Physics I
Class 16
Conservation of
Angular Momentum
16-1
Important Announcements
Exam 2 will be 7 PM on October 29 in DCC 308 and 318.
Conflict exams at 8, 9, and 10 PM on October 29 and
8 AM on October 30 in SC 2C06.
If you get extra time, you need to come to a conflict exam.
Review session on October 27 at 7 PM in DCC 330.
Review activity for October 28/29 classes:
Similar format, extended time, on WebAssign in near future.
All sections will have the same time frame to do this assignment.
The due date/time for everyone will be October 29 at 7 PM.
 Try to do as many of the problems as possible before class.
 Work with your team in class on problems you couldn’t do.
 Attendance is mandatory if you want activity credit.
16-2
Angular Momentum of a Particle
Review
center of rotation (defined)

r


p  mv
Angular momentum of a particle
once a center is defined:
  
l  r p
(What is the direction of angular
momentum here?)
Once we define a center (or axis) of rotation, any object with a
linear momentum that does not move directly through that point
has an angular momentum defined relative to the chosen center.
16-3
Angular Momentum of a Particle
Angular Momentum of an Object
For a solid object, each atom has its own angular momentum:
  

l i  ri  p i  ri  ( m i v i )
The direction is the same as the direction of angular velocity.
The magnitude is

 
 
2
| l i |  | ri | | p i | sin( )  m i | ri | | v i |  m i ri  ri   m i ri
so
 
2
l i   m i ri
The total angular momentum, summing all atoms, is

 

2
L   l i   m i ri  I 
16-4
How Does Angular Momentum of
a Particle Change with Time?
Take the time
 derivative of angular momentum:


dl
d   d r   dp
 ( r  p) 
p  r 
dt dt
dt
dt
Find each term separately:
so

dr   
 p  v  p  0 (Why?)
dt

 dp  

r
 r  Fnet  net (Why?)
dt

dl 
 net (Newton’s 2nd Law for angular momentum.)
dt
16-5
Angular Momentum of a Particle:
Does It Change if  = 0?
Y
(0,0)

r

r
X
(red)
(blue)
(0,–3)
(4,–3)


p  m v = 1 kg m/s (+X dir.)
The figure at the left shows the same
particle at two different times. No forces
(or torques) act on the particle.
Is its angular momentum constant?
(Check magnitudes at the two times.)
Blue angle:  = 90º
l = r p sin() = (3) (1) sin(90º) = 3 kg m2/s
Red angle:  = arctan(3/4) = 36.87º
l = r p sin() = (5) (1) sin(36.87º) = 3 kg m2/s


[r sin()] is the component of r at a right angle to p . It is
constant.
It is also the distance at closest approach to the center.
16-6
Conservation of
Angular Momentum
Take (for example)
two rotating objects that interact.

dl1 

 on 1 from 2  ext on 1
dt
dl2 

 on 2 from 1  ext on 2
dt
The total angular
momentum


 is the sum of 1 and 2:
dL d l 1 d l 2 



 ext on 1  ext on 2 (Why?)
dt
dt
dt
If there are no external torques, then
dL
0
dt
16-7
Example 1
An ice skater spins at 6 rad/sec with out-stretched hands.
Her rotational inertia is 50 kg m2. She then pulls her arms
in, thereby changing her rotational inertial to 40 kg m2.
What is her angular speed now?
No external torque, so L remains constant
I before before  L  I after after
after
I before before 50  6


 7.5 rad/sec
I after
40
16-8
Example 2
A wheel is rotating freely with an angular speed of 30 rad/sec on
a shaft whose rotational inertia is negligible. A second wheel,
initially at rest and with twice the rotational inertia of the first is
suddenly coupled to the same shaft. What is the angular speed of
the resultant combination of the shaft and two wheels?
No external torque, so L remains constant
I1 before  L  I1 after  I 2 after
after 
I1 before I1 before 1 30


 10 rad/sec
I1  I 2
I1  2 I1
3
16-9
Class #16
Take-Away Concepts
  
1. Angular momentum of a particle (review): l  r  p .
2.
Newton’s 2nd Law for angular momentum:
3.
Conservation of angular momentum (no ext. torque):

dl 
 net
dt

dL
0
dt
16-10
Parallel Axis Theorem
(for Activity, not on Exam!)
This helps if we need to calculate the moment of inertia
when an object rotates off center. d = distance off center.
16-11
Activity #16 - Conservation of
Angular Momentum
Objective of the Activity:
1.
2.
3.
Think about conservation of angular momentum.
Use conservation of momentum to predict the
change in rotational speed in a simple system.
Compare measurements with predictions.
16-12