A uniform straight rod of length L and a uniform solid disc of radius R have the same radius of gyration. The rod rotates about an axis passing through its centre and perpendicular to its length, while the disc rotates about one of its diameters. If the ratio of the length of the rod to the radius of the disc is x , the value of x is _____.

Step-by-step solution

The moment of inertia of the rod about an axis passing through its centre and perpendicular to its length is I_ rod = mL^2 12 . Its radius of gyration squared is k_ rod ^2 = L^2 12 . The moment of inertia of the solid disc about its diameter is I_ disc = mR^2 4 . Its radius of gyration squared is k_ disc ^2 = R^2 4 . Since the radii of gyration are equal: L^2 12 = R^2 4 L^2 R^2 = 12 4 = 3 L R = 3 Comparing this with x , we get x = 3 . Answer: 3