Rotational Equilibrium and
Dynamics
Rotational Inertia and
Angular Momentum
Video

http://www.youtube.com/watch?v=tcs93mPn
91E&feature=related
Inertia vs. Rotational Inertia

Inertia is the resistance of an object to changes in its
state of motion.
–

Objects in motion stay in motion and objects at rest stay at
rest unless acted upon by unbalanced forces.
Rotational Inertia is the resistance of an object to
changes in its rotational state of motion.
–
Rotating objects tend to keep rotation and non-rotating
objects tend to stay non-rotating unless acted upon by
unbalanced torques.
(Note: Rotational inertia is often called moment of inertia.)
Inertia vs. Rotational Inertia

Recall that inertia depends on the mass of the
object.
–
i.e. massive objects have a lot of inertia, whereas objects
with small mass have little inertia.
Like inertia in the linear sense, rotational inertia
depends on the mass of the object, however,
rotational inertia is unique in that it also depends
on the distribution of the mass.
(By distribution of the mass, we mean where the mass
is concentrated in the object.)

Rotational Inertia
Rotational inertia depends on:
 Total mass of the object
 Distribution of the mass
The farther the mass is from the axis of rotation, the larger the
rotational inertia or the harder it is to rotate.
Rotational inertia (I) is proportional to (mass) x (distance)2
Iamr2
(What are the units of rotational Inertia?)
Rotational Inertia

Rotational Inertia is different for different objects. Some of the
common rotational inertias are shown here. (see p. 285 for a
more complete list)
Demo: Inertia Sticks
Demo: Hammer Balance
When is the hammer easier to
balance on your finger?
A
B
Demo: Drop the Stick
Two meter sticks stand upright
against a wall; one has a
mass on the end.
Which stick will swing down and
hit the floor first?
Why?
Demo: Hoop & Disk Racing
Roll a hoop and a disk down a ramp;
which wins the race?
Check Yourself:

Why do tightrope
walkers carry a
long pole when
walking across
the high wire?
Angular Momentum

As we have just discovered, rotating objects have inertia. Because of
this, rotating objects also possess momentum associated with its
rotation.

There are two types of momentum:
–
Linear momentum: momentum associated with an objects linear motion


–
p=mv
(linear momentum)=(mass)x(velocity)
Angular momentum: momentum associated with an objects rotational
motion


L=Iw
(angular momentum)=(moment of inertia)x(angular speed)
(Notice the form of the equations is the same)
Angular Momentum




L=Iw
w is the Greek letter omega and represents the
angular speed of the rotating object.
Angular speed is the ratio of the angular
displacement (in radians) to the time.
w=Dq/Dt where the angular displacement is
measured in radians.
(what are the units of angular speed?)
(what are the units of angular momentum?)
Calculating Angular Speed

A young boy swings a yo-yo around his head
such that it makes three revolutions every
second. What is the angular speed of the
yo-yo?
Angular Speed v. Linear Speed

Angular speed is related to linear speed by
the following equation:
–
vt=rw
-(tangential speed)=(distance from axis)x(angular speed)
Angular Speed v. Linear Speed

Tangential speed varies with the distance
from the axis, angular speed does not.
Conservation of Angular Momentum:


Recall that linear momentum is conserved if
the net external force acting on an object or
system of objects is zero.
Similarly, angular momentum is conserved
if the net external torque acting on an object
or system of objects is zero.
Demo: Conservation of Angular
Momentum
Demo: Bicycle gyro
Zero
Rotation
CounterClockwise
Rotation
Clockwise
Rotation
Similar to
recoil
Sample Problem

A 65 kg student is spinning on a merry-goround that has a mass of 5.25x102 kg and a
radius of 2.00 m. She walks from the edge of
the merry-go-round toward the center. If the
angular speed of the merry-go-round is
initially 0.20 rad/s, what is its angular speed
when the student reaches a point 0.50 m
from the center?
Think about it:

Racing identical pop cans. What happens?