A uniform solid cylinder of radius R and length L has a moment of inertia I₀ about its central axis. A cylindrical portion of radius R / 3 and length L , whose axis is parallel to the central axis and at a distance of 2R / 3 from it, is considered. The moment of inertia of this specific portion about the original central axis is I_ p . The ratio I₀ I_ p is ________.

Step-by-step solution

Let the mass of the original solid cylinder be M . Its moment of inertia about the central axis is I₀ = 1 2 MR^2 . The considered cylindrical portion has the same length but a radius of R/3 . Since mass is proportional to the square of the radius for a given length and uniform density, the mass of this portion is m_ p = M ( 1 3 )^2 = M 9 . The moment of inertia of this portion about its own central axis is: I_ cm = 1 2 m_ p ( R 3 )^2 = 1 2 ( M 9 ) R^2 9 = MR^2 162 The axis of this portion is parallel to the origina