bullet Angular Momentum of Rigid Solid

 

The angular momentum of the small element of mass with respect to the center of mass can be calculated as with .

The total angular momentum of the solid will be

but the magnitude of the velocity for the element of mass is related to the angular velocity by . Therefore, the total angular momentum of the solid is

where is the moment of inertia of the solid with respect to the center of mass.

Correspondingly, the angular momentum with respect to any other point can be calculated considering that , see drawing on the left. Remember that for a fix point , the vector is constant and independent of the time (the velocity does not change). Therefore, the angular momentum of the element of mass with respect to point  is

Therefore, the total angular momentum of the solid with respect to point is

where correspond to the linear momentum of the center of mass, . Then, the total angular momentum of the solid is

The previous expression presents the total angular momentum of a solid object with respect to an arbitrary point . If , the rotational angular momentum is the same independent of the selected point for calculating the angular momentum. Thus, the rotational angular momentum of a solid, when turning on an axis going through its center of mass, is an intrinsic property of the object. It is called the spin angular momentum. On the other hand, if , the added term to the total angular momentum can be called the "orbital" angular momentum.

 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com