# Atwood Machines

1 Fixed Pulley

Let’s begin our analysis by studying the fundamental Atwood machine – a fixed pulley with two masses.

**Question**

Find the accelerations of the masses and the tension in the string.

We solve problems involving Atwood machines by using F=ma equations and an equation of conservation of string.

**F=ma equations**

**Conservation of String**

**Accelerations**

**Tension**

We can use the idea of effective mass of a fixed pulley to simplify our analysis in fixed pulley combinations.

**Effective Mass of a Fixed Pulley**

1.1 Fixed pulley combination

**Question**

Find the accelerations of the masses and the tension in the string.

**F=ma equations**

We can use the idea of effective acceleration of a fixed pulley to simplify finding the equation for conservation of string in fixed pulley combinations.

**Effective Acceleration of Fixed Pulley**

**Conservation of String**

**Accelerations**

**Tension**

2 Free Pulley

Now let’s analyze systems with free pulleys in which they are not attached to supports.

**Question**

Find the accelerations of the masses and the tension in the string.

**F=ma equations**

**Conservation of String**

**Accelerations**

**Tension**

2.1 Strings

**Question**

Find the accelerations of the masses and the tension in the string.

**F=ma equations**

**Conservation of String**

**Accelerations**

**Tension**

2.2 Strings

We can generalize this to strings.

**Question**

Find the accelerations of the masses and the tension in the string.

**F=ma equations**

**Conservation of String**

**Accelerations**

**Tension**

2.3 Free Pulley In Between

**Question**

Find the accelerations of the masses and the tension in the string.

**F=ma equations**

**Conservation of String**

**Accelerations**

**Tension**

2.4 Massless Free Pulley

**Equivalence of Free Pulleys**

The subject of massless free pulleys is an interesting topic in Atwood machines. We can use a key idea derived here to solve more interesting problems.

**Question**

Find the acceleration of the free pulley.

**Procedure**

Assume the free pulley has mass . This is equivalent to the case with a free pulley derived above.

We have . The **force on the pulley must be zero** so . We can use the first equation and third equation to solve for . Note that we cannot use the second equation.

**Question**

Find the accelerations of the masses and the tension in the string.

**Accelerations**

Note that the net force on the left massless pulley must be zero.

**Tension**

The techniques developed here can be applied to any Atwood Machine problem.

For more information see: Introduction to Classical Mechanics (David Morin)