---
title: "Connected Particles and Pulleys"
description: "Two bodies joined by a string share one acceleration and pull on each other equally and oppositely. Write F = ma for each body, then solve the two equations together for the acceleration and the tensi"
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source: https://lightmysky.com/learn/mathematics/connected-particles-and-pulleys-mt_pZ6zV0RFdC.md
retrieved: 2026-09-12
---

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# Connected Particles and Pulleys

Two bodies joined by a string share one acceleration and pull on each other equally and oppositely. Write F = ma for each body, then solve the two equations together for the acceleration and the tension.

Subject: Mathematics · Area: Mechanics · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/connected-particles-and-pulleys-mt_pZ6zV0RFdC

## Ready when they can

- Write the equation of motion for each of two connected masses
- Solve the pair for the shared acceleration and the tension
- Explain why the tension is the same throughout a light string over a smooth pulley

## Lesson: One string, one shared pull

Two bodies joined by a light string over a smooth pulley share one acceleration and one tension. The light string cannot keep any extra force and the smooth pulley adds no friction, so the pull matches all along. Write F = ma for each body with the same a and T, then solve the pair.

**Example.** Hang 6 kg and 2 kg either side with g = 10: weights 60 N and 20 N. For the heavy side 60 - T = 6a, and for the light side T - 20 = 2a. Net pull 40 N on 8 kg gives a = 5 m/s squared, and T = 20 + 2 times 5 = 30 N.

Always start with one equation of motion per body, each with its own forces but shared unknowns. Adding the equations kills T and leaves a, then substituting back finds T. Take the direction of motion as positive for each body and keep it fixed.

**Tip.** Check answers by asking what each side feels: tension 30 N sits between 20 N and 60 N, holding the light side back and the heavy side up. If T ever lands outside the two weights, the algebra slipped. Equal sides would balance at zero acceleration.

**Recap.** Give each body its own F = ma with shared a and T, then solve the two equations together.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Simultaneous Equations](https://lightmysky.com/learn/mathematics/simultaneous-equations-mt_mqgu72aCMz)
- [Newton's Second Law in Two Dimensions](https://lightmysky.com/learn/mathematics/newtons-second-law-in-two-dimensions-mt_yNUFF3lxlj)