1. Center of Mass Conservation of Linear Momentum
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. 0 / 100
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.
(A), metal, because it would transfer more energy to the block.
(B), rubber, because it would transfer more momentum to the block (higher impulse).
Makes no difference
Sorry, I don’t 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: P=mV (conserved, doesn’t 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? V1 V2 M1 M2 M1 M2
INITIAL:
FINAL:
Conservation of Momentum:
In this problem, K2 = 2K1

8. Solution, Exploding object.M1 M2 V1 V2
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
Conceptual Checkpoint.
What are the following quantities, in order?
Work, Impulse, Power, Force
Work, Power, Impulse, Force
Work, Power, Impulse, Energy
Work, Energy, Power, Impulse
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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 v2=0, M=m.
If V2=V1, 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.”
Let’s 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.
R = .25 m
R = .50 m
R = .75 m
R = 1.0 m
Cross-Tab Label

19. Forces and Center of Mass
Suppose
Fcm F2 F1