**Emmette Cox
Product Management Coordinator for Physical Science
**

September 2015

The Atwood machine, or Atwood’s machine, is a standard problem involving dynamic forces that is usually covered in first-year physics.

Atwood’s machine is simply a single, fixed pulley system set up as a counterweight as in Figure 1. The goal of the problem is to solve for an equation that defines the acceleration, (**a**), of the system. This problem is an ideal way to introduce physics students to more challenging problems and help them apply prior knowledge of Newton’s second law, free-body diagrams, and kinematics.

Depending on the course, the difficulty of the solution varies as more variables, such as friction or the mass of the pulley, are considered. For the simplest case, the following assumptions are made:

- Friction is ignored.
- The mass of the string and the pulley are ignored.
- The string is inelastic.

To find the solution, examine the free body diagram:

Figure 1This is an ideal system, where the pulley itself has no mass, and the tension is the same everywhere in the rope. The mass of m

_{1}is greater than the mass of m_{2}, indicated by the mathematical expression m_{1}>m_{2}.The weight of mass m

_{1}, which is m_{1}g,accelerates the system as mass m_{1}falls and mass m_{2}is raised.Gis the acceleration due to gravity, 9.8 m/s^{2}. The symbols m_{1}and m_{2}represent the masses in kilograms.Tis the tension in Newtons. Force is measured in Newtons, where 1N = 1kg x m/s^{2}.The sum of the forces on mass m

_{1}are given by:m

_{1}a =m_{1}g – TThe sum of the forces on mass m

_{2}are given by:m

_{2}a=T- m_{2}gNotice that positive acceleration is assigned to the direction of motion, not the up direction. Since tension,

T, is present in both equations, solve both equations forTand set them equal to each other.

T= m_{1}g- m_{1}a

T= m_{2}a+ m_{}1aCombine like terms and solve for the acceleration

a.m

_{1}g- m_{1}a= m_{2}a+ m_{2}g

m_{1}a+ m_{2}a= m_{1}g- m_{2}g

a(m_{1}+ m_{2}) =g(m_{1}- m_{2})

The value for

acan be found by substituting 9.8 m/s^{2}forg, the acceleration due to gravity, and the values for masses m_{1}and m_{2}, which can be measured.Measuring the acceleration directly with a stopwatch is difficult. However, students can use their knowledge of the kinematics equation to predict the time for mass m

_{1}to reach the ground. Test the solution by setting up a system where m_{2}rests on the ground in the start position. Use one of the kinematics equations to find the time for mass m_{1}to descend to the ground.Depending on available resources and equipment, you can set this experiment up several ways, and it is a great inquiry-based activity for students to design an experiment to test their results. If your classroom is equipped with probeware, photogates, or a motion sensor, you can measure the acceleration directly. The following procedure can help get you started.

Measuring accelerationDuring this procedure, you will confirm the acceleration by calculating the time for the Atwood machine to move through its entire range of motion and come to rest using a basic pulley. Before beginning, you should calculate the amount of friction in the system.

## Materials

- String
- Tape Measure or Meter Stick
- Low Friction Pulley
- Timer or Stopwatch

## Step 1

Calculate acceleration for the system adjusted for the force of friction in the pulley.

Without friction, the acceleration in the system is shown by the equation:

## Step 2

Construct an Atwood machine using a low friction pulley, some string, and some weights. The weights should be able to move freely as the pulley rotates without colliding with each other. Suspend the pulley from a ring stand or other support. Mass m

_{2}should rest on the floor. Results will be better when mass m_{1}has a longer drop.## Step 3

Record the values of masses m

_{1}and m_{2}, and measure and record the height of mass m_{1}from the bottom of the mass to the floor.## Step 4

Calculate the time it takes for the system to move through the full range of motion. If the system starts out with the lighter mass, m

_{2}, resting on the floor, and h is the distance from the bottom of the heavier mass, m_{1}, to the floor, measured in meters, then the position of the higher mass as a function of time can be found using this kinematic equation:

Where

ais the acceleration calculated for the system, t, is the time for mass m_{1}to reach the floor from the starting position h, and v_{1}is the initial velocity of the system. If the position starts from rest, v_{1}is zero, and the time, t, can be found by rearranging the equation.

## Step 5

Confirm the calculations. Allow the system to start from rest with mass m

_{2}on the ground. Measure the time for mass m_{1}to reach the ground.## Step 6

Calculate the percent difference between your theoretical value of time and the measured value of time.

Additional resources

- Atwood Machine (item #752019)
- Visual Scientifics Atwood Machine (item #753174)
- Neulog™ USB Bridge Module (item #369733)
- Neulog™ Wi-Fi™ Communication Module (item #369735)
- Neulog™ Motion Sensor (item #369642)

The Problem with Pulleys

A resource to help you build and test working models of pulley systems commonly used in physics problems.