A uniform solid cylinder and a uniform solid sphere, having the same mass and the same radius, are free to rotate about their respective central axes. If the same constant torque is applied to both for the same time duration starting from rest, what will be the ratio of the final angular velocity of the solid cylinder to that of the solid sphere?

Step-by-step solution

Let the mass of both objects be M and their radius be R . The moment of inertia of a uniform solid cylinder about its central axis is I_ cyl = 1 2 MR^2 . The moment of inertia of a uniform solid sphere about its central axis is I_ sph = 2 5 MR^2 . From Newton's second law for rotation, the angular acceleration is given by = I . Since both objects start from rest, their final angular velocity after time t is = t = t I . Given that the applied torque and time t are the same for both, the angular velocity is inversely