Physics I
Class 14
Introduction to
Rotational Motion
14-1
Definitions
Angular Position:
Angular Displacement:

 (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?
14-2
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.
14-3
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.
14-4
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 
14-5
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
14-6
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 the moment of inertia.
14-7
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 moment of inertia for point objects, but we will
give you a formula for a solid object or just give you its moment of inertia.
I for a solid sphere:
I for a spherical shell:
I  52 M R 2
I  23 M R 2
14-8
Characteristics of the
Moment of Inertia
The moment of inertia of an object depends on
 Its mass.
 Its shape.
 The axis of rotation.
 NOT the angular velocity or acceleration.
The moment of 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
moment of inertia is the sum of each individual moment of inertia.
I   Ii
14-9
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.
14-10
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.
14-11
Class #14
Take-Away Concepts
1.
2.
Definitions of rotational quantities: , , .
Centripetal and tangential acceleration.
 mi ri1 2
4. Rotational kinetic energy: K  2 I 


5. Introduction to torque:   I 
3.
Moment of inertia: I 
6.
Correspondence
x
mI
v
F
2
a
14-12
Activity #14
Introduction to Rotation
Objective of the Activity:
1.
2.
Think about rotation concepts.
Try changing the moment of inertia
 of a simple
object and see how that affects   I  .
14-13