Rolling Without Slipping

Release a hoop, disk, sphere, and spherical shell together down a ramp. They finish in a fixed order every time — and it has nothing to do with their mass or size.

Rolling Without SlippingLive

a shape race down the ramp

Controls

Presets

Incline angle (°)

All these shapes share the same mass and radius, yet a solid sphere beats a hoop every time. Acceleration = g·sinθ / (1 + I/mR²): mass concentrated near the axis (small I) accelerates faster — and the result is independent of mass and radius.

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Data Inspector

Hoop accel1.12 m/s²

Disk accel2.24 m/s²

Sphere accel2.40 m/s²

Shell accel2.01 m/s²

Governing equation

a = \frac{g sin θ}{1 + I / m R ^{2}} = \frac{9.81 sin 2 0 ^{\circ}}{1 + I / m R ^{2}} = \frac{3.36}{1 + I / m R ^{2}} m/s^{2}

Reading this result: At 20° the solid sphere leads at 2.40 m/s² while the hoop trails at 1.12 m/s² — a 2.14× gap fixed entirely by mass distribution, independent of the incline and of every shape mass and radius.

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How it works

For an object rolling without slipping, a = g·sinθ / (1 + I/mR²). The bracket depends only on how mass is distributed: a solid sphere (I = 0.4mR²) accelerates fastest, a hoop (I = mR²) slowest. Mass and radius cancel out entirely. Educational tool.

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Frequently asked questions

Is this rolling without slipping ramp tool really free?▾

Yes. Rolling Without Slipping runs entirely in your browser using your device's own compute, so local use is free forever. You only pay Compute Tokens if you scale a job to the cloud.

Do I need to install anything?▾

No. Everything runs client-side in a modern browser — no downloads, no license, no account required to start.

Can I save or share my simulation?▾

Create a free account to save projects, and use a shareable embed or minted DOI to publish a live, interactive version anywhere.

How accurate are the results?▾

The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.