compare the three types of motion-linear motion and rotational motion and the coupled motion of a double pendulum

A physical system's inertia isn't always a constant value; it can depend on its configuration. The generalized inertia tensor provides a framework for describing this relationship. For a single particle moving in a straight line, the inertia tensor is constant and directly proportional to the mass of the particle. A greater mass requires more force to achieve the same acceleration. For a rigid object rotating, the inertia tensor is also constant (as long as the object doesn't deform). A greater moment of inertia requires more torque to achieve the same angular acceleration. While some components of the generalized inertia tensor are constant, the coupling term continuously changes as the pendulum swings. This term depends on the angles of the two arms, demonstrating that the inertia of the system is not fixed but is instead a function of its dynamic configuration.

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