Center of Mass

1
Chapter 9 4-7

• Center of Mass
• Conservation of Linear Momentum

2
Main Concepts

• Linear momentum is Mass times Velocity
• Momentum is a vector quantity
• Momentum is conserved in the absence of an
  outside force
• Momentum is changed by an Impulse
• Impulse is Force times Time

3
Momentum is a vector quantity
Initial
mV
Final
-mV
Change in momentum
DP
4
A block of wood is struck by a bullet. Is the
block more likely to be knocked over if the
bullet is (A) metal and embeds itself in the
wood, or if the bullet is (B) rubber and bounces
off the wood? The mass of the two bullets is the
same.
0 / 100

1. (A), metal, because it would transfer more energy
   to the block.
2. (B), rubber, because it would transfer more
   momentum to the block (higher impulse).
3. Makes no difference
4. Sorry, I dont believe in guns.

Cross-Tab Label
5
Relationship of Momentum and Kinetic Energy
6
Consequences of Momentum Conservation in Elastic
and Inelastic Collisions

• Linear momentum is conserved (unchanged) in a
  collision
• Kinetic energy is only conserved in an elastic
  collision
• Linear momentum can stay the same, even though K
  changes!

Before
V
M
M
After INELASTIC collision
V/2
M
M
Momentum PmV (conserved, doesnt change before
and after)
Kinetic Energy
Initial K ½ mV2
Final K ½ (2m) (V/2)2 ¼ mV2
7
Problem Exploding Object
An object initially at rest breaks into two
pieces as the result of an explosion. One piece
has twice the kinetic energy of the other piece.
What is the ratio of the masses of the two
pieces? Which piece has the larger mass?
V2
V1
M1
M2
M1
M2
FINAL
INITIAL
Conservation of Momentum
In this problem, K2 2K1
8
Solution, Exploding object.
What is the ratio of the masses? Which piece has
the larger mass?
So
Larger mass has smaller K.
Use P1 P2
9
Conceptual Checkpoint.
0 / 100
What are the following quantities, in order?

1. Work, Impulse, Power, Force
2. Work, Power, Impulse, Force
3. Work, Power, Impulse, Energy
4. Work, Energy, Power, Impulse

Cross-Tab Label
10
Putting Momentum and Energy to Work
The masses m and M are known, and the incident
velocity. Can we determine how high the pendulum
moves after the inelastic collision?
11
Work backwards from desired result
If you knew the kinetic energy of the block and
bullet after the collision, you could use
conservation of energy to convert the kinetic
energy into potential energy, and find the height
H. If you new the velocity of the block and
bullet after collision, you could find the
kinetic energy. You can use conservation of
momentum to find the final velocity!
12
Work this one out!
13
A real-world example Ion scattering
Light scattering atom.
Vi
m
Vf
M
Heavy atoms in substrate target
VT
You select the scattering atom mass m, and the
incident velocity Vi. If you can measure the
scattered velocity Vf, can you tell what is the
mass of the target, M?
14
Ion scattering can it be done?
M
m
Initial
V1
M
m
V
Final
V2
Apply conservation of momentum and energy.
UCF Heavy Ion Backscattering Spectrometer (HIBS)
After some algebra.
Is this reasonable? If v20, Mm. If V2V1, M
goes to infinity.
15
The Rocket Problem
Professor Goddard does not know the relation
between action and reaction and the need to have
something better than a vacuum against which to
react. He seems to lack the basic knowledge
ladled out daily in high schools." (1921
New York Times editorial about Robert Goddard's
revolutionary rocket work.)
A rocket engine emits a certain mass of fuel per
unit time. This results in a force, which is
called Thrust.
m
This is an important case where the change in
momentum comes about because of the change in
mass.
v
16
Center of Mass What it is, and why it matters
The center of mass is the point at which the
external forces acting on an object appear to
act.
Lets look at some examples.
17
Center of Mass of a Mobile
The center of mass can be found by the following
process
For example, for the mobile,
Numerator is the moment
Total mass
18
Find the Center of Mass 2nd try
0 / 100
A mass of 1 kg is located at the origin of a
meter stick. A mass of 3 kg is at the other end
of the meter stick. Where is the center-of-mass
located?
3
1
1m
The choice of origin does not affect result.

1. R .25 m
2. R .50 m
3. R .75 m
4. R 1.0 m

Cross-Tab Label
19
Forces and Center of Mass
Suppose
Fcm
F2
F1