Classical Mechanics
Review 2: Units 1-11
Conservation of Energy
 E   f k d  W NC ( except
Dynamics of the Center of Mass
Perfectly Inelastic Collisions:

a CM 

vf 

F Net , External
friction )
0
M Total


m 1 v1 i  m 2 v 2 i
m1  m 2
Mechanics Review 2 , Slide 1
Example: Work and Inclined planes
The block shown has a mass of 1.58 kg. The coefficient of
friction is µk = 0.550 and θ = 20.0o. If the block is moved 2.25
m up the incline, calculate (including signs):
(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.
For a constant Force:

W  F   r  F  r cos q

r2


W  F dr
M

r1
q
Mechanics Review 2 , Slide 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 0.5 m wide that has a
coefficient of kinetic friction of 0.2. 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.
What is the speed of the box at the bottom of the ramp?
What is the maximum distance the spring is compressed by the
box?
2.5 kg
k=250 N/m
h=1.5 m
mk = 0.4
d = 0.50 m
E  W f   fk d
Mechanics Review 2 , Slide 3
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 when it is at an
angle ϕ.
(b) Find the normal force N at that angle.
(c) At what angle does the block "fly off" the hill?
(d) With what speed will the block hit the ground?
E  0
Mechanics Review 2 , Slide 4
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?
E  W f   fk d
θ
m2
HH
=2
m
=2
2m
kg
v  v  2 ad
2
f
2
i
Mechanics Lecture 19, Slide 5
Example: Two Ice-Skaters
Two ice-skaters of mass m1 and m2, each having an initial
velocity of vi in the directions shown, collide and fall and slide
across the ice together. The ice surface is horizontal &
frictionless.
1. What is the speed of the skaters after the collision?
2. What is the angle θ relative to the x axis that the two skaters
yy
travel after the collision?


xx
m 1 v1 i  m 2 v 2 i

vf 
m1  m 2
v
vx  v y
q  tan
2
1
2
(v y / v x )
V
Mechanics Review 3, Slide 6
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 of the spring?
(b) What is the position of the equilibrium of the block-spring
system?
1kg
E  0
vi
E i  mgh 
k
a  0   Fy  0
1
2
2
mv i
E f  mg (  x ) 
1
kx
2
2
Mechanics Review 2 , Slide 7
Example: Collision on a Vertical Spring
A vertical spring with k is standing on the ground supporting a
m2 block. A m1 block is placed at a height h directly above the
m2 block and released from rest. After the collision the two
blocks stick together.
(a) What is the speed of the two blocks right after the
collision?
(b) What is the maximum compression of the spring?
(c) What is the position of the equilibrium after the collision?
 E  0  v i  2 gh
2

vf 


m 1 v1 i  m 2 v 2 i
m1  m 2
 E  0   K   U g   U sp  0
a  0   Fy  0
Mechanics Review 3, Slide 8
Example: Pendulum
L
v
Conserve Energy from initial to final position
mgL 
1
mv
2
v
2 gL
2
Mechanics Review 2 , Slide 9
Example: Pendulum
T
L
2
v
a
L
a 
v
2
L
v
2 gL
mg
T  mg 
mv
L
2
T  mg 
mv
2
L
Mechanics Review 2 , Slide 10
Example: Pendulum
v v  2 2
ghgh
hh
Conserve Energy from initial to final position.
mgh 
1
mv
2
v 
2 gh
2
Mechanics Review 2 , Slide 11
Example: Pendulum
v
h
2 gh
r
a
v
2
r
T
T  mg 
mv
r
2
T 
mv
mg
2
 mg
r
Mechanics Review 2 , Slide 12
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 of 12 N.
Find the speed of the block after it has moved 3.0 m if the
surfaces in contact have μk = 0.15. What if the force F is
applied at an angle θ = 40⁰?
 E   f k d  W NC ( except
friction )
Mechanics Unit 9, Slide 13
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) Determine the spring constant
(b) Find the speed of the projectile as it
moves through the equilibrium position
of the spring.
 E  0   K   U g   U sp  0
Mechanics Review 2 , Slide 14
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 an inclined surface of 20o and is
connected to a spring of spring constant k = 120 N/m. The system is
released from rest when the spring is unstretched. The coefficient of
kinetic friction of the inclined surface and is 0.12.
Find the work of friction when m2 has fallen a distance D = 0.2 m.
Find the speed of the blocks at that time.
At what distance d does the acceleration of the blocks becomes zero?
E  W f   fk d
k
20o
m1
20o
E i  m 2 gd  m1 gd sin q
Ef 
m2
1
2
( m1  m 2 ) v 
2
1
kd
2
2
a  0   Fy  0
Mechanics Review 2 6, Slide 15
Example: Spring and Mass
A spring is hung vertically and an object of mass m is
attached to its lower end. Under the action of the load mg the
spring stretches a distance d from its equilibrium position.
1. If a spring is stretched 2.0 cm by a 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?
 F y  kd  mg  0
d
W s   F s dx  
0
1
kd
2
2
Mechanics Unit 7, Slide 16
Example: Ballistic Pendulum
m
v
M
H
A projectile of mass m moving horizontally with speed v strikes
and sticks to a stationary mass M suspended by strings of
length L. Subsequently, m + M rise to a height of H.
Given H, what is the initial speed v of the projectile?
Mechanics Unit 11, Slide 17
Ballistic Pendulum
m
v
M
H
mv  ( m  M )V
V
2
 2 gH
V 
m
(m  M )
v
Conservation
of momentum
Conservation of energy after the collision
Combine the two equations
v
mM 
2 gH 

 m 
Mechanics Unit 11, Slide 18