Physics 1D03 - Lecture 10 1

Rotation of a Rigid Body (Chapter 10)

Each particle travels in a circle. The speeds of the particles differ, but each one completes a full revolution in the same time.

We describe the rotational motion using angle, angular velocity, and angular acceleration:

Physics 1D03 - Lecture 10 2

Units: by convention, angles are measured in radians.

or rsrs r

r

s

arc length

2 rad = 360o

Angular velocity has units of rad/s or s-1

Angular acceleration has units of rad/s2 or s-2

(The radian is a ratio of two lengths, and not really a unit. Some equations will require angles to be in radians.)

Physics 1D03 - Lecture 10 3

tdtd

t

tdtd

t

t

angle (“theta”):

angular velocity (“omega”):

angular acceleration (“alpha”):

0 reference axis

(radians)

(rad/s)

(rad/s2 )

Physics 1D03 - Lecture 10 4

Linear and angular quantities

rdtd

rrdtd

dtds

vt )(

r P

0

circular path of point P

v

s

Distance:

s = r

Tangential Velocity:

A particle P travels in a circle of radius r. The velocity is tangential to the circle and perpendicular to the radius.

Tangential Acceleration:

rdt

dr

dt

drr

dt

d

dt

dva t

t 2

2

)(

Physics 1D03 - Lecture 10 5

Quiz

The Earth rotates on its axis. How does its angular velocity vary with location?

a) is larger at the equator, and smaller near the polesb) is smaller at the equator, and larger near the polesc) is the same at the equator and near the poles

Physics 1D03 - Lecture 10 6

22

rrv

ar

The tangential component at is equal to the rate of increase of speed. There is also a radial (centripetal) component, due to the change in direction of v:

rs rvt

rat

ar

at

Pa

These relations require angular quantities to be measured in radians (or rad/s, etc.).

In simpler notation:

Physics 1D03 - Lecture 10 7

Quiz

Several pennies are placed on a turntable. As the angular velocity of the turntable is slowly increased, which penny slides first?

Physics 1D03 - Lecture 10 8

Quiz

High-speed CD-ROM drives sometimes specify that they use a “constant linear velocity” method of recording.What does this mean for the rotation rate in revolutions per minute as the write head moves from the inner tracks to the outer tracks?

Physics 1D03 - Lecture 10 9

iif t

222

22

1

if

iif tt All for constant only !

These expressions are should remind you of relations for constant linear acceleration: replaces x, replaces v, replaces a.

Constant angular acceleration:

advv

attvxx

atvv

if

ii

i

2

21

22

2

All for constant a only !

Physics 1D03 - Lecture 10 10

dtddtd

, Since

tdtt

if 0

(Extra-derivation) Constant angular acceleration: In this special case, there are relations among , , and that are analagous to the relations among x, v, and a for linear motion with constant linear acceleration.

if t (constant ; “initial” at t = 0)or

; , Since dtddtd

t

i

t

if dtt dt0

0

)(

iif tt 22

1or

Physics 1D03 - Lecture 10 11

Example: A computer disc has a linear velocity of 1.3m/s. What is the angular velocity when at the innermost track where r=23mm.

Physics 1D03 - Lecture 10 12

Example: A computer disc starts from rest and reaches a final rotation rate of 7200 rev/min after 10 seconds. Assuming constant angular acceleration, through how many revolutions does it turn?

Physics 1D03 - Lecture 10 13

Summary

• Rotational motion can be described by angle, angular velocity, and angular acceleration

• For constant angular acceleration (a special case!), kinematical relations are similar to those for linear motion with constant linear acceleration

Physics 1D03 - Lecture 10 14

Practice problems: Chapter 10

Problems 1, 5, 11

(5th ed) Problems 5, 7, 11