Angular Momentum about an Arbitrary Axis

Angular momentum at an arbitrary point P is determined by taking the cross product of the position vector and linear momentum vector for each mass element. The velocity of each mass element combines its translational velocity and relative velocity from the body's rotation. Integrating these components over the entire mass yields the total angular momentum about point P.

When point P is selected as the center of mass of the body, the first integral in the angular momentum equation becomes zero because the position vector becomes zero. This simplification means the total angular momentum reduces to only the angular momentum at the center of mass, eliminating the moment due to linear momentum term.

If point P is chosen to be a fixed point in the inertial reference frame, the linear velocity term vanishes from the angular momentum equation. This simplification allows direct calculation of angular momentum without accounting for translational motion, making the analysis focus solely on rotational effects about that fixed point.

For any arbitrary point not at the center of mass or a fixed point, the angular momentum consists of two terms. The first term represents the moment due to linear momentum of the entire body, while the second term provides the angular momentum at the center of mass of the object.

In a rotating rigid body, each mass element experiences both translational motion of the body and rotational motion about the center of mass. The total velocity combines the translational velocity of the body and the relative velocity caused by the body's rotation, which is essential for accurately calculating angular momentum.

By substituting the velocity equation into the angular momentum equation, expanding the cross product, and integrating over the entire mass, the contributions from all individual mass elements are combined. This integration process yields the total angular momentum about point P, accounting for the distributed mass throughout the rigid body.

The inertial reference frame provides a non-accelerating coordinate system in which angular momentum is measured. All calculations of position vectors, velocities, and angular momentum for the rigid body are performed relative to this inertial frame, ensuring that the resulting angular momentum values are accurate and meaningful for rigid body dynamics analysis.