N after the mass has reached a speed of 6 m/s. At what power is work
being done on the mass if this is true?
27
Name:_______________ Date:________
Regents Physics Mr. Morgante
More Work, KE, PE in Springs & Pendulum Problems
1. A spring is stretched for a distance of 1m by 10N force. What is the potential energy stored in the spring (Think F=kx)?
2. A spring with a spring constant of 3 N/m is stretched a distance of 4m. What is
the PEs?
3. A 30-kg box is pulled at constant velocity of 4 m/s across a rough surface.
15 N
θ = 25º
+x
a) Calculate the horizontal component of the pulling force, Fx. (Show mag. & dir.)
b) Calculate the friction force, Ff Sketch the vector on the diagram(Show mag. &
dir.)
c) Calculate the work against friction in 90 seconds (Show mag. & dir.):
d) Calculate the work done by friction in 90 seconds (Show mag. & dir.):
(OVER)
28
4. Answer the following questions based on the diagram below:
Pt. B
h
Pt. A
mass of pendulum ball = 2 kg h = 2 m length of string = 4 m
a) What is the PEmax?
b) What is the maximum velocity of the ball when it reaches Pt. A after being
released from Pt. B?
c) What is the KEmax?
5. Find the potential energy stored in the spring below. The mass of the object is 40
dg and the distance the spring is elongated 2m. Show all of your work.
Spring
m
h=2m
m
a) What is the max. velocity and KEmax of the mass?
b) What is the max PEg of the mass?
29
NAME________________________________ DATE________
Regents Physics Mr. Morgante
Nonideal Mechanical Systems
Objective: Investigators will analyze nonideal mechanical system and develop
solutions based on evidence.
Definition:
Nonideal mechanical system:________________________________________________
Sketch: vi = 0 m/s
m = 40 kg
Point A
L = 3 m
40O h = 0
1. A 40-kg box initially at rest slides down a 3 m long inclined plane that is elevated 40º
to the horizontal. At point A, the bottom of the plane, the velocity of the block is found to
be 4 m/s.
a) Calculate the gravitational potential energy of the box at the top of the inc. plane
a)________________
b) What is the grav. potential energy of the box at the bottom of the plane?
b)________________
c) What is the kinetic energy of the box at the bottom of the plane?
c)________________
d) Has ET been conserved in this system? Yes No
Explain: __________________________________________________________
e) If ET = PE + KE + Q, What is the probable value of Q in this system?
(over) e)________________
30
f) What condition probably caused the increase in Q? f)________________
g) How much work was done by friction in this system? g)________________
h) Calculate the average friction force acting on the box:
Magnitude Direction
h)_______ ________
i) Sketch the box on the plane and show friction force direction, Normal force exerted on
the box by the plane, Fg on box:
Calculate Fg
Magnitude Direction
_______ ________
j) Neglecting friction on the inclined plane, calculate the theoretical speed of the box
at the bottom of the plane:
j)______________
k) What is the magnitude of the normal force acting on the box?
k)_______________
l) Additional comments
31
NAME________________________________ DATE________
Regents Physics Mr. Morgante
Law of Conservation of Energy Practice
1. A 3.0-kg mass free-falls from rest a distance of 10m to the ground (h=0)
Use g = 10 m/s2
; neglect air resistance; Complete the table below
Time
Dist.
from
ground
(m)
10 9 8 7 6 5 4 3 2 1 0
PEg
KE
Velocity
2. Using the strip of graph paper provided, Label x-axis 0 to 10m, y-axis Energy
Using RED pencil plot PEg, BLUE pencil KE, GREEN pencil ET
Attach graph here
3. Explain how this exercise helps illustrate the Law of Conservation of Energy
32
Name:_______________________ Date:__________
Regents Physics Mr. Morgante
Energy Stored in Head-On Collisions
The graph below shows the kinetic energy of a moving cart vs. time as it collides with
the spring bumper of a fixed cart. The mass of the moving cart is 1.0 kg.
KE (J) vs. Time (s)
012345678
0 1 2 3 4 5 6 7
Time (S)
KE
(J)
KE (J)
1. Determine the time at which the spring reaches its maximum compression.
2. The graph shows that the kinetic energy of the cart on rebounding from the spring
bumper is less than before the collision. Explain a possible cause.
3. The cart compresses the spring and then rebounds. Why is the graph not drawn as
the graph shown below?
(OVER)
33
KE (J) vs. Time (s)
-3-2-1012345678
0 1 2 3 4 5 6 7 8
Time (s)
KE
(J
)Series1
4. If the KE of the cart at t=1.0 seconds is 8 J calculate the speed of the cart.
The graph below relates the force exerted on a spring to its compression.
Force (N) vs. Compression (m)
0
1
2
3
4
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Compression (m)
Fo
rce (
N)
Series1
5. Use the graph to find potential energy stored in the spring when it is compressed
by 0.04 m.
34
Name:__________________________ Date:_________
Regents Physics Graphs and Work Worksheet Mr. Morgante
Find the work done in each of the cases pictured below. Show All Work!!!!!!
0
1
2
3
4
5
6
7
0 5 10 15 20 25 30
Displacement (m), Left
Fo
rce
(N
)
Series1
0
5
10
15
20
25
0 0.1 0.2 0.3 0.4 0.5 0.6
Displacement (m), North
Fo
rce
(N
)
Series1
02468
1012141618
1 2 3 4 5 6 7
Fo
rce
(N
)
Displacement (m), West
Series1
0
2
4
6
8
10
12
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Spring Elongation (m)
Fo
rce (
N)
Series1
Z:\Physics\Regents Physics\Class Material\Unit 3 Work & Energy 1-11-10.doc