Angular Motion

Chapter 10

Figure 10-1Angular Position

Figure 10-2Arc Length

Figure 10-3Angular Displacement

Figure 10-4Angular Speed and Velocity

Angular Speed is a Vector!

We use a “right hand rule” to determine the vector direction of a rotation. Using your right hand, curl your fingers in the direction of the rotation. Your thumb points in the direction of the rotation.

Works for angular acceleration as well.

Figure 10-5Angular Acceleration

Summary of angular motions.

t

t

Angular position, radians, measure counter-clockwise.

Angular velocity, radians per second.

Angular acceleration, radians per second squared.

Note that radians are a dimensionless quantity.

Radians = Degrees * /180

Example: 180 degrees = 3.14 radians

Linear and Rotational Motion Compared

2

2

1mvK

amF

vmP

t

va

t

xv

x

2

2

1

IK

IT

IL

t

t

Position

Velocity

Acceleration

Momentum

Force/Torque

Kinetic Energy

Figure 10-7Angular and Linear Speed

Conceptual Checkpoint 10-1How do the angular speeds compare?

V=r

How do the linear speeds compare?

Figure 10-8Centripetal and Tangential Acceleration

IMPORTANT:For uniform circular motion, The centripetal acceleration is:

r

vac

2

For constant angular speed, at = 0. Then, the acceleration is RADIAL, inwards.

Figure 10-9Rolling Without Slipping

Figure 10-11Velocities in Rolling Motion

Figure 10-10Rotational and Translational Motions of a Wheel

Figure 10-12Kinetic Energy of a Rotating Object

2

2

1mvK

But… rv

So…

22

2

2

2

12

12

1

mr

rm

mvK

Define the moment of inertia, I…

2mrI

(it’s different for different shapes!)

2

2

1 IKROT

Moment of Inertia

i

iirmI 2

Vi

Mi

Ri

I

RM

RM

VM

KK

ii

i

iiii

iii

ii

2

22

22

2

2

1

2

1

2

1

2

1

Rigid body. Break up into small pieces Mi. What is the angular speed of each piece?

Rotational force: Torque

Torque is the “twisting force” that causes rotational motion. It is equal to the magnitude of the component of an applied force perpendicular to the arm transmitting the force.

F

RA

The torque around point A is T = R x F

Example: torque’s in balance

2r 4f

2mm