Physics 170 - MechanicsLecture 22

Rotational Kinematics

1

A Particle inUniform Circular Motion

For a particle in uniform circular motion, the velocity vector v remains constant in magnitude, but it continuously changes its direction. However, at each position it is tangent to the circular path. For this reason, it is called the tangential velocity of the particle.

2

Angular Position θ

3

Angular Position θ

Degrees and revolutions:

4

Angular Position θ

Arc length s, measured in radians:

5

Angular Velocity ω

6

Rotational Period T

Example: Find the period of a music phonograph record that is rotating at 45 RPM.

7

Angular Acceleration α

8

Example: Decelerating Windmill

As the wind dies, a windmill that had been rotating at ω = 2.1 rad/s begins to slow down at a constant angular acceleration of α = −0.45 rad/s2.

How long does it take for the windmill to come to a complete stop?

9

Summary of Angular Variables

10

Rotational vs. Linear KinematicsAnalogies between linear and rotational kinematics:

11

Example: Thrown for a Curve To throw a curve ball, a pitchergives the ball an initial angularspeed of 36.0 rad/s. When thecatcher gloves the ball 0.595 slater, its angular speed hasdecreased (due to air resistance)to 34.2 rad/s.(a) What is the ball’s angular acceleration, assuming it to be constant?(b) How many revolutions does the ball make before being caught?

12

Connections BetweenLinear & Rotational Quantities

The tangential velocity vt is zero at the center of rotation and increases linearly with r.

13

Connections BetweenLinear & Rotational Quantities

Question: Two children ride a merry-go-round, with Child 1 at a greater distance from the axis of rotation than is Child 2.

How do the angular speeds ω1,2 of the two children compare?

(a) ω1>ω2 (b) ω1=ω2 (c) ω1<ω2

14

Connections BetweenLinear & Rotational Quantities

Question: Two children ride a merry-go-round, with Child 1 at a greater distance from the axis of rotation than is Child 2.

How do the angular speeds ω1,2 of the two children compare?

(a) ω1>ω2 (b) ω1=ω2 (c) ω1<ω2

15

Connections BetweenLinear & Rotational Quantities

16

Connections BetweenLinear & Rotational Quantities

This merry-go-round has both tangential and centripetal acceleration.

Speeding up

17

Example: Time to Rest A pulley rotating in the counterclockwise direction is attached to a mass suspended from a string. The mass causes the pulley’s angular velocity to decrease with a constant angular acceleration α = −2.10 rad/s2.

(a) If the pulley’s initial angular velocity is ω0 = 5.40 rad/s, how long does it take for the pulley to come to rest?

(b) Through what angle does the pulley turn during this time?

18

Example: A Rotating Crankshaft

A car’s tachometer indicates the angular velocityω of the engine’s crankshaft in rpm. A car stopped ata traffic light has its engine idling at 500 rpm. When thelight turns green, the crankshaft’s angular velocity speeds upat a constant rate to 2,500 rpm in a time interval of 3.0 s.

How many revolutions does the crankshaft make in this time interval?

19

5.0 s41.9

41.9

Example: CD Speed

Find the angular speed ω that a CD must have in order to give it a linear speed vt = 1.25 m/s when the laser beam shines on the disk(a) at 2.50 cm from its center, and(b) at 6.00 cm from its center.

Unlike old phonograph records that turned with a constant angular speed (like 33 1/3 rpm), CDs and DVDs turn with a variable ω that keeps the tangential speed vt constant.

20