Problem: Compare the motions of:

a. a body sliding without friction
b. a cylinder rolling down the incline without slipping
c. a sphere rolling down the incline without slipping

a) For a body which is sliding down an incline without friction, the size and shape are irrelevant since no rotation is occurring. The free body is:

Using Newton's Second Law for the direction parallel to the plane, I get:

Since the height that the body "falls" through is h, we can use either the constant acceleration equations or the conservation of energy method to compute the speed at the bottom of the incline. (If the incline were a curve, the acceleration would be variable and only the energy method would be valid.)

Using conservation of energy:

b & c) It will be more efficient to solve the problem of a rolling body in general, and then specialize to a cylinder or a sphere later.

The only difference between the free body for this case and case a is that there must be friction for rolling to occur; this is shown by f in the free body. (Without f, the body slides without rotation, and the problem is the same as case a.)

Using conservation of energy,

For the body sliding without friction, it is essentially a point mass since its size is irrelevant. If you'll substitute I = 0 (a point mass has no center of mass moment of inertia) into equation 1, you'll get the same result as part a.

For the cylinder,

For the sphere,

As we anticipated, the velocities of the rigid bodies are less than that for the point mass, since energy has to go into rotating the bodies and not as much is available for producing center-of-mass velocity.