---
title: "Atwood Machine Interactive Lab | Sandbox Physics"
description: "In an Atwood machine, why are the rope tensions on the two sides unequal? Because the pulley has rotational inertia and the rope has weight."
canonical_url: "https://sandboxphysics.com/labs/atwood-machine/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/atwood-machine/"
last_modified: "2026-10-27"
---

M014 · Rope constraints and work

# Atwood Machine

**In an Atwood machine, why are the rope tensions on the two sides unequal?** Because the pulley has rotational inertia and the rope has weight. Changing the pulley's spin takes a torque, and that torque can only come from different tensions on the two sides. As the rope runs over, its own weight moves from one side to the other and changes the imbalance.

Run an Atwood machine whose rope follows its real path over the wheel, and compare an ideal setup with a heavy pulley, a heavy rope and a damped axle. Read the tension at both masses and at both sides of the rim, sample the position, and fit an acceleration to the measurements.

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

The pulley is rigid and fixed. The rope stays taut, does not stretch and does not slip on the rim. The heavy rope has uniform mass per length. Axle drag is viscous, with no static friction. The run ends at the limit of travel, before anything hits. Shapes and rope thickness are schematic, and elasticity, air drag and material calibration are left out.

Physics tutorial

## A rope constraint does not guarantee equal tension

**Background** Two hanging masses turn a fixed pulley. As the left mass rises, the right one falls, and the pulley and every piece of a heavy rope gain kinetic energy too.

**Why it matters** OpenStax derives Newton's laws for connected bodies and the work done in rotation. This rig extends those laws to a uniform heavy rope and an axle with viscous drag; the derivation is recorded in the repository.

Start with the essentials

- **Focus question**: How much acceleration does the ideal prediction miss?
- **One-sentence intuition**: Both the weight imbalance that drives the motion and the inertia that resists it must include the rope. A spinning pulley needs a torque, and unequal tensions at the rim supply it.

### Core mathematical model

#### One coordinate for the whole rope

$$
\ell_1=\ell_0-q,\quad\ell_2=\ell_0+q,\quad\ell_0=2\,\mathrm m,\quad\omega=-\dot q/R
$$

The coordinate is positive when the left side moves up. Both hanging lengths stay positive throughout the run, and the rope over the top semicircle keeps a constant length.

#### Inertia and driving force

$$
\left[m_1+m_2+\lambda(2\ell_0+\pi R)+\frac I{R^2}\right]\ddot q+\frac B{R^2}\dot q=(m_2-m_1)g+2\lambda gq
$$

A heavy rope adds translational inertia, and its weight imbalance changes with position. The viscous axle torque opposes the spin; it is not Coulomb friction at the axle.

#### Rope forces on the masses

$$
T_1=m_1(g+\ddot q),\qquad T_2=m_2(g-\ddot q)
$$

These are the upward arrows on the two masses. Each mass is treated on its own, without any of the rope.

#### Tension at the rim includes the hanging rope

$$
T_{1r}=(m_1+\lambda\ell_1)(g+\ddot q),\quad T_{2r}=(m_2+\lambda\ell_2)(g-\ddot q)
$$

The difference between the two rim forces turns the wheel, accelerates the rope over the top arc and overcomes the axle loss. The rim forces are equal only when the wheel is massless and frictionless and the rope is massless too.

#### Energy balance with the rope included

$$
\Delta K+\Delta U=-\int_0^t\frac B{R^2}\dot q^2\,\mathrm dt,\quad \Delta U=(m_1-m_2)g\Delta q-\lambda g(q^2-q_0^2)
$$

Kinetic energy includes both masses, the whole rope and the spinning pulley. The power lost in the bearing is integrated on its own, not worked out from the energy difference.

#### Fit the measurements, not a theoretical answer

$$
q_i\simeq c_0+c_1t_i+\tfrac12a_{\mathrm{fit}}t_i^2
$$

The fit needs at least three samples. It returns a single acceleration, and that number depends on the model: with a heavy rope or damping the motion is not exactly quadratic. The RMS value measures how far the positions sit from the fit; it does not guarantee an accurate acceleration.

### Common difficulties

#### One rope, different tensions

**Typical misconception** The rope has the same tension everywhere.

**Better mental model** Give the pulley some inertia or the rope some mass, then compare the four tension readings.

#### Equal masses can still move

**Typical misconception** Equal masses always stay still.

**Better mental model** With a heavy rope, release the masses away from the center. The side with more rope hanging is also heavier.

#### A good-looking fit can still be biased

**Typical misconception** A small position RMS proves the acceleration is correct.

**Better mental model** Change the time span and look at the instantaneous acceleration. Sensor noise and a changing acceleration are different kinds of error.

### Run the experiment

1. 01

   #### Reproduce the ideal result

   Choose Equal tension. Open Position sensor and more readings, set the sensor noise to zero, and press Complete record.

   **What to observe:** The quadratic fit matches the constant acceleration, and the rope forces on the two masses agree.

2. 02

   #### Add a heavy pulley

   Choose Spin needs torque and compare the end times.

   **What to observe:** The acceleration drops, the two rim tensions move apart, and the rotational energy grows.

3. 03

   #### Unbalance the rope

   Choose Longer side wins with equal masses, then reverse the release offset.

   **What to observe:** The motion reverses direction even though the masses are still equal.

4. 04

   #### Check a measurement

   Add axle drag, change the sample period and noise, then export the positions.

   **What to observe:** Tell apart three effects: sensor noise, the error of assuming constant acceleration, and energy lost in the bearing.

Physical conclusion

Compare the fit to the measured positions with the instantaneous acceleration, and make sure the energy balance includes every moving part.

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)