---
title: "Rolling Race Laboratory Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/rolling-race/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/rolling-race/"
last_modified: "2026-10-22"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Rolling is a friction constraint

**Background** A sphere and a hoop need different amounts of torque to spin up. Friction at the contact supplies that torque, and the same friction also changes how fast the center speeds up.

**Why it matters** OpenStax University Physics, section 11.1, derives rolling motion and how much static friction it needs. This race checks whether that friction is available before applying the rolling formula.

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 misconception** Friction must always oppose the mass-center velocity.

**Better mental model** If 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 misconception** A visible rotation proves pure rolling.

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

#### The finish line ends the recording

**Typical misconception** The bodies stop at the finish line.

**Better mental model** Each 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

1. 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.

2. 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.

3. 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.

4. 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.

5. 05

   #### Remove the friction torque

   Choose Frictionless tie.

   **What to observe:** All four finish together and no contact heat accumulates.

6. 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.

Physical conclusion

Released from rest with good grip, the sphere wins. Without friction, all four centers of mass accelerate equally, and a spinning body can keep slipping.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)