Angular Momentum

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.

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Force and TorqueF = ma and = 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

Ft = mv 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

t

Area under the curve = I

I is the change in angular momentum produced

by the resultant torque () acting for t.

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

• ITotal = Constant

Spinning Figure Skater

(Big I, small ) (Small I, big )

http://youtu.be/FyHyni1-zYE

Turntable Example

ITotal = I11 + I22 = Constant

Bicycle Wheel Person and top of turntable

1. ITotal = 50 + 0 = 50 ArbitraryValues

2. ITotal = 0 + 50 = 50

3. ITotal = -50 + 100 = 50

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 ExampleThe 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.

Friction Force

C of G

Notice how the right arm and leg rotate counter-clock wise to prevent the trunk from pitching forward.

Long Jump Hitch Kick

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 MomentumThe 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

Large (I) about longitudinal axis prior to jump

This means an external torque can be applied to his body for longer time

Torque

Small (I) means less I

Torque

Time

Large (I) means more I