ROLLING
a = g·sinθ/(1+β) — shape alone runs the race. Mass and radius cancel, bitwise.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A ball, a can, a ring — release them together at the top of a ramp and they do not tie. Rolling forces the object to spend energy on two motions at once, sliding forward and spinning, and the balance between them is fixed entirely by shape. Not mass, not size — shape.
Two motions
Every joule of dropped height splits between translation and rotation. Rolling without slipping locks the two together — v = ωR — so the spin energy becomes just ½βmv², where β = I/mR² is the shape factor. The mass and the radius both cancel out of the answer, which is why a marble and a boulder of the same shape tie exactly.
Solve
One factor runs the whole race. The acceleration is g·sinθ divided by (1+β), and the final speed carries the same penalty. The energy always divides in the ratio 1 to β — so a hoop, with β = 1, sinks half of everything into spin, while a solid sphere, β = 0.4, keeps five-sevenths for getting down the hill.
The descent
The body rolls down at a constant acceleration, its spokes turning in lockstep with its progress — v = ωR, frame after frame. The energy bars hold their ratio the entire way: the translational share races ahead while the rotational share, fixed by β, takes its cut. Change the shape and watch the split, and the finish line, move.
Audit
The energy books close exactly: half v-squared times (1+β) equals gh at the bottom, at every angle and every shape. The translational-to-rotational split is 1 to β independent of mass, radius, gravity, and slope. And the acceleration carries no m and no R — the deepest and most counterintuitive fact of the rolling race, verified to the last bit.