Question

A string is wrapped around a solid cylinder with mass M and radius R. The free end of the string is held in place, allowing the cylinder to fall. Recall that the moment of inertia of a solid cylinder rotated about its center is given by MR2/2. All answers to this problem should be symbolic, purely in terms of the variables M, R, and g. (a) Find the linear acceleration (in m/s2) of the cylinder and the tension in the string (in Newtons) as the cylinder falls. (b) Now suppose the cylinder is hollow instead of solid. The moment of inertia of a hollow cylinder rotated about its center is given by MR2. What is the acceleration and tension in this case?

Answer 1

I will use (a / R) for alpha the angular acceleration

T R = I a / R torque equals angular acceleration for cylinder

M g - T = M a linear acceleration of center of mass

T = M (g - a) = I a / R^2 from first equation

If I = 1/2 M R^2 then M ( g - a) = M a / 2 from above

or g = 3 a / 2 and a = 2 g / 3

Also we have T = M (g - a) = M (g - 2 g / 3) = g / 3

Substitute I = M R^2 for the hollow cylinder

Looks like a = g/2 for hollow cylinder