Torque

changing rotational motion

§ 10.1–10.2

Example Problem

If the pulley has no mass:

a. What is the machine’s change in potential energy from its initial to its final states?

b. With what speed will the heavier mass hit the ground?

m1 m2

h

v0 = 0

mp

r

Example Problem

If the pulley has mass mp and radius r:

c. What is the pulley’s kinetic energy when its tangential speed is v?

d. What is the kinetic energy of the two masses traveling at speed v?

e. With what speed will the heavier mass hit the ground?

m1 m2

mp

h

r

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

• An influence causing angular acceleration

• angular analogue of Newton’s second law:

net = I– = torque– I = moment of inertia– = angular acceleration

• units?

Torque Definition

= Fl– F = tangential component of force– l = lever arm or moment arm

• units?

Lever Arm

Shortest distance from line of action to point of interest

Source: Young and Freedman, Figure 10.2

Poll Question

Force P is applied to one end of a lever of length L. What is the magnitude of the torque about point A?A. PL sin .

B. PL cos .

C. PL tan .

D. PL sec .

E. PL cot .

F. PL csc .

Source: Young and Freedman, Test Your Understanding §10.1

Torque Vector

Turning influence = torque

= radius force

= r F

Units: Nm (not J)

Source: Hewitt, Conceptual Physics

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°

Reconcile

= PL cos = PL sin (90° – )

Source: Young and Freedman, Test Your Understanding §10.1

90° –

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)

Poll Question

What is the direction of the torque about point O from force F1?

Source: Young and Freedman, Figure 10.2

A. B. C. D. E.

F.

Poll Question

What is the direction of the torque about point O from force F2?

Source: Young and Freedman, Figure 10.2

A. B. C. D. E.

F.

F

Point of Interest

Can define the torque about any point• a point on the rotation axis• the center of mass

• the origin• an observer

• Same force, same line of action, different “axes”

Adding Torques

• Net torque is zero

Source: Hewitt, Conceptual Physics

Whiteboard Work

A 10,000-N truck is stalled 1/4 of the way across a 100-m bridge.

a.What torque does its weight apply about the far support?

r

Whiteboard Work

b. What upward force must the near support provide to cancel the truck’s torque about the far support?

r

F

Whiteboard Work

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

F

r

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

Poll Question

A spool rests on a surface with sufficient friction to keep it from slipping. Which direction does it rotate when the cord is pulled as indicated?

A. Clockwise.

B. Counterclockwise.

r

Poll Question

A spool rests on a surface with sufficient friction to keep it from slipping. Which direction does it rotate when the cord is pulled as indicated?

A. Clockwise.

B. Counterclockwise.

r

Poll Question

A spool rests on a surface with sufficient friction to keep it from slipping. Which direction does it rotate when the cord is pulled as indicated?

A. Clockwise.

B. Counterclockwise.

r