Acceleration Down a Frictionless Incline

Strip friction away and the down-slope force mg sin θ divided by the mass m leaves a = g sin θ — the mass cancels completely, so a marble and a bowling ball slide identically. A 30° ramp yields 9.80665 × 0.5 ≈ 4.90 m/s², exactly half of free fall. This is Galileo's "diluted gravity": by 1604 he had shown that distances on an incline still grow as t², and he extrapolated to θ = 90°, where a = g and the ramp becomes free fall.

Watch the assumption — sliding, not rolling. A ball that rolls without slipping must also spin up its own moment of inertia, so a solid sphere manages only (5/7)g sin θ and a hoop just (1/2)g sin θ. That difference is the whole point of the classic race down a ramp, where a solid cylinder always beats a hollow one of identical mass and radius.