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What are the ways to create a mathematical model for the moment of inertia of a football? Can the moment of inertia of the football be simplified to two cones stack against each other?

I'm trying to find the angular momentum of the football by using $L=I \omega$

Is the angular momentum of football actually conserved?

What are the ways one can obtain the angular velocity on a rotating football?

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Just to clarify, I assume from your description that you are talking about an American football. –  Colin McFaul Jul 3 '12 at 16:50

2 Answers 2

Angular velocity is "rotations per second" or minute. One can simple convert this into radians per second, which is common units in mechanics. One rotation is just $2\pi$ radians.

To calculate $I$ of a given shape, one should integrate over space or use precalculated formula:

Angular momentum is conserved if no interaction with other bodies. A football can interact, for example with air, causing and others.

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Generally, moment of inertia isn't scalar $I$ but tensor $\tilde{I}$. The exact equation is

$$\vec{L} = \tilde{I} \vec{\omega}$$

which means that generally vectors $\vec{I}$ and $\vec{\omega}$ do not have the same direction! The expression you wrote

$$\vec{L} = I \vec{\omega}$$

where $I$ is simply a scalar is that it is valid only if the body rotates only around one of the three principal axes of moment of inertia.

For all other rotations things become increasingly complex and you have to use Euler's equations. Angular momentum $\vec{L}$ is of course always conserved if there are no external moments, and $\vec{L}$ is therefore constant, but taking Euler's equations angular velocity $\vec{\omega}$ generally is not constant!

Since football is highly symmetric body, it is easy to identify its principal axes of moment of inertia and it is obvious that $I_{xx} = I_{yy} > I_{zz}$.

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