Angular Momentum

Lecturer: Professor Stephen T. Thornton

Reading Quiz:Can an object moving in a straight line ever have a nonzero angular momentum?

 

A) Always B) Never C) Sometimes

Answer: C

Sometimes, because it depends upon the axis of rotation around which you want to find the angular momentum. There is no angular momentum when the object passes through the rotation axis, because the moment arm is zero. There is angular momentum when the moment arm is nonzero (see left sketch).

Last TimeRotational kinetic energyObjects rolling – energy, speedRotational free-body diagramRotational work

Today

Angular momentum

Vector (cross) products

Torque again with vectors

Unbalanced torque

Angular Momentum of Circular Motion

This particle has linear momentum. We can also say it has an angular momentum with respect to a given point, in this case the center of the circle.

2

unit: kg m / s

L I

In the case of the particle moving around the circle, let’s look more carefully at the angular momentum.

2( )

But note that ,

so we have

vL I mr mvr

r

mv p

L pr

This is another way to

determine angular momentum.

The Angular Momentum of Non-Tangential Motion

sinL pr pr

m

m

sin sinL mv r prq q= =

A particle moving in any direction can have angular momentum about any point.

pO

O

O

O

Angular Momentum in Linear and Circular Motion

The L in each view is constant. If are the same, then L is the same.

,v r

L mv r̂=

Change in angular momentum

L = IL = I, divide by t

torque

LI I

t t

dLI

dt

This equation looks similar to Newton’s 2nd law. It is sometimes called Newton’s 2nd law for rotation.

looks like dp

F madt

Conservation of angular momentum

Note what happens when there is no torque. L = 0, and angular momentum is constant.

L dL

t dt

net, ext if 0i fL L

Note similarity to conservation of linear momentum when Fnet,ext = 0.

net, ext if 0i fp p F

Conceptual Quiz: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, how do her rotational inertia, her angular momentum and her rotational kinetic energy change?  A) They all increase.B) They all remain the same.C)They all decrease.D) Rot inertia decreases, angular momentum remains constant, and her KE increases.E)Rotational inertia and angular momentum decrease, KE decreases.

Answer: DAngular momentum must be conserved. No torque. Rotational inertia decreases, because radius decreases. Only D is possible.

How does KE increase?

I goes down, goes up, L constant.

But K will increase because of .

2 1 1( )2 2

12

I LK I

http://www.youtube.com/watch?v=AQLtcEAG9v0

Vector Cross Product; Torque as a Vector

The vector cross product is defined as:

The direction of the cross product is defined by a right-hand rule:

sinC AB q= ´ =A B

= ´C A B

The vector (cross) product can also be written in determinant form:

Some properties of the cross product:

Conceptual QuizThe direction of the vector cross product is along the direction

A)

B)

C)

D)

E)

j i

i

i

j

k

k

Answer: E

For a particle, the torque can be defined around a point O:

Here, is the position vector to the point of application of force relative to O.

r

t = ´r F

·

Torque can be defined as the vector product of the position vector from the axis of rotation to the point of action of the force with the force itself:

t = ´r F

Torque

A right-hand rule gives the direction of the torque.

t = ´r F

The Right-Hand Rule for Torquer

r

r

F

t = ´r F

Yo-yo demo

Rotate

Rotate

v v

No motion

r

r

F

F

torque in

torque outO O

Angular Momentum of a Particle

The angular momentum of a particle about a specified axis (or point) is given by:

L r p

The Right-Hand Rule for Angular Momentum

L I

L r p

p

Let’s do this demo!

L I

Let’s do this demo!

L I

No torque: L is conserved. I decreases, therefore ω must increase.

A Rotational Collision – Angular momentum will be conserved here.

Angular Momentum of a Particle

If we take the derivative of , we find:

Since

we have:

L

( )dL d dr dp dp

r p p r rdt dt dt dt dt

0

dp dLr F r

dt dt´ = ´ =å

dL

dtt =å

Opposite Particles. Two identical particles have equal but opposite momenta, and , but they are not traveling along the same line. Show that the total angular momentum of this system does not depend on the choice of origin.

p

p

Conceptual Quiz:When a large star burns up its fuel, the gravitational force contracts it to a small size, even a few km. This is called a neutron star. When neutron stars rotate at high speed, even 100 rev/sec, they are called pulsars. They have more mass than our sun. What causes the high rotational angular velocity?

 

A) Friction of gas particles B) Conservation of angular

momentum

C) The dark force

D) Conservation of energy

Answer: B

Just like our own sun, these stars rotate about their own axis. As gravity contracts the particles closer and closer, the density becomes huge. There are no torques, so angular momentum must be conserved. L=I, so as I decreases, must increase.