A solid object of uniform density with mass M, radius R, and moment of inertia I rolls without slipping down a ramp at an
angle θ to the horizontal. The object could be a hoop, a disk, a sphere, etc.
(a) Carefully follow the complete analysis procedure explained in earlier chapters, but with the addition of the Angular
Momentum Principle about the center of mass. Note that in your force diagram you must include a small frictional
force f that points up the ramp. Without that force the object will slip. Also note that the condition of nonslipping
implies that the instantaneous velocity of the atoms of the object that are momentarily in contact with the ramp is
zero, so f <>
speed of the object, since the instantaneous speed of the contact point is vCM − ωR.
(b) The moment of inertia about the center of mass of a uniform hoop is MR2, for a uniform disk it is (1/2)MR2, and
for a uniform sphere it is (2/5)MR2. Calculate the acceleration dvCM/dt for each of these objects.
(c) If two hoops of different mass are started from rest at the same time and the same height on a ramp, which will
reach the bottom first? If a hoop, a disk, and a sphere of the same mass are started from rest at the same time and
the same height on a ramp, which will reach the bottom first?
(d) Write the energy equation for the object rolling down the ramp, and for the point-particle system. Show that the
time derivatives of these equations are compatible with the force and torque analyses.