Angular Momentum about an Arbitrary Axis

Imagine a rigid body with a mass denoted as 'm', which has its center of mass at point G and is rotating around an inertial reference frame. The angular momentum at an arbitrary point P can be calculated by taking the cross product of the position vector and linear momentum vector for each individual mass element.

The velocity of a mass element comprises its translational velocity and the relative velocity instigated by the body's rotation. Substituting the velocity equation into the angular momentum equation, expanding the cross product, and integrating over the entire mass yields the total angular momentum about point P.

If point P is selected as the center of mass of the body, then the first integral becomes zero as the position vector becomes zero. If point P is chosen to be a fixed point, then the linear velocity term disappears. For any other arbitrary point, the integral can be simplified. In this case, the first term provides the moment due to linear momentum, while the second term offers the angular momentum at the center of mass of the object. This approach provides a comprehensive understanding of the angular momentum in relation to varying points on a rotating body.