Rolling Race (Moment of Inertia)
Release a hoop, a solid disk, and a solid sphere from rest together — each on its own ramp curve (straight incline, brachistochrone, exponential decay, circular arc, or parabola) — and see how both the SHAPE (moment of inertia) and the PATH decide who reaches the bottom first.
Preview — sign up free to change the numbers yourself
What you can adjust
- Ramp angle
- 5 – 40 deg
- Ramp length
- 1 – 10 m
- Sphere's ramp shape
- 0 – 4
- Disk's ramp shape
- 0 – 4
- Hoop's ramp shape
- 0 – 4
A steeper ramp makes every shape roll faster, but never changes which one wins — that's decided by shape alone, not steepness.
Distance from the release point to the finish line at the bottom of the ramp.
The path the sphere rolls down — Straight incline, Brachistochrone (cycloid, mathematically the fastest possible descent curve), Exponential decay (steep drop right away, then flattening), Circular arc, or Parabola (both start level and curve down). Every shape connects the same start and finish points.
The path the disk rolls down — independent of the sphere's and hoop's choices. Same five shapes: Straight incline, Brachistochrone, Exponential decay, Circular arc, Parabola.
The path the hoop rolls down — independent of the sphere's and disk's choices. Same five shapes: Straight incline, Brachistochrone, Exponential decay, Circular arc, Parabola.
About this simulation
What it shows
A hoop, a solid disc and a solid sphere roll down their own ramps side by side. Each ramp can have a different curved shape, and the finish order is shown.
The key idea
Rolling without slipping shares the energy between moving forward and spinning. The more of the mass sits near the rim, the slower the body: acceleration is g sin θ / (1 + I/mR²), and m and R cancel out.
Try this
Keep all three on Straight incline and run it: the sphere wins, then the disc, then the hoop. Then change Hoop's ramp shape to Brachistochrone (cycloid) and compare its time.
Gravity is fixed at Earth's value, and the bodies roll without slipping on every ramp shape.