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Physics 1D03 - Lecture 31

Angular Momentum II

• • General motion of a rigid body Collisions involving rotation Text Section 11.1-11.6

Physics 1D03 - Lecture 31

Recall: • Angular momentum” is the rotational analogue of linear momentum.

m

v F p

I

w t

L

(“angular momentum”)

L = I

w t

external



dL



I



dt

|

L

| =

mrv

t

= mvr

sin f Physics 1D03 - Lecture 31

Question

Two astronauts are held together by a long rope and rotate about their common center of mass. One has twice the mass of other.

One astronaut gathers in 1/3 of the rope separating them.

Which of the following remain constant?

A) Kinetic energy B) Angular velocity C) Angular momentum D) Tension in the rope

Physics 1D03 - Lecture 31

Angular momentum of a particle:

L



r



p



r

 (

m

v

) of a rotating rigid body:

L

= I w.

In general, for a

moving

, rotating rigid body,

L



r

 (

m

v

CM

)  I

CM

ω

The first term is called the “orbital” angular momentum and the second term is the “spin” angular momentum. Example: angular momentum of a planet about the sun.

Physics 1D03 - Lecture 31

Example:

The earth (m = 6.0 x 10 24 kg, R = 6400 km) moves at speed v = 30 km/s in an orbit of radius r = 150 x 10 6 km around the sun. It also spins on its axis once per day ( ω = 7.3 x 10 -5 rad/s). The angular momentum of the earth relative to the centre of the sun is

L = mvr + I CM

ω. The “orbital” part is calculated as if the earth were a particle orbiting the sun; then we add a the angular momentum or the spinning earth relative to its own centre of mass.

Physics 1D03 - Lecture 31

Collisions:

Collisions can conserve angular momentum as well as linear momentum .

Total

linear

momentum is conserved if there is no

external

force during the collision (or if the external forces are small compared to the forces the colliding bodies exert on each other).

Total

angular

momentum is conserved if there is no

external torque

during the collision (or if the external torques are small). Angular momentum may be calculated about any axis. Usually it is convenient to use an axis through the centre of mass, unless one of the colliding objects actually rotates about some other fixed axis.

Physics 1D03 - Lecture 31

A metre stick (mass

M

, length

L

= 1m , moment of inertia

I

) is suspended from one end by a frictionless pivot at

P

. A ball of mass

m

, velocity

v 0

, strikes the other end of the (stationary) stick at right angles, and stops (final velocity of the ball is zero).

Question:

Which of the following describe the motion

of the stick

after the collision? (Answer True, False, or Maybe for each one.) A) B) C) D)

I CM I P

w w

= mv 0 L/2 = mv 0 L Mv CM = mv 0 ½ I P

w

2 = ½ mv 0 2 v 0 P

Physics 1D03 - Lecture 31

Quiz

A stick (uniform thin rod) is lying on the ice. A hockey puck hits the stick, at right angles, and the stick starts to slide. Point

P

is on the end farthest from where the puck hits. Immediately after the collision, the end

P

will start to move: A) in the same direction as v 0 B) in a direction opposite to v 0 C) at an angle (not 0 o or 180 o ) to v 0 D) It depends where the puck hits

v 0 P CM

Physics 1D03 - Lecture 31

Example:

A 2.0kg disk moving at 3.0m/s hits a 1.0kg stick lying flat on a frictionless surface.

The moment of inertia of the stick is I=1.33kg m 2 .

Assuming an elastic collision, find the speeds of the disk and stick after the collision and the rotational speed of the stick.

2m

v 0

Physics 1D03 - Lecture 31

Example: Sticky clay of mass m and velocity v hits a cylinder of mass M and radius R. Find the angular speed of the system after the collision. Is energy conserved?

Physics 1D03 - Lecture 31

Summary

In general, for a rigid body,

L



r

 (

m

v

CM

)  I

CM

ω

In collisions, angular momentum will be conserved it there is no external torque.

Physics 1D03 - Lecture 31