Classical Mechanics
Review 3, Units 1-16
Review for midterm #3
Mechanics Review 3, Slide 1
Important Equations
1

  mi vi
M

FNet , Ext

M
1

 

Center of mass
rCM   mi ri vCM
aCM
M
Collisions

Elastic Collisions: PTotal  0 and KTotal  0  v1,i - v2,i  - (v1, f - v2, f)

Inelastic Collisions: PTotal  0 but K Total  0



m1v1i  m2v2i 

 vCM
PTotal  0  v f 
Perfectly Inelastic Collisions:


m1  m2

p

F

t

F
dt
Impulse of a Force

avg
Rigid Bodies
  
2
Moment of Inertia: I Axis   mi ri Torque:   r  F  = rF sin


Dynamics: FNet , Ext  MaCM and τNet  Iα
Relations Between Angular and Linear Quantities:
s  R , v  R , at  R
Parallel Axis Theorem: IAxis = ICM + MD2
Rolling Motion:
xcm = Rq, vcm = Rw, acm = Ra
Mechanics Review 3, Slide 2
Example: Elastic Collision in 1D
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 is the velocity of the center of mass after the collision?
What are the velocities of the two objects after the collision?
V1 = 1 m/s
V2 = -1 m/s
g
M
m1v1i + m2 v2i
vcm,i =
= vcm, f
m1 + m2
m1v1i + m2 v2i = m1v1 f + m2 v2 f
M/2
Frictionless surface
Because the collision is elastic:
v1i  v2i  - (v1f  v2f )
Mechanics Review 3, Slide 3
Example: Explosion
A rocket of mass m = 50 kg is fired vertically upward. At the
instant it reaches an altitude of 1000 m and a speed of 300 m/s
it explodes into three fragments of equal mass m/3. One
fragment moves upward with a speed of 450 m/s. The second
has a speed of 240 m/s and is moving east.
What are the velocity components of the third fragment
immediately after the explosion? What is the velocity of the
center of mass right after the explosion?

PTotal  0
m
50kg
mvi,x = å vi,x =
(0 + 240m / s + vx ) = 0
3
i=1 3
3
m
50kg
mvi,y = å vi,y =
(450m / s + 0 + vy ) = 50kg (300m / s)
3
i=1 3
3
Mechanics Unit 10, Slide 4
Example: Perfectly Inelastic Collision in 2D
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 ϕ relative to the x-axis that the two skaters
travel after the collision?

PTotal  0
yy
x
x
(m1 + m2 )v fx = m1v1ix + m2 v2ix
(m1 + m2 )v fy = m1v1iy + m2 v2iy
v f = v 2fx + v 2fy f = tan-1 (v fy / v fx )
V
Mechanics Review 3, Slide 5
Example: Collision in 2D
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 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 v1 and v2.

PTotal  0
mv = mv1 cosqw + mv2 cosqb
0 = mv1 sinqw - mv2 sinqb
Mechanics Unit 13, Slide 6
Example: Impulse
Given: m, H, h, Δt. Find the average force Favg on the ball during
the collision
Dp = m ( v f - vi )
t  time during bounce
H
h
Dp
Favg =
Dt
= reading on scale
vi
vf
Scale
vi  - 2 gH
v f   2 gh
Example: Elastic Collision with Spring
A block of m1 = 1.60 kg initially moving to the right with a speed of 4.0
m/s on a frictionless horizontal track collides with a spring k = 600 N/m
attached to a second block of m2 = 2.10 kg initially moving to the left
with a speed of 2.50 m/s.
A. Find the velocities of the blocks after the elastic collision.
Answer: v1f =-3.38 m/s, v2f = 3.12 m/s
B. During the collision at the instant block 1 moves with a velocity of
+3.00 m/s determine the velocity of block 2. Answer: v2f = -1.74 m/s.

PTotal  0
in both A. and B.
in part A: K is conserved:
Mechanics Unit 13, Slide 8
Example: Perfectly Inelastic Collision
A vertical spring with spring constant 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
m1
blocks right after the collision?
(b) What is the maximum compression
h
of the spring? (Assume there is no
initial compression.)
m2
2
E  0  vi  2 gh
Dp = 0 ® (m1 + m2 )v f = m1vi
DE = 0 ® Ei = E f
k
Ei = Ki +Ugi E f =Ugf +Uspf
Mechanics Review 3, Slide 9
Example: Angular Kinematics
1 2
(i)    0  0t   t
2





t
(ii)
0
2
2



(iii)
0  2 
Using (ii)  
Using (i)
 - 0
t
1
2
  t 2
Mechanics Review 3, Slide 10
Example: Angular Kinematics
(iv) I DISK
1
 MR 2
2
1 2
(v) K  I
2
Use (iv)
Use (v)
Mechanics Review 3, Slide 11
Example: Angular Kinematics
(vi)
d  R
v  R
(viii) aT  R
(vii)
v2
2
a



R
(ix) c
R
Use (viii)
Use (ix)
Mechanics Review 3, Slide 12
Example: Angular Kinematics
(vi)
d  R
v  R
(viii) aT  R
(vii)
v2
2
a



R
(ix) c
R
Use (vii)
Use (vi)
Mechanics Review 3, Slide 13
Example: Torque

  RF sin 
  900
  00
  900 - 360  540
Mechanics Review 3, Slide 14
Example: Torque
  RF sin 
Direction is perpendicular
to both R and F, given by
the right hand rule
x  0
y  0
 z   F  F  F
1
2
3
Mechanics Review 3, Slide 15
Example: Torque
(i) I DISK 
(ii)
1
MR 2
2
  I
(iii) K 
1 2
I
2
Use (i) & (ii)
Use (iii)
Mechanics Review 3, Slide 16
Example: Torque and parallel Axis Theorem
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 ICM = ML2/12
(a) What is the moment of inertia of the rod about its left end?
(b) What is the initial angular acceleration?
(c) What is the initial tangential acceleration of its center of
mass?
I = ICM + M(L/2)2 = ML2/3
τNet  Iα  MgL/2
at  L / 2
Mechanics Review 3, Slide 17
Example: Atwood's Machine with Massive Pulley
A pair of masses are hung over a massive disk-shaped
pulley as shown. For a disk ICM =1/ 2 MR2 . Note that for a
massive pulley T1 is not equal to T2.
y
Find the acceleration of the blocks.
x
M
We can use Dynamics:

R
For the hanging masses use F  ma
net
-m1g  T1  -m1a
-m2g  T2  m2a
For the pulley use
net  I
a 1
T1R - T2R  I  MRa
R 2
T2
T1
a
I
R
m2
m1
a
m1g
a
m2g
Mechanics Review 3, Slide 18
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.
1. Use energy methods to find the speed of the mass after
it has fallen a distance h.
2. Find the angular speed of the wheel
at that time.
3. What is the acceleration of the block?
4. What is the angular acceleration of the
wheel?
Ei = E f ® mgh = Iw + mv
1
2
v2  2ah
2
1
2
2
vR
a  R
Mechanics Review 3, Slide 19
Example: Falling Disk
A disk of radius R and mass M has a string wrapped around it.
The string is suspended from a fixed point and the sphere is
released from rest. The moment of inertia of a disk about its
center of mass is I = (1/2) MR2. Calculate:
(a) The angular acceleration of the disk.
(b) The tension in the string.
(c) What is the speed of the center of mass of the disk after it has
fallen a distance H?
 cm  I cm  TR  I cm
2
vcm
 2acm H
 Fy  Macm  Mg - T  Macm
M
R
=15rad/s
acm  R
Mechanics Review 4, Slide 20