Angular Momentum
Definition:
  
Lrp
(for a particle)
L  rp sin 
Units:


 
L   Li   ri  pi

L
(for a system of particles)
depends on the choice of origin,
since it involves the position of vector
of the particle relative to the origin
[ L]  kg  m 2 / s  N  m  s  J  s
Angular momentum and second Newton's law



   
dL d  
dr   dp 
 r  p  
 pr
 v  mv   r  F  
dt dt
dt
dt



 dL
dp
 
compare with: F 
dt
dt
Conservation of angular momentum


FAB   FBA

FAB
r AB sin  ABr BAsin  BA
  
  rF sin 
  r F




dL A dLB 

 AB   BA 

  AB   BA  0
dt

If  ext

rAB

FBA

rBA r sin 
dt



dL
0 
 0  Li  L f
dt
When the external torque acting on a system is
zero, the total angular momentum is conserved
compare with
conservation of
linear momentum
Angular momentum for a rigid body rotation around a symmetry axis
z

L1

r1
L  Rp sin   Rp  Rmv  Rmr
L x1  L x 2  0

L2

r2
L z1  L z 2  L sin   Rmr sin   mr 2
L  L z   Liz   mi ri 2  I


R1   R 2
L  I


L  I
L z  I z
Kinetic energy:
K rot
2
L
 12 I 
2I
2
Conservation of angular momentum
for a rigid body rotation around a symmetry axis

If  ext  0 

 ext


dL dI 

   I
dt dt
I ii  I f  f
Noether’s Theorem
So far we have learned about three
conserved quantities
Energy
Momentum
Angular Momentum
Conserved quantities are precious in
theoretical physics because…
According to Noether’s Theorem,
for each conserved there exists a
symmetry of the laws of physics
which “generates” it.
Emmy Amalie Noether
1882-1935
Example: A person of mass 75 kg stands at the center of a rotating
marry-go-round platform of radius 2.0 m and moment of inertia 900 kgm2.
The platform rotates without friction with angular velocity 2.0 rad/s.
The person walks radially to the edge of the platform.
Calculate the angular velocity when the person reaches the edge.
m = 75 kg
r = 2.0 m
Ip = 900 kgm2
ω1 = 2.0 rad/s
ω2 - ?
L  I
I1 = Ip
I2 = Ip + mr2
Ip ω1 = (Ip + mr2 )ω2
L1 = L2
2 
I
Ip
p
 mr
2

1
900kg  m 2
900
2 
2.0rad / s 
2.0rad / s  1.5rad / s
2
2
1200
900kg  m  75kg  2.0m 


Example: A uniform stick of mass M and length D is pivoted at the center.
A bullet of mass m is shot through the stick at a point halfway between
the pivot and the end. The initial speed of the bullet is v1 and its final
speed is v2. What is the angular speed ω of the stick after the collision?
External forces: weight of the stick and force on the stick by the
pivoting axle produce no torque. Weight of the bullet is negligible.
No external torque → Angular momentum conserved
D
D
mv1  mv2  I z
4
4
D
Lz before  mv1
4
D
Lz after  mv2  I z
4
v1
D
z 
I
1
12
D/4
MD 2
mD v2  v1 
4I
3m v2  v1 

M
D
ω
v2
Example: What if instead of a stick we have a thicker block so the bullet
embeds itself in it?
Lz
Lz
before
after
D
 mv1
4
v 
D
4
z
2
2


D
D
  


D
 mv  I z  m   z  I z  m    I  z
4
4
  4 

D
12m v1
4
z 

1
1
mD 2  MD 2 3m  4M D
16
12
mv1
Total linear
momentum ptotal
is not conserved,
because
Fnet,ext  0
v1
ω’
D
D/4
Example: A student sits on a rotating stool and holds a rotating
horizontal bicycle wheel by a rod through its axis. The stool is initially
at rest. The student flips the axis of rotation of the wheel by 180°.
What happens to the stool?
A. It rotates in the same direction
as the wheel after the flip.
B. It rotates in the same direction
as the wheel before the flip
C. Nothing! Why would it rotate at all?
Lw
Ltotal
Ls+
s
Lw
Example: A force is applied to a dumbbell for a certain period of time,
first as in (a) and then as in (b). In which case does the dumbbell
acquire the greater center-of-mass speed?
1) case (a)
2) case (b)
3) no difference
4) It depends on the rotational
inertia of the dumbbell.
In which case does the dumbbell acquire the
greater energy?
Example: A spherical shell rotates about an axis through its center of
mass. It has an initial radius Ri and angular speed ωi. By applying a radial
force, we can cause the sphere to collapse to Rf = Ri/3. What is the ratio of
the final and the initial angular speed, ωf/ωi ?
Li  Iii
Lf  I ff
Iii  I ff
f
Ii
MRi 2
Ri 2


 2 9
2
i
If
MRf
Rf