Which ring will win the race to the bottom and why?

Question

Suppose two rings are at the top of a ramp. The rings have the same mass, but one ring has a much larger radius than the other. Which ring will win the race to the bottom and why? (Hint: consider the potential energy, translational kinetic energy, and rotational kinetic energy of each ring.)

Answer 1

potential energy = m g h

translational kinetic energy = .5 m v²

rotational kinetic energy = .5 I ω² = .5 (m r²) (v / r)² = .5 m v²

rotational inertia for a ring (I) = m r²

angular velocity (ω) = v / r

Potential energy at the top will equal the sum of the translational and rotational kinetic energy at the bottom:

PE = KE_{translational} + KE_rotational

mgh = .5mv² + .5mv²

mgh = mv²

v² = mgh / m = gh

v = √gh

Based on the equation above, the velocity of each ring will only be based on the gravitational constant and height of the ramp. Since the radius doesn't determine the velocity, both rings will hit the bottom at the same time.