State and prove the theorem of parallel axes.

This theorem of parallel axes is a significant principle in rotational dynamics that lets us calculate the moment of inertia of a rigid body about an axis parallel to one that passes through its center of mass. This theorem states that the moment of inertia about any parallel axis is the sum of the moment of inertia about the center of mass axis and the product of the body’s mass and the square of the distance between the two axes.

Understanding this theorem better can be accomplished by considering a rigid body consisting of several point masses. When calculating the moment of inertia about the center of mass, each mass contributes based on its distance from that axis. When shifting to a parallel axis located a certain distance away, the positions of the masses change accordingly. The new moment of inertia is then calculated by summing the contributions from all masses, taking into account their new distances from the parallel axis.

This theorem simplifies the process of finding moments of inertia for complex shapes, so engineers and physicists can analyze rotational behavior in objects. The moment of inertia about the center of mass is first determined, and from that, it gives a very straightforward method of determining it about any parallel axis, thus enriching the understanding of rotational motion in mechanical systems.

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