Chapter 9: 4-7

Center of Mass

Conservation of Linear Momentum

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

VmP

tFIP

Momentum is a vector quantityInitial

Final

mV

-mV

Change in momentum:

xmV

xmVxmV

PPP INITFINAL

2

ˆ)(ˆ)(

P

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.

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 difference4. Sorry, I don’t believe in guns.

Relationship of Momentum and Kinetic Energy

m

PK

mVK

VmP

2

2

1

2

2

222

22

2

1

2

22

mVm

Vm

m

mV

m

PK

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!

M M

VBefore

M M

V/2After INELASTIC collision

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

Problem: Exploding ObjectAn 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?

M1 M2 M1 M2

V1V2

INITIAL: FINAL:

0

0

i

i

K

P

12

1

21

2

22

1122

22

KK

m

P

m

PK

xVmVmP

f

f

Conservation of Momentum: 1122 VmVm

In this problem, K2 = 2K1

Solution, Exploding object.

111222 PVmVmP

12 2KK

What is the ratio of the masses? Which piece has the larger mass?

2

1

121

222

1

2

2/

2/2

m

m

mP

mP

K

K

So

22

1 m

m

12

1

21

2

22

1122

22

0

KK

m

P

m

PK

xVmVmP

f

f

Use P1 = P2

Larger mass has smaller K.

M1 M2

V1V2

Conceptual Checkpoint.

1. Work, Impulse, Power, Force

2. Work, Power, Impulse, Force

3. Work, Power, Impulse, Energy

4. Work, Energy, Power, Impulse

What are the following quantities, in order?

TP

VF

TF

LF

/

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?

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!

Work this one out!

A real-world example: Ion scattering

Heavy atoms in substrate target

Light scattering atom.

m

M

Vi Vf

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?

Ion scattering: can it be done?

UCF Heavy Ion Backscattering Spectrometer (HIBS)

mM

V1

mM

V2

V

Initial:

Final:

After some algebra….

21

1

2

1vmK

vmP

init

init

222

2

2

1

2

1MVvmK

vmMVP

final

final

Apply conservation of momentum and energy.

21

21

vv

vvmM

Is this reasonable?If v2=0, M=m.If V2=V1, M goes to infinity.

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.)

m

v

A rocket engine emits a certain mass of fuel per unit time. This results in a force, which is called Thrust.

thrustFt

P

t

vm

This is an important case where the change in momentum comes about because of the change in mass.

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.

Center of Mass of a Mobile

The center of mass can be found by the following process:

...

...

21

2211

mm

RmRmRCM

For example, for the mobile,

21

2211

mm

XmXmXCM

Numerator is the “moment”

Total mass

Find the Center of Mass: 2nd try

1. R = .25 m2. R = .50 m3. R = .75 m4. R = 1.0 m

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?

1 3

1m

...

...

21

2211

mm

RmRmRCM

The choice of origin does not affect result.

Forces and Center of Mass

21

2211

mm

XmXmXCM

Suppose

221121 )( amamamm cm

21 FFFcm

F1

Fcm

F2