# Angular Momentum of a Rotating Rigid Body

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The angular momentum of a rigid body with angular velocity is given by , where is the inertia tensor. This Demonstration shows the rotation of an axially symmetric ellipsoid rotating about a fixed angular velocity vector . The body axes , , (indicated by the red, green, and blue spheres) and the angular momentum rotate as functions of time. The space axes , , are indicated by the red, green, and blue arrows. The body’s height and radius* *can be adjusted, as can the angular velocity .

Contributed by: Frederick W. Strauch (October 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The inertia tensor for a body of mass , uniformly distributed through an ellipsoid , is

; here we have set * *and *.*

This tensor is used to calculate the angular momentum in the body frame (with body axes , , fixed), which is then transformed to the space frame (with space axes , , fixed) by a rotation matrix. The inertia tensor and rotation are covered in the references below, and in the Wikipedia entry for "Moment of inertia".

References

[1] J. R. Taylor, "Rotational Motion of Rigid Bodies," *Classical Mechanics*, Herndon, VA: University Science Books, 2005 pp. 367–416.

[2] S. T. Thornton and J. B. Marion, "Dynamics of Rigid Bodies," *Classical Dynamics of Particles and Systems*, Pacific Grove, CA: Brooks/Cole, 2004 pp. 411–467.

## Permanent Citation