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```									Angular Momentum of a rigid object rotating about a fixed axis

L  I

dL d  I 

dt   dt
But for any rigid object the rotational
inertia is a constant

dL    d
I
dt    dt
dL
 I
dt
       
Newton’s            dL                                 dp
Second Law                    Analogous to   Fnet 
dt                                 dt
What if the system is isolated and closed?

Isolated – no external torques            Closed – no change in the mass

dL
 
dt
dL
0
dt
 L  constant

Law of Conservation of Angular Momentum

In any closed, isolated system, the
angular momentum is constant
Conservation of Angular Momentum Examples

1. The spinning volunteer.

Li  L f
Iii  I f  f

Ii   i  I f
f
Conservation of Angular Momentum Examples

2. Stabilizing a Frisbee®.

The classic Frisbee® has a heavy outer ridge which increases its
rotational inertia and is then spun resulting in a large angular
momentum which resists changes to its motion.
Conservation of Angular Momentum Examples

3. Two disks are mounted on low friction bearings on the same axle and can be
brought together so that they couple and rotate as one unit. The first disk, with
mass 2.0 kg and radius 0.50 m, is set spinning at 450 rev     . The second disk,
min
with mass 4.0 kg and radius 0.50 m, is set spinning at 900. rev     in the same
min
direction as the first. They then couple together.

a. What is their angular speed after coupling?

Li  L f
l1i  l2i  L f
I11i  I 22i  I1  I 2  f
 1 mR 2    1 2m R 2    1 mR 2  1 2m R 2 
         1i              2i                       f
2           2               2         2          
1i  22i  3 f
Conservation of Angular Momentum Examples

3. Two disks are mounted on low friction bearings on the same axle and can be
brought together so that they couple and rotate as one unit. The first disk, with
mass 2.0 kg and radius 0.50 m, is set spinning at 450 rev     . The second disk,
min
with mass 4.0 kg and radius 0.50 m, is set spinning at 900. rev     in the same
min
direction as the first. They then couple together.

a. What is their angular speed after coupling?

1i  22i  3 f
1i  22i
f 
3

f 
450 rev
min

 2 900. rev
min

3
 f  750 rev min
Conservation of Angular Momentum Examples

3. Two disks are mounted on low friction bearings on the same axle and can be
brought together so that they couple and rotate as one unit. The first disk, with
mass 2.0 kg and radius 0.50 m, is set spinning at 450 rev       . The second disk,
min
with mass 4.0 kg and radius 0.50 m, is set spinning at 900. rev       in the same
min
direction as the first. They then couple together.
b. If instead the second disk is set spinning at 900.rev        in the opposite direction
min
of the first disk' s rotation, what is their angular speed after coupling?

f 
450 rev
min

 2  900. rev
min

3

 f  450 rev min
Conservation of Angular Momentum Examples

2. Two children, each with mass M, sit on opposite ends of a narrow board with length
L and mass M. The board is pivoted at its center and is free to rotate in a horizontal
circle without friction. (Treat the board as a thin rod.)
a. What is the rotational inertia of the board plus the children about a vertical axis
through the center of the board?
l
2                      I total  I board  2 I child
M                              M                                                2
I total
1       l
 Ml  2M  
2
l
12        2
1   1 2
I total     Ml  Ml
2
12   2
7
I total    Ml 2
12
Conservation of Angular Momentum Examples

2. Two children, each with mass M, sit on opposite ends of a narrow board with length
L and mass M. The board is pivoted at its center and is free to rotate in a horizontal
circle without friction. (Treat the board as a thin rod.)
b. What is the magnitude and direction of the angular momentum of the system if it
is rotating with angular speed ωo in a clockwise direction as seen from above?

L  I
M                              M
7
l                                  L  Ml 2o           Downward
12
Conservation of Angular Momentum Examples

While the system is rotating, the children pull themselves toward the center of the
board until they are half as far from the center as before.
c. What is the ratio of the new rotational inertia to the initial rotational inertia?
l
2
4                      7
I i  Ml 2
1       l
I f  Ml  2M  
2
12                    12        4
M                     M                                     5
I f  Ml 2
l                                               24
5
If         Ml 2
 24
Ii      7
Ml 2
12
If       5

Ii     14
Conservation of Angular Momentum Examples

While the system is rotating, the children pull themselves toward the center of the
board until they are half as far from the center as before.
d. What is the resulting angular speed in terms of ωo?
l
Li  L f
4
Iio  I f  f
M                    M
l                                          Ii
 f  o
If

14
 f  o
5
Conservation of Angular Momentum Examples

While the system is rotating, the children pull themselves toward the center of the
board until they are half as far from the center as before.
e. What is the change in kinetic energy of the system as a result of the children
changing their position? (From where does the added kinetic energy come?)
l
4           Ek  Ekf  Eki
1         1
M                    M    Ek  I f  f  I ii2
2
2         2
l                                                  2
1 5   2  14   1  7 Ml 2  2
Ek   Ml  o                  o
2  24    5  2  12        
*Note: L = constant                                   The added energy comes
21 2 2
Ek  Ml o
from the work done by the
Ek = increases
children when pulling
40                        themselves forward.

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