Physics I
Class 13
Introduction to
Rotational Motion
Rev. 25-Feb-04 GB
13-1
Definitions
Angular Position:
Angular Displacement:

 (normally
  in radians)
    0
Average or mean angular velocity
isdefined as follows:

   0 

avg 

t  t 0 t
Instantaneous angular velocity
 or just
 “ angular velocity”:
 d 

  lim

t 0 t
dt
Wait a minute! How can an angle have a vector direction?
13-2
Direction of Angular Displacement
and Angular Velocity
•Use your right hand.
•Curl your fingers in the direction
of the rotation.
•Out-stretched thumb points in
the direction of the angular
velocity.
13-3
Angular Acceleration
Average angular acceleration is defined as follows:

 avg
 

  0 


t  t0
t
Instantaneous angular acceleration or just “angular
acceleration”:



 d  d 2 

  lim

 2
t  0 t
dt dt
The easiest way to get the direction of the angular acceleration is
to determine the direction of the angular velocity and then…
 If the object is speeding up, angular velocity and acceleration
are in the same direction.
 If the object is slowing down, angular velocity and acceleration
are in opposite directions.
13-4
Equations for Constant 
1.
  0  t  t 0 
2.    0  0 ( t  t 0 )  2  ( t  t 0 )
1
1




(0  )( t  t 0 )
3.
0
2
4.    0  ( t  t 0 )  2  ( t  t 0 )
1
5.
2
x
v
a
2
2  02  2  0 
We recommend always using radians for angles.
13-5
Relationships Among
Linear and Angular Variables
MUST express angles in radians.
s  r
v  r
a tangential   r
a centripetal
v 2 2 r 2


 2 r
r
r
The radial direction is defined to be +
outward from the center.
a radial  a centripetal
13-6
Energy in Rotation
Consider the kinetic energy in a rotating object. The center of
mass of the object is not moving, but each particle (atom) in the
object is moving at the same angular velocity ().
K   12 m i v i   12 m i 2 ri  12 2  m i ri
2
2
2
The summation in the final expression occurs often when
analyzing rotational motion. It is called rotational inertia or
the moment of inertia.
13-7
Rotational Inertia
or Moment of Inertia
For a system of discrete “point” objects:
I   m i ri
2
For a solid object, use an integral where  is the density:
I    r 2 dx dy dz
We may ask you to calculate the rotational inertia for point objects, but we will
give you a formula for a solid object or just give you its rotational inertia.
R
I for a solid sphere:
I for a spherical shell:
I  52 M R 2
I  23 M R 2
13-8
Characteristics of
Rotational Inertia
The rotational inertia of an object depends on
 Its mass.
 Its shape.
 The axis of rotation.
 NOT the angular velocity or acceleration.
The rotational inertia is a measure of how difficult it is to get an
object to start rotating or to slow down once started.
For two or more objects rotating around a common axis, the total
rotational inertia is the sum of each individual rotational inertia.
I   Ii
13-9
Introduction to Torque
For linear motion, we have “F = m a”. For rotation, we have


  I
The symbol “” is torque. We will define it more precisely next
time.
Torque and angular acceleration are always
in the same direction in Physics 1 because
we consider rotations about a fixed axis.
13-10
Correspondence Between
Linear and Rotational Motion
x
v
a
mI
F
K  12 I 2


  I
You will solve many rotation problems
using exactly the same techniques you
learned for linear motion problems.
13-11
Class #13
Take-Away Concepts
1.
2.
Definitions of rotational quantities: , , .
Centripetal and tangential acceleration.
 mi ri 1 2
4. Rotational kinetic energy: K  2 I 


5. Introduction to torque:   I 
3.
Rotational inertia: I 
6.
Correspondence
x
mI
v
F
2
a
13-12
Class #13
Problems of the Day
___1. Lance Armstrong is riding his bicycle due east in the
Tour de France bicycle race. What is the direction of
the angular velocity of his bicycle wheels?
A)
B)
C)
D)
E)
F)
North.
South.
East.
West.
Up.
Down.
13-13
Answer to Problem 1 for Class #13
The answer is A, north.
The direction of the angular velocity vector is along the axis of
rotation. The axis of rotation of a wheel of a bicycle ridden in a
straight line (not turning) is horizontal and at a right angle to the
direction of motion. That means it must be either north or south.
Using the right hand to curl the fingers in the direction of rotation
of the wheel, the thumb points north.
13-14
Class #13
Problems of the Day
2. A carousel takes 8 seconds to make one revolution
when turning at full speed. It takes exactly one revolution
to reach its full speed from rest. What is the magnitude of
its angular acceleration while speeding up, assuming that
its angular acceleration is constant?
13-15
Answer to Problem 2 for Class #13
We can use units of rev for angles in this problem, converting later
to radians if we need to.
The initial angular speed of the carousel is zero. The final is one
rev divided by 8 seconds or 0.125 rev/s. The total rotation covered
is 1.0 rev. Equation 5 on slide 5 gives us :
 = (0.125 rev/s)2/(2 x 1 rev) = 7.8125 x 10–3 rev/s2.
13-16
Activity #13
Introduction to Rotation
Objective of the Activity:
1.
2.
Think about rotation concepts.
Try changing the rotational inertia
 of a simple
object and see how that affects   I  .
13-17