What is the ratio of the moments of inertia of two rings of radii r and nr about an axis perpendicular to their plane and passing through their centers?

The moments of inertia of two rings, one with radius r and the other with radius nr, can be compared when studying their rotational dynamics about an axis perpendicular to their planes and passing through their centers. The moment of inertia is a measure of an object’s resistance to angular acceleration when a torque is applied. For rings, mass and the square of radius can be used to calculate moments of inertia, so in that case, for our first ring: the radius is taken as r . For the second ring, in this case a multiple of first, its radius is taken to be nr.

When the moments of inertia for each ring are calculated, the ring with radius nr will have a moment of inertia proportional to the square of its radius compared to the first ring. Hence, if we find the ratio of their moments of inertia, we would get that the moment of inertia for the second ring is proportional to the square of the scaling factor n . Thus, the ratio of the moments of inertia of the two rings comes out to be 1 : n² . This also explains how mass distribution and geometry influence rotational motion.

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