A rolling body accelerates at g sin(theta) divided by (1 + k), where k says how far its mass sits from the axis. A hoop gets exactly half the acceleration of a frictionless slide, a solid sphere keeps five sevenths — and mass and radius cancel out entirely.
A rolling object has to spin up as well as move forward, and both come out of the same gravity. How the energy splits depends only on the shape, not the size or the weight, so a marble and a cannonball released together finish together, and a hoop released with either of them loses. The order is fixed: solid sphere, solid cylinder, hollow sphere, hoop. It never changes with angle.
It assumes rolling without slipping, which needs friction of at least tan(theta) times k/(1+k) — about 0.17 for a sphere on a 30 degree slope but 0.29 for a hoop. On a slippery slope the body slips instead, stops spending energy on spin, and arrives sooner. Air resistance and rolling resistance are also ignored, which matters for anything light or very fast.
Estimate with the rule, then check it against the calculator that models it properly.
Open Inclined Plane →A rolling body accelerates at g sin(theta) divided by (1 + k), where k says how far its mass sits from the axis. A hoop gets exactly half the acceleration of a frictionless slide, a solid sphere keeps five sevenths — and mass and radius cancel out entirely. A rolling object has to spin up as well as move forward, and both come out of the same gravity.