Rotation of Rigid Bodies

Ms. Mikaela Irene Fudolig

Physics 71 Lecture

Rigid Bodies and Rotation

Rigid bodiesbodies with a definite size and shapedo not deform

Rotationfor now, we consider rotation about a fixed axis

Example 1

Identify the axis of rotation for each of the following rotating objects:

Describing Rotational Motion

Use polar coordinates, (r,)!

s r is in

RADIANS!

Signs of :

+ if rotation is CCW

– if rotation is CLOCKWISE

Exercise 1

A girl is on a merry-go-round, 5 meters from the center of the circular platform. What is her angular displacement if she travels a total distance of 2 meters after some time t?

Angular velocity,

Analogous to the linear velocity, vRate of change in angular displacement

Angular velocity,

Average angular velocityInstantaneous angular velocityAngular speed

magnitude of the angular velocity

Average Angular Velocity

The average angular velocity over a time interval t = tf - ti is

ˆ av kt

*assumes

that the rotation axis is the z-axis.*

(Instantaneous) Angular velocity

rate of change of the angular displacement

VECTOR!

ˆ d

kdt

*assumes

that the rotation axis is the z-axis.*

Angular velocity,

Direction along the axis:UPWARDS (+), if

rotation is CCWDOWNWARDS (-), if

rotation is CLOCKWISE

Angular velocity,

CAUTION:

“Clockwise” and “CCW” are NOT directions for the ANGULAR VELOCITY!!!

Angular speed,

magnitude of the angular velocity

d

dt

Angular acceleration,

Analogous to the linear acceleration, aHow fast the angular velocity changes

2

2

ˆ

ˆ

dk

dt

dk

dt

*assumes that the rotation axis is the z-axis.*

Angular acceleration,

Direction along the axis:SAME DIRECTION

AS , if rotation SPEEDS UP

OPPOSITE DIRECTION AS , if rotation SLOWS DOWN

Exercise 2

You put a DVD in the DVD drive of your laptop, and it starts rotating clockwise.Draw a diagram of the rotating DVD, indicating

direction of rotation.Draw the direction of the angular velocity

vector.Draw the direction of the angular acceleration

vector.

Linear vs. Angular Quantities

Linear Angular

x,s

v

a

Motion with Constant Angular Acceleration

2 20

0

2

2t

Analogous to motion with constant linear acceleration (aka Chapter 2)

0

20

1

2

t

t t

Exercise 3

Y&F ex.9.3. A DVD is slowing to a stop. At t=0, its angular velocity is 25.7rad/s and its angular acceleration is a constant –10.0rad/s2.What is the disc’s angular velocity at

t=0.300s?What angle does the line PQ make with the

+x-axis at t=0.300s (I.e., what is (t=0.300s)?

Linear Kinematics of a rotating particle

A particle in a rotating body moves in a circle:

Linear/tangential velocity of a rotating particle

From s=r, we can get the expression for the linear speed of a rotating particle a constant distance r from the axis of rotation:

v rThe direction of the linear velocity vector is TANGENT TO ITS CIRCULAR PATH, parallel to the infinitesimal displacements!

Acceleration of a rotating particle

Tangentialchange in linear speed

Radialchange in direction of the linear velocity

Tangential acceleration

The tangential acceleration of a rotating particle a distance r from the axis of rotation is:

tan

da r r

dt

Direction: parallel or antiparallel to the linear velocity

Radial acceleration

The radial acceleration of a rotating particle a distance r from the axis of rotation is:

22

rad

va r

r

Direction: pointing towards the axis of rotation

Exercise 4Y&F 9.26. An electric ceiling fan with

blades 0.750m in diameter is rotating with constant angular acceleration 0.900rev/s2 about a fixed axis with an initial angular velocity of 0.250rev/s. After 2.00s:what is the angular velocity of the fan?what is the linear speed of the tip of a fan

blade?

Exercise 4

Y&F 9.26. A electric ceiling fan with blades 0.750m in diameter is rotating with constant angular acceleration 0.900rev/s2 about a fixed axis with an initial angular velocity of 0.250rev/s. After 2.00s:what is the radial acceleration?what is the tangential acceleration?