Linear and Angular Momentum

Linear and Angular momentum

Linear Momentum

• Concept: how difficult it is to stop a moving object.

• Linear momentum is a vector quantity.

• Changing momentum requires unbalanced force.

• If we have a system of particles, then the total momentum is defined as the vector sum of the individual particles’ momenta:

Linear momentum

• A 200g ball is moving with constant speed of 2m/s. How much is the linear momentum of this ball?

(A)400 kgm/s

(B) 4kgm/s

(C) 0.4 kgm/s

(D)100 kgm/s

• A 200g ball is moving with constant speed of 2m/s. How much is the linear momentum of this ball?

(A)400 kgm/s

(B) 4kgm/s

(C) 0.4 kgm/s

(D)100 kgm/s

Note that you need to convert mass into kilograms. The correct answer is C.

• In the previous problem, assume that the ball hits a wall and reflects back with the same velocity. How much is the change in the momentum of the ball in this collision?

(A)Zero

(B) 0.4kgm/s

(C) 0.8kgm/s

(D)0.2kgm/s

• In the previous problem, assume that the ball hits a wall and reflects back with the same velocity. How much is the change in the momentum of the ball in this collision?

(A)Zero

(B) 0.4kgm/s

(C) 0.8kgm/s

(D)0.2kgm/s

Note that the momentum is a vector. So even though the magnitude of the momentum stays the same, its Direction changes, so Δp=0.4-(-0.4)=0.8kgm/s The correct answer is C.

• In the previous problem, which statement is valid during the collision of the ball and the wall?

(A)Wall exerts a bigger force on the ball

(B) Ball exerts a bigger force on the wall

(C) Ball and wall exert the same force on one another

(D)Ball and wall don’t exert any force on one another during collision

• In the previous problem, which statement is valid during the collision of the ball and the wall?

(A)Wall exerts a bigger force on the ball

(B) Ball exerts a bigger force on the wall

(C) Ball and wall exert the same force on one another

(D)Ball and wall don’t exert any force on one another during collision

Remember Third law? The correct answer is C.

Conservation of linear momentum

• If there is no unbalanced external force acting on a system of particles, then the total momentum of that system remains unchanged.

• Examples: billiard balls. Jumping off a stationary boat.

Conservation of linear momentum

When the man jumps out of the boat the boat moves backward.

System=man + boat

Initial momentum = 0

Final momentum also needs to be zero, so the boat finds a negative momentum to cancel the momentum of the person.

Cannon Ball!

F Dt = m Dv

For the same Force (amount of powder), why is the speed of

a cannon ball greater when fired from a longer cannon barrel?

Interaction Time

F Dt = m Dv

The longer cannon barrel gives the cannon ball a larger

impulse and therefore more momentum. The Force (F) is

allowed to act for a longer time Dt to build up velocity (Dv).

F Dt = m Dv

Impulse and Momentum acceleration = acceleration

a = a

F = Dv F Dt = m Dv

m Dt

Impulse Momentum

Impulse and Momentum F = Dv F Dt = m Dv

m Dt

Impulse Momentum

If a change in velocity (momentum) occurs over a short time,

a large force results.

If the change in velocity (momentum) occurs over an extended

time, a small force results. • Recall the Egg Toss Game

• A Boxer Bobs and Weaves His Head

• Bending Legs Upon a Parachute Landing

Conservation of Momentum Momentum is a conserved quantity, that is, for any isolated

system, the total momentum remains unchanged.

Momentum = mass x velocity P = m v




Consider the following collision:

Before After

m M V

v





Before After

m M m M V

v

v’

V’





Before After

Total Momentum

MV + mv = total momentum = MV’ + mv’

m M m M V

v

v’

V’

v

M

m

Total Momentum Before:

M V + m v

60 kg ( 0 km/hr) + 20 kg (10 km/hr) = 200

Ice Ball Toss

Momentum After (must be identical to momentum before)

= 200

= (M+m) v’

200 = (M+m) v’

200 = (60+20) v’

v’ = 200/80 = 2.5 km/hr

Ice Toss

What is the total momentum of the debris from a firecracker?

Before After

M V = 0 = total momentum before

Total Momentum After = m1v1 + m2v2 + m3v3 + …

Conservation of Momentum

m1

m2

m4

m3

What is the total momentum of the debris from a firecracker?

Before After

M V = 0 = total momentum before

Total Momentum After = m1v1 + m2v2 + m3v3 + …

= 0

Conservation of Momentum

m1

m2

m4

m3

Rifle Shot

Let mbullet = 0.3 kg, Mrifle = 5 kg, and vbullet = 370 m/s

mbulletvbullet + MrifleVrifle = 0.3 kg (370 m/s) + 5kgVrifle

0 = 0.3 kg (370 m/s) + 5kgVrifle

Rifle Shot If momentum is conserved, why doesn’t a rifle kill you upon

recoil after firing a bullet?

Before: mbulletvbullet + MrifleVrifle = 0

Rifle Shot If momentum is conserved, why doesn’t a rifle kill you upon

recoil after firing a bullet?

Before: mbulletvbullet + MrifleVrifle = 0

After: = 0


Rifle Shot



0 = 0.3 kg (370 m/s) + 5kgVrifle

-0.3(370) = 5 kg Vrifle

Vrifle = - 0.3(370)/5 = - 2.2 m/s

Rifle Shot Let mbullet = 0.3 kg, Mrifle = 5 kg, and vbullet = 370 m/s


0 = 0.3 kg (370 m/s) + 5kgVrifle

-0.3(370) = 5 kg Vrifle

Vrifle = - 0.3(370)/5 = - 22.2 m/s

Shoulder aches, BUT your alive!

Mriflevrecoil = mbulletVbullet

• In this picture, if the person jumps off the boat with a speed of 3m/s, how much would the speed of the boat be, assuming that the mass of the person is three times smaller than the mass of the boat?

• In this picture, if the person jumps off the boat with a speed of 3m/s, how much would the speed of the boat be, assuming that the mass of the person is three times smaller than the mass of the boat?

See if you can show that the answer is 1m/s.

Train Link An train engine runs into a stationary box car

weighing 4x more than itself to link up. If the engine

was traveling 10 mph before link up, how fast does

the train move after?





MOMENTUM BEFORE = MOMENTUM AFTER

MVBC + 0 = (M + 4M) VAC





MOMENTUM BEFORE = MOMENTUM AFTER

MVBC + 0 = (M + 4M) VAC

M(10) = (5M) VAC

10 = 5 VAC

2 = VAC

Angular Momentum L Angular Momentum: A combination of...

m Mass

v Speed of Rotation

r Mass Position (with respect to rotational axis)

L = m v r

• Conservation Examples:

– Spins of Dancers or Ice Skaters

– Those Funky Coin Vortexes in Stores

– Tops and Gyroscopes

– Riding a Bicycle

Angular momentum and torque

• Angular momentum is a measure of how difficult it is to stop a rotating object. It first defined for rotating bodies, but even object moving on a straight line can have angular momentum with respect to specific observers.

• Angular momentum=mass*velocity*distance from axis of rotation.

• How can we change angular momentum? • It can be changed by applying “torque”.

Torque

• Torque is the quantity which changes the angular momentum. Torque is a twisting action that produces rotational motion or a change in rotational motion.

torque

If force is parallel to r, it doesn’t have any torque.

Conservation of angular momentum

• If there is no unbalanced external torque, then the angular momentum of the system remains unchanged.

• In some situations, the net force acting on the object is not zero, but force is parallel to radius. So, the torque created by that force is zero. As a result angular momentum is conserved.

Faster, Closer

Conservation of Angular Momentum: L = L

m V r m v R

Using conservation of angular momentum, explain why Earth slows down, when its farther from the Sun.

• A comet at its farthest point from the Sun is 900 million miles, traveling at 6000 mi/h. What is its speed at its closest point of 30 million miles away?




Copyright © Houghton Mifflin Company. All rights reserved.

3-43

Conservation of Angular Momentum

• Figure Skater – she/he starts the spin with arms out at one angular velocity. Simply by pulling the arms in the skater spins faster, since the average radial distance of the mass decreases.

• m1v1r1 = m2v2r2

• m is constant; r decreases;

• Therefore v increases

Section 3.6

Which statement is valid? (A) If there is no unbalanced torque, total angular momentum changes. (B) unbalanced torque changes the total angular momentum. (C) Any unbalanced force can change the angular momentum.

Which statement is valid? (A) If there is no unbalanced torque, total angular momentum changes. (B) unbalanced torque changes the total angular momentum. (C) Any unbalanced force can change the angular momentum.

If angular momentum changes, there is unbalanced torque. B is correct.

• What will happen for a figure skater, when she pulls in her arms during a spin?

(A)She will spin faster

(B) She will spin more slowly

(C) Nothing is going to happen

(D)She will fall

• What will happen for a figure skater, when she pulls in her arms during a spin?

(A)She will spin faster

(B) She will spin more slowly

(C) Nothing is going to happen

(D)She will fall

Angular momentum stays conserved. Since the average “r” gets smaller, angular speed has To go up to keep the angular momentum conserved. As a result she will spin faster. A is correct.

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Linear and Angular Momentum