M052 · Rolling / friction constraints

Rolling Race Laboratory

Release a solid sphere, disk, thin hoop and custom annulus on the same incline. Adjust the slope, surface friction and initial rim spin; compare finish events, contact velocity and energy. The custom annulus makes the required static friction visible, while an initially slipping release can settle into pure rolling.

Recorded time $0\,\mathrm{s}$

Sphere acquired finish time $\text{Awaiting crossing}$

Disk acquired finish time $\text{Awaiting crossing}$

Hoop acquired finish time $\text{Awaiting crossing}$

Annulus acquired finish time $\text{Awaiting crossing}$

Sphere predicted finish time $0\,s$

Disk predicted finish time $0\,s$

Hoop predicted finish time $0\,s$

Annulus predicted finish time $0\,s$

Annulus recorded distance $0\,m$

Annulus recorded center speed $0\,\mathrm{m\,s^{-1}}$

Annulus recorded angular speed $0\,\mathrm{rad\,s^{-1}}$

Annulus bottom contact speed $0\,\mathrm{m\,s^{-1}}$

Annulus signed contact friction $0\,N$

Annulus normal force $0\,N$

Annulus required rolling friction magnitude $0\,N$

Annulus available static friction $0\,N$

Annulus minimum static coefficient for rolling $0$

Actual kinetic coefficient $0$

Annulus integrated contact heat $0\,J$

Maximum energy defect across all four lanes $0\,J$

DESIGN / PREDICT / VERIFY

Is the sphere always the fastest?

Mass of each body

m

1.00\,\mathrm{kg}

Outer radius of each body

R

0.25\,\mathrm{m}

Incline angle

\gamma

18.00\,{}^{\circ}

Distance to finish along incline

L

5.00\,\mathrm{m}

Static friction coefficient

\mu_s

0.50\,

Kinetic-to-static coefficient ratio

\mu_k/\mu_s

0.70\,

Custom annulus inner-radius ratio

R_{\mathrm{in}}/R

0.65\,

Initial downslope speed

v_0

0.00\,\mathrm{m\,s^{-1}}

Initial signed rim speed

R\omega_0

0.00\,\mathrm{m\,s^{-1}}

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

Drag the amber slope knob below the tracks. Initial rim speed is positive for counterclockwise spin; pure rolling downhill needs opposite signs of rim speed and center speed. The four lanes share physical scales. Lower grip until static friction fails, then compare the finish order and contact heat.

Rigid axisymmetric bodies, one straight incline, constant gravity and ideal Coulomb contact. Rolling resistance and air drag are absent. Kinetic friction is a fraction of the static coefficient. Constant-acceleration arcs are solved exactly between slip-zero events; the simulator admits rolling only when the required static friction is available. An ideal noiseless event clock records the first forward finish crossing, then each lane stops recording without modeling a physical stop.

Start with the essentials

Focus question: Does the usual finish order survive on a slippery surface?
One-sentence intuition: No slip requires both a zero bottom-contact velocity and an achievable friction force. Motion of the mass center alone cannot determine the contact regime.

Core mathematical model

Translation and rotation

ma=mg\sin\gamma+f,\qquad I\dot\omega=Rf,\qquad v+R\omega=0

Downhill translation is positive; counterclockwise rotation is positive. Friction is signed along the incline.

Admissible rolling

\beta=\frac{I}{mR^2},\quad a=\frac{g\sin\gamma}{1+\beta},\quad \mu_s\ge\frac{\beta}{1+\beta}\tan\gamma

The inequality is a feasibility condition, not the magnitude of actual friction.

Sliding contact

s=v+R\omega,\qquad f=-\mu_kN\,\operatorname{sgn}(s),\qquad N=mg\cos\gamma

At a release with zero slip, the solver uses the impending slip direction when static friction is insufficient.

Energy with contact heat

K=\frac12mv^2+\frac12I\omega^2,\quad Q=\int\mu_kN|s|\,\mathrm{d}t,\quad K+Q-K_0=mgx\sin\gamma

Ideal static contact dissipates no energy even though it redistributes energy between translation and rotation.

Common difficulties

Friction acts on slip

Typical misconceptionFriction must always oppose the mass-center velocity.

Better mental modelAn initially spinning release can make friction accelerate the center downhill while reducing contact slip.

A rotating body can slide

Typical misconceptionA visible rotation proves pure rolling.

Better mental modelInspect bottom-contact velocity and the static-friction budget.

Finish is a recording boundary

Typical misconceptionA body physically stops at the finish line.

Better mental modelThe instrument ends that lane’s recording at its exact crossing and leaves the last acquired state on screen.

Run the experiment

01
Race with enough grip
Record all finish events from the rolling preset.
What to observe: The sphere leads, then the disk; the annulus and hoop depend on their mass distributions.
02
Cross the constraint
Lower static friction and inspect the custom annulus threshold.
What to observe: Slip becomes nonzero when the rolling requirement is no longer feasible.
03
Remove contact torque
Choose the frictionless preset.
What to observe: All four finish together and no contact heat accumulates.
04
Catch a spinning release
Choose the slipping-release preset and scrub through contact capture.
What to observe: Slip reaches zero at an exact event; rolling then persists if static friction is sufficient.