Classical Mechanics
Review 3, Units 1-16
Today:
Review for midterm #3
Mechanics Review 3, Slide 1
Important Equations

1. Elastic Collisions: PTotal  0 K Total  0
v1,i - v2,i  - (v1, f - v2, f )
2. System of particles – Rigid bodies:
Dynamics:
FNet  MaCM τNet  Iα
Relation between Angular and Linear quantities:
s  r , v  r , at  r
1
1
2
2
, U g  MghCM
Energy: K  I CM   MvCM
2
2
Parallel Axis Theorem: I = ICM + MD2
Rolling motion: vCM = R aCM = αR
Mechanics Review 3, Slide 2
Example: Elastic Collision
Two masses approach each other with equal and opposite
velocities as measured in the lab reference frame. The mass
moving to the right is twice as massive as the one moving to
the left. The collision between them is elastic.
What is the velocity of the center of mass before the collision?
What are the velocities of the two objects after the collision?
What is the velocity of the center of mass after the collision?
V1 = 1 m/s

vcm
V2 = -1 m/s
g
M
M/2
Frictionless surface


m1v1i  m2v2i

m1  m2

PTotal  0
v1,i - v2,i  - (v1, f - v2, f )
Mechanics Review 3, Slide 3
Example: Rotating Rod, I
A rod of length L and mass M is attached to a frictionless pivot
and is free to rotate in the vertical plane. The rod is released
from rest in the horizontal position. The moment of inertia of the
rod about the center of mass is I = ML2/12
(a) What is the moment of inertia of the rod about the pivot?
(b) What is the initial angular acceleration?
(c) What is the initial tangential acceleration of its center of
mass?
I = ICM + MD2
τNet  Iα  MgL/2
at  L / 2
Mechanics Lecture 17, Slide 4
Example: Rotating Rod, II
A rod of length L and mass M is free to rotate on a pin. The rod
is released from rest in the horizontal position. The moment of
inertia of the rod about the center of mass is I = ML2/12
(a) What is its angular speed when the rod reaches its lowest
position? (b) Determine the tangential speed of the center of
mass and of the lowest point on the rod when it is in the vertical
position.
v  r
Mechanics Review 3, Slide 5
Example: Rolling up a Plane
A uniform sphere of mass M and radius R rolls without slipping
up a plane inclined at an angle  with the horizontal. The initial
center-of-mass velocity of the sphere is v. ICM = 2MR2/5.
(a) What is the frictional force exerted by the plane on the sphere?
(b) What is the maximum height, H, above the horizontal, the
sphere can reach?
 Fx  Macm  f s - Mg sin   Macm
 cm  I cm  f s R  I cm
acm  -R
V
M
R
fs


1
1
2
2
I cm  Mvcm
 MgH
2
2
Mechanics Review 4, Slide 6
Example: Atwood's Machine with Massive Pulley
A pair of masses are hung over a massive disk-shaped
pulley as shown.
y
Find the acceleration of the blocks.
x
For the hanging masses use F  ma
-m1g  T1  -m1a
-m2g  T2  m2a
a
For the pulley use   I  I
R
a 1
T1R - T2R  I  MRa
R 2
1
(Since I  MR 2 for a disk)
2
M

R
T2
T1
m2
m1
a
m1g
a
m2g
Mechanics Review 3, Slide 7
Example: Billiard Balls
A white billiard ball with mass m = 1.65 kg is moving directly to
the right with a speed of v = 3.22 m/s and collides with a black
billiard ball with the same mass that is initially at rest. The two
collide elastically and the white ball ends up moving at an angle
above the horizontal of θw = 41° and the black ball ends up
moving at an angle below the horizontal of θb = 49°. Find the
final speeds of the balls and the final total energy of the system.
Px  0
Py  0
Mechanics Unit 13, Slide 8
Example: Falling Rod
A rod of length L and mass M is pivoted about a horizontal,
frictionless pin through one end. The rod is released, almost from
rest in a vertical position. The moment of inertia of the rod about
its center is I = ML2/12. At the instant the rod makes an angle of θ
with the vertical find:
(a) The rotational kinetic energy of the rod.
(b) The angular acceleration of the rod.
(c) The speed of the center of mass of the rod.
Ei  E f  Mg ( L / 2)(1 - cos )  I 
1
2 P
 P  I P  Mg( L / 2) sin   I P
2
f
L
M
θ
m
P
vcm   f (L / 2)
Mechanics Review 4, Slide 9
Example: Pulley and Mass
A wheel of radius R, mass M, and moment of inertia I is
mounted on a frictionless horizontal axle. A light cord
wrapped around the wheel supports an object of mass m.
Find the angular acceleration of the
wheel, the linear acceleration of the
object and the tension T.
FNet  mg – T  ma
τNet  Iα  TR
a  αR
Mechanics Review 3, Slide 10
Example: Pulley and horizontal mass
A block of mass m1 = 1 kg sits atop a plane with coefficient of
kinetic friction 0.2 and is connected to mass m2 = 2 kg through a
string that goes over a pulley of mass M = 4 kg and radius
R = 0.2 m. The pulley rotates about its axis without friction and the
string moves over the pulley without slipping. The system starts at
rest and mass m2 falls through a height H = 2 m. The moment of
inertia of the pulley is I = MR2/2
What is the velocity of mass m2 just before it hits the ground?
M, R
m1
Rpul =
m2
HH
=2
m
=2
2m
kg
E  - f k H
1 2 1
K  I  (m1  m2 )v 2
2
2
v  R
Mechanics Review 3, Slide 11