Conservation of Angular Momentum8.01W10D2You

ng and Freedman 10.5-10.6

Announcements

Math Review Night Next Tuesday from 9-11 pm Pset

10 Due Nov 15 at 9 pm No Class Friday Nov

11 Exam 3 Tuesday Nov 22 730-930 pm W011D1

Rolling without Slipping, Translation and

Rotation Reading Assignment Young and Freedman

10.3-4

Time Derivative of Angular Momentum for a Point

Particle

- Angular momentum of a point particle about S
- Time derivative of the angular momentum about S
- Product rule
- Key Fact
- Result

Torque and the Time Derivative of Angular

Momentum Point Particle

- Torque about a point S is equal to the time

derivative of the angular momentum about S .

Conservation of Angular Momentum

- Rotational dynamics
- No torques
- Change in Angular momentum is zero
- Angular Momentum is conserved

Concept Question Change in Angular Momentum

- A person spins a tennis ball on a string in

a horizontal circle with velocity (so that

the axis of rotation is vertical). At the point

indicated below, the ball is given a sharp blow

(force ) in the forward direction. This

causes a change in angular momentum in the

- direction
- direction
- direction

Concept Question Change in Angular Momentum

- Answer 3. The torque about the center of the

circle points in the positive -direction.

The change in the angular momentum about the

center of the circle is proportional to the

angular impulse about the center of the circle

which is in the direction of the torque

Table Problem Cross-section for Meteor Shower

- A meteor of mass m is approaching earth as

shown on the sketch. The distance h on the sketch

below is called the impact parameter. The radius

of the earth is Re. The mass of the earth is me.

Suppose the meteor has an initial speed of v0.

Assume that the meteor started very far away from

the earth. Suppose the meteor just grazes the

earth. You may ignore all other gravitational

forces except the earth. Find the impact

parameter h and the cross section ph2 .

Angular Momentum for System of Particles

- Treat each particle separately
- Angular momentum for system about S

Conservation of Angular Momentum for System of

Particles

- Assumption all internal torques cancel in pairs
- Rotational dynamics
- No external torques
- Change in Angular momentum is zero
- Angular Momentum is conserved

Angular Impulse and Change in Angular Momentum

- Angular impulse
- Change in angular momentum
- Rotational dynamics

Kinetic Energy of Cylindrically Symmetric Body

- A cylindrically symmetric rigid body with moment

of inertia Iz rotating about its - symmetry axis at a constant angular velocity

- Angular Momentum
- Kinetic energy

Concept Question Figure Skater

- A figure skater stands on one spot on the ice

(assumed frictionless) and spins around with her

arms extended. When she pulls in her arms, she

reduces her rotational moment of inertia and her

angular speed increases. Assume that her angular

momentum is constant. Compared to her initial

rotational kinetic energy, her rotational kinetic

energy after she has pulled in her arms must be - the same.
- larger.
- smaller.
- not enough information is given to decide.

Concept Question Figure Skater

- Answer 2. Call the rotation axis the z-axis.

When she pulls her arms in there are no external

torques in the z-direction so her z-component of

her angular momentum about the axis of rotation

is constant. The magnitude of her angular

momentum about the rotation axis passing through

the center of the stool is L I?. Her kinetic

energy is - Since L is constant and her moment of inertia

decreases when she pulls her arms, her kinetic

energy must increase.

Constants of the Motion

- When are the quantities, angular momentum

about a point S, energy, and momentum constant

for a system? - No external torques about point S angular

momentum about S is constant - No external work mechanical energy constant
- No external forces momentum constant

Rotational and Translational Comparison

Quantity Rotation Translation

Force

Torque

Kinetic Energy

Momentum

Angular Momentum

Kinetic Energy

Demo Rotating on a Chair

- A person holding dumbbells in his/her arms spins

in a rotating stool. When he/she pulls the

dumbbells inward, the moment of inertia changes

and he/she spins faster.

Concept Question Twirling Person

- A woman, holding dumbbells in her arms, spins

on a rotating stool. When she pulls the dumbbells

inward, her moment of inertia about the vertical

axis passing through her center of mass changes

and she spins faster. The magnitude of the

angular momentum about that axis is - the same.
- larger.
- smaller.
- not enough information is given to decide.

Concept Question Twirling Person

- Answer 1. Call the rotation axis the z-axis.

Because we assumed that there are no external

torques in the z-direction acting on the woman

then the z-component of her angular momentum

about the axis of rotation is constant. (Note if

we approximate the figure skater as a symmetric

body about the z-axis than there are no

non-z-components of the angular momentum about a

point lying along the rotation axis.)

Demo Train B134

- (1) At first the train is started without the

track moving. The train and the track both move,

one opposite the other. - (2) Then the track is held fixed and the train

is started. When the track is let go it does not

revolve until the train is stopped. Then the

track moves in the direction the train was

moving. - (3) Next try holding the train in place until

the track comes up to normal speed (Its being

driven by the train). When you let go the train

remains in its stationary position while the

track revolves. You shut the power off to the

train and the train goes backwards. Put the power

on and the train remains stationary.

A small gauge HO train is placed on a circular

track that is free to rotate.

Table Problem Experiment 6

A steel washer, is mounted on the shaft of a

small motor. The moment of inertia of the motor

and washer is Ir. Assume that the frictional

torque on the axle is independent of angular

speed. The washer is set into motion. When it

reaches an initial angular velocity ?0, at t 0,

the power to the motor is shut off, and the

washer slows down during the time interval ?t1

ta until it reaches an angular velocity of ?a at

time ta. At that instant, a second steel washer

with a moment of inertia Iw is dropped on top of

the first washer. The collision takes place over

a time ?tcol tb - ta.

- What is the angular deceleration a1 during the

interval ?t1 ta? - What is the angular impulse due to the frictional

torque on the axle during the collision?

c) What is the angular velocity of the two

washers immediately after the collision is

finished?

Experiment 6 Conservation of Angular Momentum

AppendixAngular Momentum and Torque for a

System of Particles

Angular Momentum and Torque for a System of

Particles

- Change in total angular momentum about a point S

equals the total torque about the point S

Internal and External Torques

- Total external torque
- Total internal torque
- Total torque about S

Internal Torques

- Internal forces cancel in pairs
- Does the same statement hold about pairs of

internal torques? - By the Third Law this sum becomes

The vector points from the

jth element to the ith element.

Central Forces Internal Torques Cancel in Pairs

- Assume all internal forces between a pair of

particles are directed along the line joining the

two particles then - With this assumption, the total torque is just

due to the external forces - No isolated system has been encountered such

that the angular momentum is not constant.