Moment of inertia of a ring of mass M and radius R about an axis passing through the centre and perpendicular to the plane is l. What is the moment of inertia about its diameter?

The moment of inertia about the axis of rotation is merely a function of the distribution of its mass with regards to that axis; for a given mass and radius for a ring, it represents how the same mass is concentrated at the distance uniformly distributed from its center. However, if we were to consider it about one of its diameters, things would change based upon the principles of rotational dynamics.

By the perpendicular axes theorem, for any planar object, the moment of inertia about an axis perpendicular to its plane equals the sum of its moments of inertia about two mutually perpendicular axes lying in the plane and passing through the same point. In the case of a ring, the two perpendicular axes in its plane are its diameters, and they are identical due to symmetry. Therefore, the moment of inertia about a single diameter is half of that about the perpendicular axis through its center.

This relation further shows that moment of inertia is dependent on the geometry as well as the orientation of the axis of rotation. For the ring, the moment of inertia about its diameter comes out to be half the moment of inertia.

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