Presets
Playback
Ramp
Angle, how far the bodies roll to the finish line, and gravity (any planet).
Friction
Ramp friction for the rolling bodies and the sliding block, separately.
Clock
Playback speed of the race.
Camera
Layers
Quality
Model checks
The science
What this lab shows
Five bodies of the same mass and the same outer radius are released together from the top of a ramp: a solid sphere, a solid cylinder, a thin spherical shell, a thin hoop, and a block that slides instead of rolling. Mass and radius cancel out of the motion; only the way the mass is spread around the spin axis decides the winner.
1. Rolling without slipping
Write the moment of inertia as I = k m r2 (sphere k = 2/5, cylinder 1/2, shell 2/3, hoop 1). Along the slope m a = m g sinθ - f and about the centre I α = f r with a = α r for rolling. Eliminating the friction force gives a = g sinθ / (1 + k): 5/7, 2/3, 3/5 and 1/2 of g sinθ, against the full g sinθ for a frictionless slider. The time to roll a distance L is t = √(2L/a) and the speed at the bottom is v = √(2 g L sinθ / (1 + k)).
2. Where the energy goes
Dropping a height h = L sinθ releases m g h. A rolling body keeps the share 1/(1+k) as motion down the slope and k/(1+k) as spin. A hoop spends half of its energy on spinning, so it is slow; a sphere spends only 2/7.
3. The friction needed, and when the body slips
The static friction required for rolling is f = m g sinθ k/(1 + k). The ramp can supply at most μ m g cosθ, so rolling without slipping needs μ ≥ k tanθ / (1 + k). On a slicker ramp a body slides while it turns: a = g (sinθ - μ cosθ) and the spin rises only as α r = μ g cosθ / k. Friction then turns part of the energy into heat. The block slides only if tanθ > μb.
Model checks on this page
(1) The finish times of all five bodies from a separate step-by-step stick/slip integration are compared with the closed forms. (2) At the bottom, translational plus spin energy plus the heat made by friction equals m g L sinθ. (3) The critical friction coefficient found by bisection on the numerical simulation is compared with k tanθ/(1+k) for all four rolling shapes.
What to try
Run the Classic race and note that the order sphere, cylinder, shell, hoop never changes with mass, size or gravity. Use Icy ramp to make the hoop and shell slip first: the critical coefficient is highest for the hoop. Tilt the ramp to 40 degrees and watch the friction arrows grow. Make the block's friction larger than tanθ and it stays put.
Honest simplifications
Rigid bodies on a rigid plane, point contact, constant friction coefficient, no rolling resistance, no air drag, bodies released from rest at the same instant. The sphere, cylinder, shell and hoop are drawn with the same outer radius (0.15 m).