Angular Momentum
Moment of Momentum

To continue the analysis of
rotational motion, we must also
extend the idea of momentum.
p

Define an angular momentum L.
• Based on the momentum p = mv
• Includes a lever arm
• Follows the rules for torque
r
L  rp sin 
Momentum Cross Product

Angular momentum is a
vector.
• Vector cross product of the
lever arm and momentum.
• Direction follows the righthand rule
• Magnitude from sine rule
L
p
r
  
Lrp
L  rp sin 
Law of Angular Inertia

The time derivative of
angular momentum vector is
the net torque vector.

By the law of reaction, all
internal torques come in
canceling pairs.
• Add external torques


 
L   L i   (ri  pi )



 dpi dri 
dL
  (ri 

 pi )
dt
dt dt

  

dL
  (ri  Fi  vi  mvi )
dt

 

dL
  (ri  Fi )   i
dt


dL
  ext
dt
Applying Torque

An external torque changes angular momentum.
• dL/dt = 
L
w
p
L+rpsin
w
Starting the Ride


A child of 180 N runs at 4.5
m/s and hops on the edge of
a merry-go-round with radius
2.0 m.
If the motion is perpendicular
to the radius, what is angular
momentum?

• W = mg, m = W/g = 18 kg.
• p = mv = 81 kg m/s

r
At right angles, all the linear
momentum contributes.
• L = rp = 160 kg m2/s

m
The child starts with linear
momentum.
The torque is due to the
friction at contact.
Spinning Mass

The moment of inertia is the analog of mass for
rotational motion.

The analog for angular momentum would be:
w
L  Iw
Initial Spin


A child of 180 N runs at 3.0
m/s and hops on the edge of
a merry-go-round with radius
2.0 m and mass of 160 kg.
What is the period of
rotation?

The moment of inertia and
the angular momentum for
the child on the ride was
found before.
• I = Id + Ic = 390 kg m2
• L = rp = 160 kg m2/s

The period is related to the
angular velocity.
• L = Iw = I(2p/T)
• T = 2p I / L = 15 s
m
r
M
Single Axis Rotation

An axis of rotation that is fixed in direction gives a
single axis rotation.
• Simplest case has the axis through the center of mass
• Angular momentum vector is parallel to the angular velocity
L
w


L  Iw
Limitations

w
p
L
There are limitations to the
relationship between angular
momentum and angular
velocity.
• Moving axis of rotation
• Asymmetric axis of rotation
r

Angular momentum and
angular velocity can have
different directions.
Angular Momentum Vector

The vector form of the law of rotational action is
generalized to use angular momentum vectors.



dw
r F  I
dt

 dL
 
dt
• Correct for all axes
• Correct for changes in direction as well as angular velocity
next