Which item gets to the bottom first?

Question

A Hoop (a car tire), and a solid sphere (a bowling ball) and a solid cylinder (a round fence post) all roll down a hill at once. Which item gets to the bottom first?

m g h = ½ m v2 + ½ I w2   w = vT / r

m g h = ½ m v2 + ½ I vT2 / r2

m g h = ½ m v2 + ½ I vT2 / r2 for Hoop I = m r2

m g h = ½ m v2 + ½ m r2 vT2 / r2

g h = ½ v2 + ½ v2

V = SQRT (g h)

Answer 1

If we let c stand for the term in front of the mR² for the various moments of inertia (c= 1 for hoop, 1/2 for cylinder, and 2/5 for a sphere), we can use energy conservation to solve for final velocity:

mgh at the top = 1/2 mv² + 1/2 Iω² at the bottom of the ramp

Since ω = v/R for a non-slipping rolling object, and I = cmR² we obtain

mgh = 1/2 (1+c) mv²

which you can solve for v. Note that the mass AND the radius of the rolling object has no effect on the final velocity (only the shape has an impact)

You actually don't have to do the calculation because, the one with the most rotational inertia will go slower.

Good luck