AP Free Response Question
2006 C3
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A thin hoop of mass
M
, radius
R
, and rotational inertia
MR
^{2}
is 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.
Topic Formulas
Description
Published Formula
angular displacement
angular momentum
angular velocity
center of mass
centripetal acceleration
friction
gravitational force (vector)
gravitational potential energy
Hooke's Law
impulse
kinetic energy
linear momentum
linear velocity and angular velocity
moment of inertia
net torque
Newton's 2nd Law
Newton's Law of Universal Gravitation
period and frequency
period of a simple pendulum
period of a spring
potential elastic energy
potential energy
power (dot product)
rate of change of momentum
rate of change of work
rotational kinetic energy
torque
uniform acceleration  displacement and instantaneous velocity
uniform acceleration  instantaneous position
uniform acceleration  instantaneous velocity
work (dot product)
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