Two identical thin circular rings, each of mass 3 kg and radius R , are fixed at the ends of a light rod of length 40 cm . The centers of the rings are attached to the ends of the rod, and the planes of the rings are perpendicular to the rod. The moment of inertia of the system about an axis passing through the midpoint of the rod and perpendicular to it is 360 10⁻³ kg m ^2 . The value of R is ____ cm.

Step-by-step solution

Let the mass of each ring be M = 3 kg and the length of the rod be L = 40 cm = 0.4 m . The axis of rotation passes through the midpoint of the rod and is perpendicular to it. Since the planes of the rings are perpendicular to the rod, the axis of rotation is parallel to the planes of the rings (i.e., parallel to their diameters). The distance from the center of each ring to the axis of rotation is d = L 2 = 0.2 m . By the parallel axis theorem, the moment of inertia of one ring about this axis is: I₁ = I_ diameter