Rolling Race and Radial Mass

Does a solid sphere always win a race down a slope? The mass distribution and available grip decide. With enough static friction, a uniform sphere beats a disk and a hoop, but a custom wheel with more mass near its axle can be faster still. If grip is insufficient, slipping changes the energy split; with no friction, all four centres accelerate equally.

Race a sphere, disk, hoop and radial-density wheel. Compare translational and rotational energy, static friction and the transition from sliding to rolling.

Gold body translational energy $4.24\,\mathrm{J}$

Gold body rotational energy $3.02\,\mathrm{J}$

Gold body signed contact slip $0\,\mathrm{m\,s^{-1}}$

Gold body signed friction $-1.26\,\mathrm{N}$

Model assumptions

Four equal-mass rigid axisymmetric bodies move on separate straight lanes under constant gravity. The sphere is solid, the disk uniform and the hoop ideally thin. The gold wheel has an optional central hole and an areal density proportional to a power of distance from its axle; its inertia comes from the corresponding mass integral. Negative density exponents without a hole have an integrable ideal centre singularity. Colour indicates relative density schematically. Visible hoop thickness is schematic. Coulomb static friction is tested before imposing pure rolling; kinetic friction is a chosen fraction of static friction. Constant-load arcs and slip-zero events are solved analytically. Downhill motion and counterclockwise rotation are positive, so the bottom-point velocity is centre velocity plus signed rim speed. There is no air drag, rolling resistance, deformation or axle torque. Each lane ends its model record at its first forward finish crossing; frozen geometry is a retained record, not a simulated brake. Sliders recompute independent trials; moving mass during a race is not simulated. All finish times are ideal model predictions, not measurements.

Focus question: Is the sphere always the first body to finish?
One-sentence intuition: The usual order assumes both a particular mass distribution and enough grip. Change either one and the order can change.

Core mathematical model

Translation, rotation and bottom-point motion

\begin{aligned}ma&=mg\sin\theta+f\\I\dot\omega&=Rf,\qquad u=v+R\omega\end{aligned}

Downhill translation and counterclockwise rotation are positive. Static or kinetic friction acts through the same contact point.

Test grip before imposing rolling

\begin{aligned}\beta&=\frac I{mR^2},\qquad a=\frac{g\sin\theta}{1+\beta}\\\mu_s&\ge\frac{\beta}{1+\beta}\tan\theta\end{aligned}

The acceleration formula is valid only when the required static friction is available. Otherwise the body slides and the kinetic coefficient is used.

A radial-density wheel

\begin{aligned}\rho(r)&\propto (r/R)^p\\\beta&=\frac{p+2}{p+4}\frac{1-a^{p+4}}{1-a^{p+2}}\end{aligned}

The inner radius divided by the outer radius is the hole ratio. The exponent changes mass distribution while total mass stays fixed. Negative exponents with no hole have an integrable ideal centre singularity.

A wheel can beat the sphere

\begin{aligned}\beta_{\rm gold}&=\tfrac13\quad(p=-1,\ a=0)\\\beta_{\rm sphere}&=\tfrac25\end{aligned}

In the pure-rolling comparison the gold wheel can have less rotational inertia per mass and radius than the uniform sphere. These are ideal density distributions, not manufacturing specifications.

Follow both kinetic energies

\begin{aligned}K_t&=\tfrac12mv^2,\qquad K_r=\tfrac12I\omega^2\\K_t+K_r+Q&=E_0+mgs\sin\theta\end{aligned}

Static contact has zero relative motion and produces no heat here. Sliding dissipates energy; the initial kinetic energy is retained for spinning or moving releases.

Common difficulties

Static friction need not dissipate energy

Typical misconceptionAny friction force must turn mechanical energy into heat.

Better mental modelIn pure rolling, the contact point is instantaneously at rest. The static force exchanges translation and rotation without heat in this rigid model.

Finish times are computed records

Typical misconceptionA frozen body at the finish has physically stopped.

Better mental modelEach lane retains the first forward-crossing frame. No brake or end collision is calculated, and every displayed finish time is an ideal model prediction.

Run the experiment

01
Start with enough grip
Use Enough grip to roll.
What to observe: The sphere finishes before the uniform disk and hoop; all four bottom points stay at rest relative to the incline.
02
Change the mass distribution
Use Put mass near the axle, then Move mass to the rim.
What to observe: The gold wheel changes its rotational energy share and can move from ahead of the sphere to near the hoop.
03
Remove the constraint
Compare A slippery race with All four centres tie.
What to observe: Insufficient grip produces slip and heat. With no friction the centres accelerate equally, although their rotations can differ.