True or False: If a sphere of mass M and radius R and a cylinder of mass M and radius R start rolling down a ramp from the same location at the same time, they both reach the bottom of the ramp at the same time.

The have the same initial PE. So since inertia is the thing that resists motion, you have to consider rotational motion. Find the velocity of each at the bottom, if the velocities are different, then they do not reach the bottom at the same time.

Cylinder:
mgh=1/2 m vf^2+1/2 I w^2
mgh=1/2 m vf^2+ 1/2 (1/2 mr^2)(vf/r)^2
mgh= 1/2 m vf^2 + 1/4 m*vf^2
mgh= m vf^2(1/2+1/4)
vf=sqrt gh*4/3

Now do the same for the sphere. Find Vf.
are the vf the same? If so, true.