Physics 1D03 - Lecture 30 1

Angular MomentumAngular Momentum

• Angular momentum of rigid bodies• Newton’s 2nd Law for rotational motion• Torques and angular momentum in 3-D

Text sections 11.1 - 11.6

Physics 1D03 - Lecture 30 2

“Angular momentum” is the rotational analogue of linear momentum.

Recall linear momentum: for a particle, p = mv .

Newton’s 2nd Law: The net external force on a particle is equal to the rate of change of its momentum.

dtd

external

pF

m Iv F p L (“angular momentum”)

To get the corresponding angular relations for a rigid body, replace:

Physics 1D03 - Lecture 30 3

Angular momentum of a rotating rigid body:

Angular momentum, L, is the product of the moment of inertia and the angular velocity.

Units: kg m2/s (no special name). Note similarity to: p=mv

Newton’s 2nd Law for rotation: the torque due to external forces is equal to the rate of change of L.

I

dtd

IdtId

dtdL )(

dtdL

external

For a rigid body (constant I ),

L = I

So, sometimes Iexternal (but not always).

Physics 1D03 - Lecture 30 4

Conservation of Angular momentum

There are three great conservation laws in classical mechanics:

1) Conservation of Energy2) Conservation of linear momentum3) and now, Conservation of Angular momentum:

In an isolated system (no external torques), the total angular momentum is constant.

Physics 1D03 - Lecture 30 5

For a symmetrical, rotating, rigid body, the vector L will be along the axis of rotation, parallel to the vector , and

L = I

L

(In general L is not parallel to , but I is still equal to the component of L along the rotation axis.)

Angular Momentum Vector

Physics 1D03 - Lecture 30 6

Angular momentum of a particle

z

rv

L

O

x

y

m

)( vrprL m

This is the real definition of L.

• L is a vector.

• Like torque, it depends on the choice of origin (or “pivot”).

• If the particle motion is all in the x-y plane, L is parallel to the z axis..

Physics 1D03 - Lecture 30 7

|L| = mrvt = mvr sinθ

For a particle travelling in a circle (constant |r|, θ=90), vt = r, so:

L = mrvt = mr2 = I

rv

m

Angular momentum of a particle (2-D):

Physics 1D03 - Lecture 30 8

As a car travels forwards, the angular momentum vector L of one of its wheels points:

A) forwardsB) backwards

C) upD) down

E) leftF) right

Quiz

Physics 1D03 - Lecture 30 9

Quiz

A physicist is spinning at the center of a frictionless turntable, holding a heavy physics book in each hand with his arms outstretched. As he brings his arms in, what happens to the angular momentum?

A) increasesB) decreasesC) remains constant

What happens to the angular velocity?

Physics 1D03 - Lecture 30 10

Example:

A student sits on a rotating chair, holding two weights each of mass 3.0kg. When his arms are extended to 1.0m from the axis of rotation his angular speed is 0.75 rad/s. The students then pulls the weights horizontally inward to 0.3m from the axis of rotation.

Given that I = 3.0 kg m2 for the student and chair, what is the new angular speed of the student ?

Physics 1D03 - Lecture 30 11

m1m2v v

R

Example

Angular momentum provides a neat approach to Atwood’s Machine. We will find the accelerations of the masses using “external torque = rate of change of L”.

O

Physics 1D03 - Lecture 30 12

m1m2v v

R

O

p1

p1

r

R

For m1 : L1 = |r1 x p1|= Rp1

so L1 = m1vR L2 = m2vR Lpulley= I = Iv/R

Thus L = (m1 +m2 + I/R2)v R

so dL/dt = (m1 +m2 + I/R2)a R

Torque, = m1gR m2gR = (m1 m2 )gR

Write = dL/dt, and complete the calculation to solve for a.

Atwoods Machine, frictionless (at pivot), massive pulley

Note that we only consider the external torques on the entire system.

Physics 1D03 - Lecture 30 13

Solution

Physics 1D03 - Lecture 30 14

Summary

)( vrprL m

i

ii prL

L = I

Newton’s 2nd Law for rotation: dtdL

external

Particle:

Any collection of particles:

Rotating rigid body:

Angular momentum is conserved if there is no external torque.