A round object with mass m and radius R rolls down a ramp that makes an angle θ with respect to the horizontal. You can assume there is static friction so that the object rolls without slipping.

Question

A round object with mass m and radius R rolls down a ramp that makes an angle θ with respect to the horizontal. You can assume there is static friction so that the object rolls without slipping. 
a) For now, take the moment of inertia of the object to be I (you do not know if it is a cylinder or a sphere or if it is hollow or solid). Derive an expression for the acceleration in terms of m, R, I, θ, and g. 
b) What is the acceleration for a thin-walled hollow cylinder? 
c) What is the acceleration for a solid cylinder? 
d) Is your answer to parts b and c the same? Why or why not? The acceleration for a block sliding down a ramp in purely linear (translational) motion is a = g sin θ - μk g cos θ, which does not depend on the size or shape of the block or its mass. Discuss how this compares to the rolling cylinders.