---
title: "Pulley Network & Mechanical Advantage Interactive Lab | Sandbox Physics"
description: "When a pulley system lets you pull with less force, where does the saved force go? Into distance."
canonical_url: "https://sandboxphysics.com/labs/pulley-network/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/pulley-network/"
last_modified: "2026-10-27"
---

M028 · Rope constraints and work

# Pulley Network & Mechanical Advantage

**When a pulley system lets you pull with less force, where does the saved force go?** Into distance. With more supporting strands you pull with less force, but you must pull proportionally more rope, so ideally the work stays the same. Real pulleys also have mass and bearing friction, so adding wheels can stop paying off.

Rig the rope over a fixed pulley, a movable pulley or a four-strand compound system. Follow the rope length, the speed of each pulley and the unequal tensions in the strands, then compare the work you put in with the energy the load gains and the energy lost in the bearings.

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

The Lab offers three fixed rope arrangements in a plane. The rope is massless, taut and cannot stretch, and it does not slip on the pulley rims. The mass of the moving assembly includes the pulleys that move with it, and every pulley has the chosen moment of inertia and viscous axle coefficient. The pulley radius is fixed at 18 cm. A frictionless guide keeps the moving carriage from pitching. The record ends before anything collides or when a strand tension first reaches zero, so a slack rope is not simulated.

Physics tutorial

## Mechanical advantage spends rope travel

**Background** A single fixed wheel redirects force. Movable wheels let several strands share a load, but the free end must travel farther.

**Why it matters** The OpenStax section on simple machines links the number of supporting strands to mechanical advantage. This Lab extends that idea to moving systems, using three drawn rope arrangements in which the rope does not slip.

Start with the essentials

- **Focus question**: Does adding more wheels always save more force?
- **One-sentence intuition**: The number of supporting strands sets the ideal force and distance ratios. Real pulleys spin at different speeds, take force to speed up and lower the rope tension a little at each wheel.

### Core mathematical model

#### Fixed rope length links the motions

$$
\Delta s_{\mathrm{in}}=n\Delta q,\qquad n\in\{1,2,4\}
$$

Upward load motion and downward motion of the free end both count as positive. In all three rope arrangements, the rope length stays constant.

#### Each pulley turns at a different speed

$$
|\omega_j|=\frac{j|\dot q|}{R},\qquad S=\sum_{j=1}^{n}j^2
$$

Number the pulleys from the anchored end toward the free end. In the compound rig, the moving pulleys get odd numbers and the fixed pulleys even numbers, and the two sets turn in opposite directions.

#### Equation of motion for the whole assembly

$$
\left(m+\frac{IS}{R^2}\right)\ddot q+\frac{BS}{R^2}\dot q=nF-mg
$$

The assembly mass includes the pulleys that move with the load. Every pulley has the same chosen moment of inertia and axle drag, whatever size it looks in the drawing.

#### Trace tension back through the rope

$$
T_n=F,\qquad T_{j-1}=T_j-\frac j{R^2}(I\ddot q+B\dot q)
$$

Together, the supporting tensions accelerate the assembly. A rope can only pull, so every tension must stay at or above zero. The taut-rope model stops when the first tension reaches zero, before it would give wrong answers.

#### Less force does not mean less work

$$
\Delta K+mg\Delta q=F\Delta s_{\mathrm{in}}-\int_0^t\frac{BS}{R^2}\dot q^2\,\mathrm dt
$$

The kinetic energy includes both the moving load and the spinning pulleys. For a very slow pull with no losses, the force equals the weight divided by the number of strands; acceleration and losses change that.

### Common difficulties

#### Count strands, not wheels

**Typical misconception** Every added wheel doubles the advantage.

**Better mental model** Only strands pulling upward on the moving assembly contribute to its support.

#### Motion changes the force balance

**Typical misconception** Input force times strand count always equals weight.

**Better mental model** When the assembly accelerates, part of the force goes into speeding it up and spinning the pulleys, so those terms cannot be ignored.

#### A rope cannot push

**Typical misconception** The equations still work if a tension turns negative.

**Better mental model** A negative tension would mean the rope pushes. The model stops the moment any strand goes slack instead.

### Run the experiment

1. 01

   #### Redirect a force

   Choose the Change direction preset and compare how far the load and the free end move.

   **What to observe:** The direction of the force changes, but the load and the free end move the same distance.

2. 02

   #### Trade force for distance

   Compare the Half the pull and Quarter the pull presets.

   **What to observe:** Both rigs drive the load the same way, but with four strands you pull with half the force over twice the distance.

3. 03

   #### Look at each pulley

   Choose Bearings cost work. Open Position sensor and more readings and press Complete record.

   **What to observe:** The pulleys spin at different speeds and the strand tensions differ. The work you put in equals the energy the load and pulleys gain plus the energy lost in the bearings.

4. 04

   #### Find where the model stops

   Lower the moving assembly mass, then raise the downward pull and the axle coefficient.

   **What to observe:** The record ends when the load reaches its travel limit or a tension first hits zero. The Lab never simulates a negative tension.

Physical conclusion

Count the supporting strands, check the tension in each one, and compare the work you put in with the energy of everything that moves.

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)