PES 1110 Fall 2013, Spendier
Lecture 33/Page 1
Today:
- Conservation of Angular Momentum (11.11)
- Quiz 5, next Friday Nov 22nd (covers lectures 29-33,HW 8)
  
 
angular moment for point particles: L  r  p  m  r  v 


angular moment for a rigid object: L  I  [kg m2/s] (dropped radians)
Conservation of Angular Momentum
For linear momentum, we proved that if there are no external forces acting on a system,
then linear momentum is conserved. This was useful for solving problems.
For angular momentum we have a similar situation. In the absence of external torques,
the total angular momentum of a system cannot change.


 
d  
d n 
t

t

L

L

L

...

L


 Li
net
1
2
3
n
dt
dt i1
If there are no external forces/torques (only internal forces) then the left hand side is zero
d n 
0   Li
dt i1
Hence
n 
 Li  constant


i 1
In the absence of external torques, the total angular momentum is conserved!
Conservation
of Angular Momentum:

L  I




I Ai Ai  I BiBi  I Af  Af  I Bf Bf
This statement describes the conservation of angular momentum. It is the third of the
major conservation laws encountered in mechanics (along with the conservation of
energy and of linear momentum).
There is one major difference between the conservation of linear momentum and
conservation of angular momentum. In a system of particles, the total mass cannot
change. However, the total moment of inertia can. If a set of particles decreases its radius
of rotation, it also decreases its moment of inertia. Though angular momentum will be
conserved under such circumstances, the angular velocity of the system might not be. We
shall explore these concepts through some examples.
Single Object Conservation
Conservation of angular momentum can occur in a single object! A change in shape
causes a change in the moment of inertia.
I ii  I f  f
Ice skaters, gymnasts and divers use this principle all the time
PES 1110 Fall 2013, Spendier
Lecture 33/Page 2
Demo: masses and rotating stage
1) Consider a spinning skater. A popular skating move involves beginning a spin with
one's arms extended, then moving the arms closer to the body. This motion results in an
increase of the speed with which the skater rotates increases. When the skater's arms are
extended, the moment of inertia of the skater is greater than when the arms are close to
the body, since some of the skater's mass decreases the radius of rotation. Because we can
consider the skater an isolated system, with no net external torque acting, when the
moment of inertia of the skater decreases, the angular velocity increases, according to the
equation
I ii  I f  f   f 
Ii
i
If
Example 1:
Your acrobatic physics professor wants to be a ballerina and stands at the center of a
turntable holding her arms extended horizontally with a 5 kg dumbbell in each hand. She
is set rotating by a student about a vertical axis making one revolution in 2.0 s. Find the
prof’s new angular velocity if she pulls the dumbbells in to her stomach.
a) Her moment of inertia (without the dumbbells) is 3.0 kg m2 when her arms are
outstretched, dropping to 2.2 kg m2 when her hands are at her stomach. The dumbbells
are 1.0 m from the axis initially and 0.2 m from it at the end. Treat the dumbbells as
particles. Neglect friction in turntable.
b) How does the kinetic energy change in this process?
PES 1110 Fall 2013, Spendier
Lecture 33/Page 3
2) A person is holding a spinning bicycle wheel on a rotating chair. The person then turns
over the bicycle wheel, causing it to rotate in an opposite direction, as shown below.
Initially, the wheel has an angular momentum in the upward direction. When the person
turns over the wheel, the angular momentum of the wheel reverses direction. Because the
person-wheel-chair system is an isolated system, total angular momentum must be
conserved, and the person begins to rotate in an opposite direction as the wheel. The
vector sum of angular momentum in a) and b) is the same, and momentum is conserved.
This example is quite counterintuitive. It seems odd that simply moving a bicycle wheel
would cause one to rotate. However, when observed from the standpoint of conservation
of momentum, the phenomenon makes sense.
PES 1110 Fall 2013, Spendier
Lecture 33/Page 4
Example 2:
Your prof is standing on a turntable that can rotate freely about a vertical axis. Your prof
is initially at rest, is holding a bicycle wheel whose rotational inertia Iwh about its central
axis is 1.2 kg m2. The wheel is rotating at an angular speed ωwh of 3.9 rev/s; as seen from
overhead, the rotation is counterclockwise. Then your prof inverts the wheel. The
inversion results in your prof, the stool, and the wheel’s center rotating together as a
composite rigid body about the stool’s rotation axis, with rotational inertia Ib = 6.8 kg m2.
With what angular speed ωb and in what direction does the composite body rotate after
the inversion of the wheel?
PES 1110 Fall 2013, Spendier
Lecture 33/Page 5
A collision between two bodies:
Example 3:
Two disks one (A), an engine flywheel with mass of 2.0 kg and radius of 0.20 m, and the
other (B), a clutch plate with mass 4.0 kg and radius of 0.10 m, attached to a transmission
shaft. Their moments of inertia are IA and IB. Initially, they are rotating with constant
angular speed ωAi = 50 rad/s and ωBi=200 rad/s respectively.
a) Find the common final angular speed ωf after the disks are pushed into contact.
b) What happens to the kinetic energy during this process?


Forces F and F are along the axis of rotation, and thus exert no torque about this axis
on either disk.
PES 1110 Fall 2013, Spendier
Lecture 33/Page 6
Example 4:
The moon is orbiting the Earth (Mmoon = 7.36 x 1022 kg, R =
3.84 x 108 m) when it is hit by an asteroid that is traveling
with velocity 620 m/s, antiparallel to the linear velocity of the
moon. (A head-on collision). The angular velocity of the
moon about the Earth is 2.66 x 10 -6 rad/s (downwards in the
diagram).
a) What is the angular momentum (magnitude and direction)
of the moon before the collision?
b) Calculate the angular momentum (magnitude and
direction) of the asteroid just before it hits the moon. Assume ma = 9.32 x 1021 kg.
c) What is the total angular momentum (magnitude and direction) before the collision?
d) If the moon keeps its radius, what is its new angular velocity about the Earth?