Rolling Race
Release a ring, disc and sphere down a ramp — and see who reaches the bottom first.
About this simulation
Choose which bodies race down an incline rolling without slipping; the one with the smallest I/mR² (the solid sphere) wins, independent of mass and radius. An energy split shows each body's total dividing into translational and rotational parts, making the counter-intuitive result click.
How to use it
- Choose the bodies to race with the Racing pills: Ring, Disc, Sphere and Shell.
- Set the Incline angle and Ramp length, then watch the race and note who wins, with the winner's time and acceleration printed beside it.
- Change the angle and check that the finishing order never changes, only the times.
- Read the KE and rot E columns in the table above the ramp to see how each body's energy is split between moving forward and spinning.
- Try the presets 'Solid vs hollow', 'Disc vs ball' and 'Sphere vs ring'.
What to notice
- Every body rolling without slipping has acceleration g sin θ/(1 + k), where k = I/mR² depends only on its shape.
- The solid sphere (k = 0.4) always wins, then the disc (0.5), the hollow shell (2/3) and the ring (1) last.
- At the bottom every body has the same total energy per kilogram; the losers have put more of it into spinning and less into moving forward.
- Neither mass nor radius appears in the result, which is why the simulation has no slider for them.
The formulas
- a = g sin θ / (1 + k), k = I / mR²
- Acceleration of a body rolling without slipping down an incline of angle θ; I is its moment of inertia, m its mass and R its radius.
- v = √( 2gh / (1 + k) )
- Speed at the bottom after descending a height h.
- KE_trans / KE_total = 1 / (1 + k)
- Fraction of the kinetic energy that goes into forward motion rather than rotation.
Questions
Which reaches the bottom first, a ring or a disc rolling down an incline?
The disc reaches the bottom first. For a disc I = ½mR², so k = 0.5, while for a ring I = mR², so k = 1. The acceleration g sin θ/(1 + k) is therefore larger for the disc. The ring has all its mass at the rim, so more of its energy goes into rotation. In the simulation, the 'Sphere vs ring' and 'Disc vs ball' presets let you compare pairs.
Why does a solid sphere roll down faster than a hollow sphere?
A hollow sphere has its mass far from the axis, giving I = ⅔mR², while a solid sphere has I = ⅖mR². With a smaller moment of inertia, the solid sphere needs less of its energy for spinning, leaving more for forward motion, so it accelerates faster. This is true whatever the masses and radii. In the simulation, the 'Solid vs hollow' preset races the two.
Does the mass of a rolling body affect how fast it rolls down an incline?
No. The gravitational force along the slope and the moment of inertia are both proportional to mass, so mass cancels from a = g sin θ/(1 + I/mR²). The radius cancels too, because I is proportional to R². Only the shape, through the ratio k = I/mR², decides the acceleration. A small steel ball and a large wooden ball roll down together.
What is the kinetic energy of a body rolling without slipping?
It is the sum of translational and rotational parts: KE = ½mv² + ½Iω². Since v = ωR for rolling without slipping, this becomes ½mv²(1 + k), where k = I/mR². For a solid sphere, 5/7 of the kinetic energy is translational; for a ring, only half. In the simulation's table, compare the KE and rot E columns at the finish: the ring's two are equal, while the solid sphere's KE is far the larger.
