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Angular Momentum
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Angular momentum is a characteristic of a
rotating body about a certain axis and is
dependent upon the moment of inertia about that
axis and the angular velocity about that axis.
Force and
F = ma and
Torque
 = I
• Newton’s Laws also apply to angular motion.
• For every linear term, there is an equivalent
angular term.
• For example, torque is the angular effect of
force. Just like a net force produces an
acceleration resisted by the mass, a net torque
produces an angular acceleration resisted by
the moment of inertia.
Linear Impulse and Angular Impulse
Ft = mv and  t = I

• A net force acting for a period of time
produces a linear impulse that results in a
change in linear momentum.
• Likewise, a net torque acting for a period of
time produces an angular impulse that
results in a change in angular momentum.
• Where angular momentum is the product of
the moment of inertia and angular velocity.

 t = I

Area under the curve =
I

t
I is the change in angular momentum produced
) acting for t.
by the resultant torque (
Conservation of Angular Momentum
• Just as linear momentum is conserved when
no external forces act on a system, angular
momentum is conserved when no external
torques act on a system.
• Understanding the above statement is
unimaginably crucial to your success.
• This would be the case when a system is
freely rotating on a turntable, or travelling
through the air.
Conservation of Angular Momentum

•  t = I
• If,

 t = 0
then
I = 0
• If there is no change in angular momentum, then
• ITotal = Constant
Spinning Figure Skater
(Big I, small )
(Small I, big )
http://youtu.be/FyHyni1-zYE
Turntable Example
ITotal = I11
I22 =
+
Person and top of turntable
Bicycle Wheel
1. ITotal = 50
2. ITotal = 0
3. ITotal = -50
Constant
+
0 = 50
50 = 50
+
+
100 = 50
Arbitrary
Values
Conservation of Angular Momentum
• Applied torques that are internal to the
system will result in changes in the angular
momentum of different parts of one system,
but the net angular momentum of the whole
system will not change.
• In aerial sports, athletes will often attain a
certain position of the trunk by rotating the
limbs in specific directions.
Long Jump Example
The forces acting on the foot at take-off produce a
torque about the jumper’s CofG. This torque will
produce forward angular momentum causing the
jumper to pitch forward in the air.
C of G
Friction Force
Long Jump Hitch Kick
Notice how the right arm and leg rotate counter-clock wise
to prevent the trunk from pitching forward.
Moment of Inertia and Angular
Momentum
• With most activities, a greater amount of
angular momentum can be put into a system
that has a high moment of inertia.
• This is partly due to the force-velocity
properties of human muscle.
• An example would be spinning a person on
a turn table with the arms outstretched as
opposed to tucked in.
Generating Angular Momentum
with an External Torque
• Generating the greatest amount of angular
momentum prior to leaving the ground is
important in such events as figure skating.
• To maximize the amount of angular
momentum generated prior to take-off, the
athlete should maximize their moment of
inertia about the axis they are trying to rotate
about.
• This would be the longitudinal axis in figure
skating.
Generating Angular Momentum
The increase in (I) means that for any given torque the athlete’s body
will take a longer time to rotate which means a longer time for the
torque to act which leads to a greater change in angular momentum
Small (I) means less I
Torque
Large (I) about longitudinal Torque
axis prior to jump
This means an external
torque can be applied to
his body for longer time
Large (I) means more I
Time