Angular Momentum about an Arbitrary Axis

Consider a rigid body of mass 'm' and a center of mass at point G, rotating in an inertial reference frame.

At an arbitrary point P, the angular momentum is determined by taking the cross product of the position vector and linear momentum vector for each mass element.

The velocity of a mass element is composed of its translational velocity and the relative velocity caused by the body's rotation.

By substituting the velocity equation into the angular momentum equation, expanding the cross product, and integrating over the entire mass gives the total angular momentum about point P.

Here if the point P is chosen as the center of mass of the body, then the first integral becomes zero. If the point P is chosen to be a fixed point, then the linear velocity term vanishes.

For any other arbitrary point, the integral can be simplified. Here the first term gives the moment due to linear momentum, and the second term gives the angular momentum at the center of mass of the object.