Rotational Motion

Lecturer: Professor Stephen T. Thornton

Reading Quiz

BonnieBonnieKlydeKlyde

Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds.

Klyde’s angular velocity is:

A) same as Bonnie’sB) twice Bonnie’sC) half of Bonnie’sD) 1/4 of Bonnie’sE) four times

Bonnie’s

Reading Quiz

BonnieBonnieKlydeKlyde

Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one complete revolution every two seconds.

Klyde’s angular velocity is:

A) same as Bonnie’sB) twice Bonnie’sC) half of Bonnie’sD) 1/4 of Bonnie’sE) four times

Bonnie’s

The angular velocityangular velocity of any point on a solid object rotating about a fixed axis is is the samethe same. Both Bonnie & Klyde go around one revolution (2 radians) every two seconds.

Last Time

Collisions – elastic, inelastic, perfectly inelastic

Center of mass

Changing mass - rockets

TodayBegin angular motion

Angular position, displacementAngular speed, velocityAngular acceleration

Similarities between translation and rotation

Pep Talk

Don’t get behind.Halfway through course.Starting most difficult part of course.

Rotational motion.Six lecturesAngular momentum and torque are the most difficult concepts of this course.

Define angular position, velocity, and acceleration – just like we did for translational motion.

= angle

SI unit: radian (rad), dimensionless

One revolution = 3600 = 2π rad

Angular Position

0

Arc Length

Arc length

s r

Sign Conventions

1 rev = 3600 = 2 rad

We will mostly use radians.

1 rad = 57.30

A radian is the angle for which the arc length on a circle of radius r is equal to the radius of the circle.

0 counterclockwise rotation

<0 clockwise rotation

r

r r

Angular Displacement

angular displacement

f i

Angular displacement and velocity

Just like with velocity, we divide angular displacement by time to find angular velocity.

Average angular velocity, av

angular displacementf i

av t

SI unit: radian per second (rad/s) = s-1

0lim

t

d

t dt

Angular Speed and Velocity

0 counterclockwise rotation

<0 clockwise rotation

Vector characteristicsMagnitude of angular velocity is the angular speed. But angular velocity is a vector.

Use right hand rule to obtain direction of angular velocity.

Curl fingers in direction of rotation, and thumb gives direction of angular velocity! (go back and check). Vector direction is perpendicular to screen.

Conceptual Quiz:You look at a bicycle as it moves from your left to your right. The angular velocity of the rear wheel is directed  A)    upB)    to the leftC) to the rightD)    towards youE)    away from you

v

Answer: E – away from you

Use the right hand rule. The angular velocity is into the screen and away from you.

Angular acceleration

av

2 -2

0

2

instantaneous angular acceleration

average angular acceleration

SI unit: rad/s s

lim =

same SI unit s

t

t

d

t dt

Vector direction follows right hand rule.

Angular Acceleration

A Pulley with Constant Angular Acceleration

0

0

0

0

0

Note:

0

d

dt t t

tt

v

t

v at

Similarities between linear and angular motion quantities ***

x

v

a

0 0

0 0 0 0

2 20 0 0 0

2 2 2 20 0 0 0

1 1( ) ( )

2 21 1

2 2

2 ( ) 2 ( )

v v at t

x x v v t t

x x v t at t t

v v a x x

Angular Quantities

The frequency is the number of complete revolutions or cycles per second:

cycles/s

Frequencies are measured in hertz:

cycles/s

The period is the time one revolution takes:

2f

wp

=

11 Hz = 1 s-

1T

f=

Linear to rotational quantitiesT is period of one revolution.

2 radangular velocity around circle

2 2

tangential speed

t

t

Tr

v r rT T

v r

Tangential speed depends on radius.

Cooling Fan. A cooling fan is turned off when it is running at 850 rev/min. It turns 1350 revolutions before it comes to a stop. (a) What was the fan’s angular acceleration, assumed constant? (b) How long did it take the fan to come to a complete stop?

An object at rest begins to rotate with a constant angular acceleration. If this object has angular velocity at time t, what was its angular velocity at the time 1/2 t?

A) 1/2 B) 1/4 C) 3/4 D) 2 E) 4

Conceptual QuizConceptual Quiz

An object at rest begins to rotate with a constant angular acceleration. If this object has angular velocity at time t, what was its angular velocity at the time 1/2t?

A) 1/2 B) 1/4 C) 3/4 D) 2 E) 4

The angular velocity is = t (starting from rest), and there is a linear dependence on time. Therefore, in half the timehalf the time, the object has accelerated up to only half the speedhalf the speed.

Conceptual QuizConceptual Quiz

An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle in the time t, through what angle did it rotate in the time 1/2 t?

A) B) C) 3D) 2 E) 4

Conceptual QuizConceptual Quiz

An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle in the time t, through what angle did it rotate in the time 1/2 t?

A) B) C) 3D) 2 E) 4

The angular displacement is = 1/2 t 2 (starting from rest), and there is a quadratic dependence on time. Therefore, in half the timehalf the time, the object has rotated through one-quarter the angleone-quarter the angle.

Conceptual QuizConceptual Quiz

Centripetal and Tangential Acceleration

2 22

cp cp

Centripetal acceleration

( )

Tangential acceleration

t

t

tt t

v ra r a

r r

v r

v r

va r r a

t t

Not uniform circular motion

v

Conceptual QuizConceptual Quiz

BonnieBonnieKlydeKlyde

A) Klyde

B) Bonnie

C) both the same

D) linear velocity is zero for both of them

Bonnie sits on the outer rim of amerry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity?

Their linear speedslinear speeds vv will be

different because v = Rv = R and

Bonnie is located farther outBonnie is located farther out

(larger radius R) than Klyde.

BonnieBonnie

KlydeKlyde

BonnieKlyde V21

V

Conceptual QuizConceptual Quiz

Follow-up:Follow-up: Who has the larger centripetal acceleration? Who has the larger centripetal acceleration?

A) Klyde

B) Bonnie

C) both the same

D) linear velocity is zero for both of them

Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity?

Do falling rigid body demo.