M052 · Rolling and friction

Rolling Race Laboratory

Does a solid sphere always win a race down a slope? Only when friction is strong enough for every body to roll without slipping from the start. Then the solid sphere wins, because it puts the smallest share of its energy into spinning. With no friction at all, every body slides down with the same acceleration, and a body that starts out spinning can keep slipping.

Race a solid sphere, a disk, a thin hoop and a custom annulus down a three-dimensional incline that you can rotate. Find each finish and the moment a body stops slipping and starts rolling, follow a point on the rim, and check the recorded energy. Change the slope, the surface friction and the starting rim spin, then compare finish times, contact velocity and energy. The custom annulus shows how much static friction rolling needs, and a body released while slipping can settle into pure rolling.

Annulus angular acceleration $0\,\mathrm{rad\,s^{-2}}$

Annulus torque about center $0\,\mathrm{N\,m}$

Annulus contact heating rate $0\,\mathrm{W}$

First predicted slip-to-roll event $\text{No transition}$

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 $0\,J$

Predict the motion, then test your design

Does a solid sphere always win a race down a slope?

Forces at their application pointsWeight is blue, support violet, friction red and string tension amber, all at one force scale per trial.Bottom material-point velocityThe teal arrow shows the signed slip relative to the surface, which is zero in pure rolling. Velocity arrows use their own scale.Follow a material point on the rimThe white marker turns with the recorded body angle. The violet hub and the string have no mass.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 knob below the tracks to change the slope. A positive initial rim speed means counterclockwise spin; rolling downhill without slipping needs the rim speed and the center speed to have opposite signs. All four lanes use the same physical scales. Lower the grip until static friction can no longer hold, then compare the finishing order and the heat generated at the contact.

The model uses rigid, axially symmetric bodies on one straight incline, with constant gravity and ideal Coulomb friction at the contact. Rolling resistance and air drag are left out. The kinetic friction coefficient is a chosen fraction of the static one. Between the moments when slip reaches zero, the acceleration is constant and the motion is solved exactly. A body rolls only when the static friction it needs is available. An ideal, noise-free clock records the first forward crossing of the finish line; after that, each lane stops recording, and no physical stop is modeled.

Start with the essentials

Focus question: Does the usual finish order survive on a slippery surface?
One-sentence intuition: Rolling without slipping needs two things: the bottom point must be at rest, and the friction required must be within what the surface can supply. Watching the center of mass alone cannot tell you whether a body rolls or slides.

Core mathematical model

Translation and rotation

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

Downhill motion and counterclockwise rotation count as positive. Friction is positive when it points down the incline.

When rolling is possible

\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 tells you whether rolling is possible. It does not give the actual friction force.

Sliding contact

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

If a body starts with zero slip but static friction is too weak, the solver sets the direction of kinetic friction from the way the contact is about to slip.

Energy, including heat at the contact

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 turns no energy into heat, although it shifts energy between translation and rotation.

Common difficulties

Friction opposes slipping, not motion

Typical misconceptionFriction must always oppose the mass-center velocity.

Better mental modelIf the body starts out spinning, friction can speed the center up downhill while it reduces the slip at the contact.

A rotating body can slide

Typical misconceptionA visible rotation proves pure rolling.

Better mental modelCheck the velocity of the bottom contact point, and compare the friction needed with the static friction available.

The finish line ends the recording

Typical misconceptionThe bodies stop at the finish line.

Better mental modelEach lane stops recording at the exact moment its body crosses, and the last recorded state stays on screen. Nothing brakes the body.

Run the experiment

01
Watch the contact on the incline
Switch to the annulus contact camera and follow the white marker on the rim. Turn the force arrows and the bottom-point velocity on and off separately.
What to observe: The body turns even while it slides. The teal slip arrow disappears only once the contact locks into rolling.
02
Find each finish
Press First finish and Next finish, then open Energy ledger at that moment.
What to observe: Each lane stops recording at its own finish, without braking. The energy chart covers only the custom annulus, up to the moment recorded so far.
03
Race with enough grip
Choose Enough grip to roll and press Record all finish events.
What to observe: The sphere finishes first, then the disk. Where the annulus and the hoop finish depends on how their mass is distributed.
04
Lower the friction until rolling fails
Lower Static friction coefficient and watch the friction threshold for the custom annulus.
What to observe: Once the surface can no longer supply the friction that rolling needs, the slip becomes nonzero.
05
Remove the friction torque
Choose Frictionless tie.
What to observe: All four finish together and no contact heat accumulates.
06
Watch a spinning body start to roll
Choose Slipping release, then rolling, and scrub through the moment the contact starts rolling.
What to observe: The slip reaches zero at a definite moment. After that the body keeps rolling, as long as static friction is large enough.

Core mathematical model

Translation and rotation

When rolling is possible

Sliding contact

Energy, including heat at the contact

Common difficulties

Friction opposes slipping, not motion

A rotating body can slide

The finish line ends the recording

Run the experiment

Watch the contact on the incline

Find each finish

Race with enough grip

Lower the friction until rolling fails

Remove the friction torque

Watch a spinning body start to roll