AP Free Response Question 2006 C3

A thin hoop of mass M, radius R, and rotational inertia MR2is released from rest from the top of the ramp of length L above. The ramp makes an angle  with respect to a horizontal tabletop to which the ramp is fixed. The table is a height H above the floor. Assume that the hoop rolls without slipping down the ramp and across the table. Express all algebraic answers in terms of given quantities and fundamental constants.

 (a) Derive an expression for the acceleration of the center of mass of the hoop as it rolls down the ramp.

 (b) Derive an expression for the speed of the center of mass of the hoop when it reaches the bottom of the ramp.

 (c) Derive an expression for the horizontal distance from the edge of the table to where the hoop lands on the floor.

 (d) Suppose that the hoop is now replaced by a disk having the same mass M and radius R. How will the distance from the edge of the table to where the disk lands on the floor compare with the distance determined in part (c) for the hoop? Briefly justify your response.