Equation of Motion for a Rigid Body

Newton's Second Law governs the translational motion of a rigid body's center of mass, point G. The sum of all external forces equals the mass times the acceleration of the center of mass. This equation describes how the body moves through space without accounting for rotation about its center.

When an external force is applied at a point other than the center of mass, it creates a moment that causes the body to rotate. This moment is calculated as the vector product of the relative position and the force applied at that point. The resulting rotation occurs about the center of mass.

The time derivative of angular momentum equals the total moment applied to the rigid body about its center of mass. By summing the moments from all external forces acting on the body, you obtain the rate of change of angular momentum. This relationship connects rotational dynamics to angular momentum about an arbitrary axis.

The vector product combines the relative position and relative velocity of a point with respect to the center of mass to express angular momentum. This mathematical operation produces a vector perpendicular to both input vectors, capturing the rotational characteristics of the body at that point.

The total moment about the center of mass is calculated by summing the moments from all points within the rigid body. Using relative acceleration definitions and vector distributive laws, you combine individual moments to determine the system's overall rotational response to applied forces.

The center of mass, point G, is the reference point for both translational and rotational equations of motion. Translational motion follows Newton's Second Law at this point, while all moments and angular momentum are calculated relative to it, simplifying the analysis of complex rigid body dynamics.

Translational motion describes how the center of mass moves under applied forces, while rotational motion describes how the body spins about that center. Together, these coupled equations fully characterize rigid body motion, with angular momentum and principle axes of inertia determining rotational behavior.