Rotational Kinematics

Circular Motion

Position: constant

Speed: constant

r

v

A Particle inUniform Circular Motion

1 circumference 2

1 period

rv

T

period the time required for one complete rotation.T

For a particle in uniform circular motion, the velocity vector v remains constant in magnitude, but it continuously changes its direction.

Angular Position Angular Position

Degrees and revolutions:

Angular Position Angular Position

Arc length s, measured in radians:

s

r

(radians) (degrees) /180 (rev) 2

Angular Velocity Angular Velocity

Sign of

Connections BetweenLinear & Rotational Quantities

Connections BetweenLinear & Rotational Quantities

Angular Acceleration Angular Acceleration

Comparison to 1-D Kinematics

Angular Linear

constant

0 t

constanta

And for a point at a distance R from the rotation axis:

x = Rv = Ra = R

2 20 02v v a x x 2 2

0 02 ( )

0v v at

20 0

1

2t t

20 0

1

2x x v t at

By convention, are positive if they are in the counterclockwise direction.

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

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

av t

2

(0) (2.1 rad/s)4.7 s

( 0.45 rad/s )f i

av

t

Angular Velocity & Acceleration ACT

The fan blade shown is slowing down. Which option describes and ?

(a) >0 and >0;

1t

(b) >0 and <0; (c) <0 and >0; (d) <0 and <0.

Rotational KinematicsRotational Kinematics

If the angular acceleration is constant:

Thrown for a CurveTo throw a curve ball, a pitchergives the ball an initial angularspeed of 157.0 rad/s. When thecatcher gloves the ball 0.795 slater, its angular speed hasdecreased (due to air resistance)to 154.7 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?

0 t

20 (157.0 rad/s) (154.7 rad/s)3.03 rad/s

(0.795 s)t

1 20 2

1 2 22

(157.0 rad/s)(0.795 s) ( 3.03 rad/s )(0.795 s)

123.9 rad 19.7 rev

t t

Wheel of MisfortuneOn a certain game show, contestants spin thewheel when it is their turn. One contestant givesthe wheel an initial angular speed of 3.40 rad/s.It then rotates through 1.25 revolutions andcomes to rest on BANKRUPT.

(a) Find the wheel’s angular acceleration,assuming it to be constant.

(b) How long does it take for the wheel tocome to rest?

2 20 2

2 2 220 0 (3.40 rad/s)

0.736 rad/s2 2(2 rad/rev)(1.25 rev)

0 t

02

0 (3.40 rad/s)4.62 s

( 0.736 rad/s )t

A Rotating Crankshaft A car’s tachometer indicates the angular velocity of the crank shaft in rpm. A car stopped at a traffic light has its engine idling at 500 rpm. When the light turns green, the crankshaft’s angular velocity speeds up at a constant rate to 2500 rpm in a time interval of 3.0 s.

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

500 rpm (2 rad/rev)/(60 s/min)=52.4 rad/si 2500 rpm 5 262.0 rad/sf i

2(262.0 rad/s 52.4 rad/s)69.9 rad/s

(3.0 s)f i

t

21

2

21 22

10 (52.4 rad/s)(3.0 s) (69.9 rad/s ) 3.0 s 472 rad 75 rev

2

f i i t t

rev

rad

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 = 2.10 rad/s2.

(a) If the pulley’s initial angular velocity is 0 = 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?

0 t

20( ) / 0 (5.40 rad/s) / ( 2.10 rad/s ) 2.57 st

1 20 2

1 2 22

(5.40 rad/s)(2.57 s) ( 2.10 rad/s )(2.57 s) 6.94 rad

t t

(c) If the radius of the pulley is 5.0 cm, through what distance is the mass lifted?

6.94 5.0 34.7s r rad cm cm

CD SpeedCDs and DVDs turn with a variable that keeps the tangential speed vt constant.

Find the angular speed and the frequency 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.

tv

r

(1.25 m/s) 50.0 12.50 cm: 7.96 rps

(0.0250 m) 2

rad revr

s rad

(1.25 m/s) 20.8 rad 1 6.00 cm: 3.31 rps

(0.0600 m) 2

revr

s rad

Rotational vs. Linear KinematicsRotational vs. Linear KinematicsAnalogies between linear and rotational kinematics:

Connections BetweenLinear & Rotational Quantities

Connections BetweenLinear & Rotational Quantities

More Connections BetweenLinear & Rotational QuantitiesMore Connections Between

Linear & Rotational QuantitiesThis merry-go-round has both tangential and centripetal acceleration.

Speeding up

ta r 2 2 /cp ta r v r

2 2t cpa a a

1tan /cp ta a

The MicrohematocritSuppose the centrifuge is just starting up, and that it has an angular speed of 8.00 rad/s and an angular acceleration of 95.0 rad/s2.(a) What is the magnitude of the centripetal, tangential, and total acceleration of the bottom of a tube?(b) What angle does the total acceleration make with the direction of motion?

22

228.00 9.07 580.5c

v rad cma r cmsr s

2295 9.07 861.7T

rad cma r cmss

2 22 2 2 2 2580.5 861.7 1039c Ta a a cm s cm s cm s

2

2

1580.5

tan 34 0.593861.7

cmoc s

cmT s

arad

a