Dynamics which rolls faster??

Question

A thin ring and a homogeneous circular disk, each of mass m and radius R, are released from rest on an inclined surface. Determine the ratio Vring/Vdisk of the velocities of their centers when they have rolled a distance D. HELP!!!

Answer 1

After rolling a distance D the two bodies have accumulated KE = PE = mgD*slope. This KE is the sum of rotation and translation energies:

KE = (Iω^2 + mv^2)/2

ω^2 = v^2/r^2

I(ring) = mr^2

I(disk) = 0.5mr^2

Thus

KE(ring) = (mr^2v^2/r^2 + mv^2)/2 = mv^2

KE(disk) = (0.5mr^2v^2/r^2 + mv^2)/2 = 0.75 mv^2

Since KE(ring) = KE(disk),

v(ring)/v(disk) = sqrt(0.75) = sqrt(3)/2

Answer 2

The cylinder will roll faster because it has a smaller moment of inertia.

Answer 3

I am not sure I am reading this correctly but if they have the same mass and radius they should roll at about the same velocity. Except, I guess for the difference in the air resistance (kind of like solid bicycle tires versus open). Also I am assuming that you are not looking for the rotational velocity at the center, as that would be zero for both.