---
title: "Capstan & Belt Friction Interactive Lab | Sandbox Physics"
description: "With a rope wrapped around a drum, how little force can hold a heavy load?"
canonical_url: "https://sandboxphysics.com/labs/capstan-friction/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/capstan-friction/"
last_modified: "2026-10-27"
---

M030 · Cable shape and friction

# Capstan & Belt Friction

**With a rope wrapped around a drum, how little force can hold a heavy load?** Very little: the holding force can fall exponentially with friction coefficient times wrap angle, so a few turns can hold a heavy load. That minimum applies only when the rope is about to slip. A rope at rest can sit anywhere between the two slip limits.

Pull on one end of a rope wrapped around a fixed drum and find the range of pulls that keeps the load still. Probe the tension along the contact with noisy readings and see that a resting rope can use less friction than the material coefficient allows.

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

The rope is massless, flexible and flat, and it rests on a fixed cylindrical drum. The Coulomb friction coefficient is the same everywhere, the rope wraps part way around the drum in one plane, and there is no V-groove to add grip. Pull stations set both end tensions. The tension profile shown is one possible static profile; static friction alone does not fix a unique one. Past either limit the model reports slip but does not simulate the motion that follows.

Physics tutorial

## Friction permits a holding range

**Background** Wrap a flexible rope around a fixed, rough drum, and a small pull on one end can hold a larger pull on the other.

**Why it matters** The force balance in MIT's capstan example gives the pull at which the rope is about to slip. This Lab shows both limits: too little pull and the load slips, too much and the rope slips the other way.

Start with the essentials

- **Focus question**: Does a rope at rest always use the same amount of friction?
- **One-sentence intuition**: Static friction has a maximum, and the rope uses all of it only when it is about to slip.

### Core mathematical model

#### The friction limit on each short arc

$$
\left|\frac{\mathrm d\ln T}{\mathrm d\alpha}\right|\leq\mu_s
$$

The rope presses on each short arc of the drum, and that normal force sets the most friction the arc can supply. The angle is measured from the effort end toward the load.

#### The range of pulls that holds the load

$$
W e^{-\mu_s\beta}\leq F\leq W e^{\mu_s\beta}
$$

With too little effort the load side slips; with too much the rope would have to slip the other way. At either equality the rope is about to slip.

#### One possible static tension profile

$$
T(\alpha)=F e^{k\alpha},\qquad k=\frac{\ln(W/F)}{\beta}
$$

The Lab picks a profile whose logarithm has a constant slope. It is allowed whenever that slope stays within the friction limit, but static contact alone does not pick one profile over the others.

#### Radius changes the force per metre and the torque

$$
n=\frac TR,\qquad |f|=\frac{|k|T}{R},\qquad \tau_{\mathrm{hold}}=R(W-F)
$$

The normal and friction forces are spread along the contact, so they are measured in force per metre. A positive holding torque opposes the positive torque that the rope applies to the fixed drum. Nothing moves, so the power is zero.

#### A fit measures friction in use, not the coefficient

$$
\ln T_i^{\mathrm{obs}}\approx b+\widehat k\alpha_i
$$

The fitted slope approaches the static coefficient only when you know from elsewhere that the rope is about to slip. Below that point, the fit measures the friction actually in use in the chosen profile.

### Common difficulties

#### Static friction is a limit, not a fixed value

**Typical misconception** A rope that holds must be using all the friction it can get.

**Better mental model** The limit only caps the friction. Many different tension profiles stay below it, and each of them holds the rope still.

#### Slip has two directions

**Typical misconception** For a fixed load, pulling harder is always safer.

**Better mental model** Past the upper limit the effort wins, and the rope slips the other way.

#### Friction in use is not the material coefficient

**Typical misconception** Fitting the tension of a held rope measures the material's friction coefficient.

**Better mental model** Below the slip limit, the fit only gives the slope of the chosen static profile.

### Run the experiment

1. 01

   #### Find the holding range

   Choose Static hold and compare the effort with both limits.

   **What to observe:** The chosen tension profile lies between the two dashed limit curves.

2. 02

   #### Cross each limit

   Try the Insufficient effort and Reverse slip demand presets.

   **What to observe:** The red curve is the balance you asked for, which friction cannot supply, so the Lab takes no readings.

3. 03

   #### Change only the radius

   Keep the wrap angle, friction coefficient and end tensions fixed, and change the drum radius.

   **What to observe:** The holding range stays the same, but the torque and the force per metre change.

4. 04

   #### Fit the tension slope

   Go back to a held trial, add noise and scan the contact.

   **What to observe:** The fitted slope comes close to the slope of the chosen profile, not to the material coefficient.

Physical conclusion

A held rope sits between two slip limits. Knowing that friction allows a static state tells you nothing about how the rope moves once it slips.

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)