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
1
Units: by convention, angles are measured in radians.
arc length
s
s
  or s  r
r
2p rad =
r

r
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
2
angle (“theta”):  t  (radians)
angular velocity (“omega”):  t   d  t  (rad/s)
dt
d
angular acceleration (“alpha”):  t    t  (rad/s2 )
dt
0

reference axis
Physics 1D03 - Lecture 10
3
Linear and angular quantities
A particle P travels in a circle of radius r. The velocity is tangential to
the circle and perpendicular to the radius.
v
Distance:
P
s = r
r
Tangential Velocity:
ds d
d
vt   (r )  r
 r
dt dt
dt
Tangential Acceleration:
s

0
circular path
of point P
dvt d
d
d 2
at 
 (r )  r
 r 2  r
dt dt
dt
dt
Physics 1D03 - Lecture 10
4
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 poles
b)  is smaller at the equator, and larger near the poles
c)  is the same at the equator and near the poles
Physics 1D03 - Lecture 10
5
In simpler notation:
s  r
vt  r
at  r
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:
at
v2
ar   r 2
r
P
a
ar
These relations require angular
quantities to be measured in
radians (or rad/s, etc.).
Physics 1D03 - Lecture 10
6
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
7
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
8
Constant angular acceleration:
 f  i  ti
 f  i  i t  1 2 t 2
All for constant  only !
 2f  i2  2
These expressions are should remind you of relations for constant
linear acceleration:  replaces x,  replaces v,  replaces a.
v  vi  at
x  xi  vi t  1 at 2
2
v 2f  vi2  2ad
All for constant a only !
Physics 1D03 - Lecture 10
9
(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.
d
Since  
, d   dt
dt
t
 f   i    dt  t
0
 f  t  i
or
Since  
(constant  ; “initial” at t = 0)
d
, d   dt;
dt
t
t
 f   i    dt   (t  i ) dt
0
0
or
 f  12 t 2  it  i
Physics 1D03 - Lecture 10
10
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
11
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
12
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
13
Practice problems: Chapter 10
Problems 1, 5, 11
(5th ed) Problems 5, 7, 11
Physics 1D03 - Lecture 10
14