When objects of the same mass and radius roll without slipping down an incline,
they do not all accelerate equally — even though gravity pulls on every point of
every object the same way. What differs is how each object's mass is
distributed relative to its rotation axis, described by the
moment of inertia I = k²mr². A larger k²
means more mass sits far from the axis, so more of gravity's pull goes into spinning
the object rather than accelerating its centre of mass down the slope.
a = g·sinθ / (1 + k²), independent of mass and radius.k² = 2/5 = 0.400 (mass concentrated near the centre) — always wins.k² = 1/2 = 0.500.k² = 1 (all mass at the rim) — always last among rolling shapes.k² = 0 effectively), so it always beats every roller.a = g·sinθ, for comparison with rolling motion.This is why a solid can of food rolls faster down a ramp than a can of the same size full of liquid that stays swirling near the bottom — thin liquid layers near the wall barely spin, effectively lowering the can's rotational drag.
A sphere, cylinder and hoop of identical mass and radius race down a 3D incline, plus a fully adjustable "mystery" shape and an optional frictionless sliding block, showing how moment of inertia — not mass — decides who wins.
For rolling without slipping, acceleration down a slope is a = g·sinθ / (1 + k²), where k² encodes how far mass sits from the rotation axis. Mass and radius cancel out entirely — only the shape's mass distribution matters.
Adjust the incline angle and watch every racer's acceleration scale together. Drag the mystery shape's k² slider between sphere-like and hoop-like values and watch it slide up or down the finishing order. Toggle a frictionless block for a non-rotating baseline.
Because a depends only on k² and θ, a bowling ball and a marble reach the bottom of a ramp at the same time — but a hollow pipe of the same size always loses to a solid rod.