Warm-up

How does an object’s momentum change if it receives a net force perpendicular to its velocity?

v

F

How about its kinetic energy?

Announcement

• Lab this week leads class. That’s on purpose.

• There will be a rotation lab next week.

Torque and Angular Momentum

Vectors of rotation

Objectives

• Determine the torque applied by a force about an axis.

• Predict an object’s response to a torque.

• Calculate an object’s angular momentum.

You push on a door. It will open easiest if you push

A. opposite the hinge.

B. at the center of the door.

C. near the hinge.

Poll Question

Torque

Turning force = torque

= radius force

= r F

Units: Nm (not J)

Vector Cross Product

• Operation symbol • Another way to multiply two vectors

• Product is a vector!• Direction of AB is perpendicular to

both A and B

Cross Product Magnitude

A B = ab sin A

B

a

b

Maximum for = 90°Zero for = 0°, 180°

Magnitude Geometrically

A

B

a

b

AB = area of parallelogram

Cross Product Direction

• Curl right-hand fingers in direction of

• Right-hand thumb points in direction of cross-product

• Not commutative

A

B

a

b

AB = –(BA)

Radius Vector

• Radius from reference point to application of force

• Strictly, reference point must be specified to determine a torque (about the point)

• Torque depends on your choice of point!

Adding Torques

• Net torque about fulcrum is zero here

• Torques are vectors

750 N

Whiteboard Work

A 10,000-N truck is stalled 1/4 of the way across a 100-m bridge. What torque does its weight apply about the far (right) support?

r

Whiteboard Work

What upward force must the near (left) support provide to cancel the truck’s torque about the far support?

r

F

Whiteboard Work

What upward force must the far support provide to support the weight of the truck?

F

r

Hint: Several ways will work:• canceling forces on the truck• canceling torques about the near support

Angular Momentum

• Torque is rotational force

• Angular momentum is rotational momentum

L = r p

• Angular momentum is a vector

• Direction by right-hand rule

Poll Question

The moon revolves around the Earth with a period of 27.322 days. If the moon were to move farther away from the Earth but maintain its angular momentum, how would its period adjust?

A. Its period would become shorter than 27.322 days.

B. Its period would become longer than 27.322 days.

C. Its period would remain 27.322 days.

Newton’s Second Law

• Force is the rate of change of momentum

• Torque is the rate of change of angular momentum

t F =

p

t =

L

Conservation of Angular Momentum

• If no outside torque, l = r p is constant.

• If r decreases, p increases!

Conservation of Angular Momentum

• Nothing can apply a torque to itself.

• Any change in one object’s angular momentum is accompanied by an opposite change in another object.

• The angular momentum of the universe never changes.

Conservation of Momentum

Q. How can linear momentum be conserved if p increases?

A. Total linear momentum is zero in a rotating system!

Conservation of Energy

Q. What happens to kinetic energy when p increases?

A. Kinetic energy increases! E = w.Work is done to pull rotating parts inward.

Gyroscopic Stability

More angular momentum

Objectives

• Determine the torque applied by a force about an axis.

• Predict an object’s response to a torque.

• Calculate an object’s angular momentum.

Angular Momentum

• Angular momentum is rotational momentum

L = r p

• Torque is the rate of change of angular momentum

t =

L

Gyroscopic Stability

Q. What happens when you apply a torque to an object in a direction different from its rotation?

A. Its axis of rotation rotates!

L0

L

Adding Angular Momentum

force

L LL

l0

L0 + L = L

Reading for Next Time

• Conservation laws– They tell us about the nature of the Universe

• Elasticity– A simple but surprisingly useful model of solid

behavior