classical mechanics review 2: units 1-9
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Classical Mechanics Review 2: Units 1-9. Example: Work and Inclined planes. The block shown has a mass of 1.58 kg . The coefficient of friction is µ k = 0.550 and θ = 20.0 o . If the block is moved 2.25 m up the incline, calculate (including signs): - PowerPoint PPT PresentationTRANSCRIPT
Classical MechanicsReview 2: Units 1-9
Mechanics Review 2 , Slide 1
Useful Formulas
Mechanics Review 2 , Slide 2
Work of a Force:
)( frictionexceptNCk WdfE Conservation of Energy:
Work-Kinetic Energy Theorem:
rdFWr
r
2
1
xdxFUx
xc
2
1
)(Potential Energy of a Conservative Force:
Mechanical Energy: UKE
If F is constant: rFW
Force of kinetic friction:
Force of static friction:
Potential Energy of Gravity:
Potential Energy of Spring:
Example: Work and Inclined planes
The block shown has a mass of m = 1.58 kg. The coefficient of friction is µk = 0.550 and θ = 20.0o. If the block is moved a distance d = 2.25 m up the incline, calculate:
(a) the work done by gravity on the block; (b) the work done on the block by the normal force; (c) the work done by friction on the block.
Mechanics Review 2 , Slide 3
q
M
0NWdmgWg )sin( q
dmgW kf )cos( q
cosrFrFW For a constant Force:
L
v
Conserve Energy from initial to final position
Example: Pendulum
Mechanics Review 2 , Slide 4
2
21 mvmgL gLv 2
mghU g 0E
L
mg
T
Example: Pendulum
Mechanics Review 2 , Slide 5
gLv 2
LmvmgT
2
mgmgmgL
mvmgT 322
2vaL L
va2
rvmmaF cy
2
Conserve Energy from initial to final position.
2v ghhh
Example: Pendulum
Mechanics Review 2 , Slide 6
2
21 mvmgh ghv 2
ghv 2
0E
mghU g
h
mgT
r
Example: Pendulum
Mechanics Review 2 , Slide 7
ghv 2
rva
2
rmvmgT
2
mgr
mghmgr
mvT 22
rvmmaF cy
2
Example: Block and spring A 2.5 kg box is released from rest 1.5 m above the ground and
slides down a frictionless ramp. It slides across a floor that is frictionless, except for a small section 0.50 m wide that has a coefficient of kinetic friction of 0.40. At the left end, is a spring with spring constant 250 N/m. The box compresses the spring, and is accelerated back to the right.
(a) What is the speed of the box at the bottom of the ramp? (b) What is the maximum distance the spring is compressed by
the box?
Mechanics Review 2 , Slide 8
2.5 kg
h=1.5 m
d = 0.50 m
k = 0.4
k=250 N/m
2
210)( bmvmghE a dmgkxmghdfE kk 2
21)(b
Example: Block with Friction A 6.0 kg block, initially at rest, is pulled to the right along a
horizontal surface by a constant horizontal force F = 12 N, applied at an angle θ = 40⁰.
Find the speed of the block after it has moved 3.0 m if the surfaces in contact have μk = 0.15.
Mechanics Unit 9, Slide 9
Example: Blocks, Pulley and Spring Two blocks m1 = 4 kg and m2 = 6 kg are connected by a string that
passes over a pulley. m1 lies on a inclined surface of 20o with coefficient of friction μk = 0.12, and is connected to a spring of spring constant k = 120 N/m. The system is released from rest when the spring is unstretched.
Find the speed of the blocks when m2 has fallen a distance D = 0.2 m. At what distance d does the acceleration of the blocks becomes zero?
Mechanics Review 2 6, Slide 10
20o
m1
m2
20o
k
qsin12 gDmgDmEi
2221 2
1)(21 kDvmmE f
01 amF 02 amFa is not constant! a = 0 when:
Example: Popgun (Spring and Gravity)
The launching mechanism of a popgun consists of a spring. when the spring is compressed 0.120 m, the gun when fired vertically is able to launch a 35.0 g projectile to a maximum height of 20.0 m above its position as it leaves the spring.
(a) Find the spring constant (A) k= 953 N/m (b) Find the speed of the projectile as it moves through the equilibrium position of the spring (A) vB = 19.7 m/s.
Mechanics Review 2 , Slide 11
00 spg UUKE
hmgU g )(21 22
ifsp xxkU
Example: Pulley and Two Masses A block of mass m1 = 1 kg sits atop an inclined plane of angle
θ = 20o with coefficient of kinetic friction 0.2 and is connected to mass m2 = 3 kg through a string that goes over a massless frictionless pulley. The system starts at rest and mass m2 falls through a height H = 2 m.
Use energy methods to find the velocity of mass m2 just before it hits the ground? What is the acceleration of the blocks?
Mechanics Lecture 19, Slide 12
H = 2 m
H m2
= 2 kg
m1
m2
θ
HgmHfE kk )cos( 1 qqsin12 gHmgHmEi
221 )(
21 vmmE f
All forces are constant so a is constant:
Example: Block slides down the hill A block of mass m starts from rest at the top of the frictionless,
hemispherical hill of radius R. (a) Find an expression for the block's speed v when it is at an
angle ϕ. (b) Find the normal force N at that angle. (c) At what angle ϕ0 does the block "fly off" the hill? (d) With what speed will the block hit the ground? (A)
Mechanics Review 2 , Slide 13
)cos1(21 2 mgRmv
RvmNmgFr
2
cos
(c) When N = 0, cosϕ0 = 2/3
gRv 2
Example: Spring and Mass
A spring is hung vertically and an object of mass m is attached to its lower end.
1. If the spring is stretched 2.0 cm from its equilibrium position by the suspended object of mass 0.55 kg what is the spring constant k?
2. How much work is done by the spring on the object as it stretches through this distance?
Mechanics Unit 7, Slide 14
k = 270 N/m Wsp = - 0.054 J
Example: Collision with a vertical spring
A vertical spring with k = 490 N/m is standing on the ground. A 1.0 kg block is placed at h = 20 cm directly above the spring and dropped with an initial speed vi = 5.0 m/s.
(a) What is the maximum compression x of the spring? (b) What is the position of the equilibrium x0 of the block-spring
system?
Mechanics Review 2 , Slide 15
1kg
k
vi
2
21
ii mvmghE 2
21)( kxxmgE f
0E
00 0 mgkxmaFy
Equilibrium means a = 0: