Question

A ball and a barrel roll down a 16m-long incline of height "h."

Assume I equals 2/5mr^2 for the ball
I equals 1/2mr^2 for the barrel.

Prove that the ball reaches the bottom first.

Answer 1

The gravitational energy available per mass is the same for both, but with the (solid) ball, a smaller share of the total energy is converted to rotational KE, leaving more for translational motion (velocity). Therefore the ball goes more rapidly down the ramp.

Here is how to prove it mathematically. Write an equation for translational PLUS rotational energy, substitute the moment of inertia I in the rotational KE part, and remember that w (angular velocity)= V/R. You should get an equation that says
Total KE = C M V^2.
Compare the values of C.