# The Atwood Machine

## Discussion:

The "ideal" Atwood machine consists of two masses, M1 and M2, connected by a massless, inelastic string which passes over a frictionless pulley. The diagram at right shows an Atwood machine, along with a free-body diagram for each mass, and the resulting equations of motion.
Alternatively, this system could be considered to be a single mass, M1 + M2, being pulled to the left by a force of magnitude M2g, and pulled to the right by a force of magnitude M1g. The net force on this object must be M1g - M2g, which by Newton's Second Law must equal the object's mass, M1 + M2, times its acceleration, a. In other words:

Solving for a gives:

You should verify that the original system of equations yields the same expression for acceleration, albeit after somewhat more algebra.

How can you determine this acceleration in practice? If you can construct a velocity vs. time graph for the motion, it should be a straight line (since the acceleration is constant) and the acceleration of the system is the slope of the line...

## Pasco "Smart Pulley" standard masses ring stand ring stand clamp c-clamp string Procedure:

### The Apparatus:

1. Use the ring-stand clamp to attach the "Smart Pulley" to the ring stand.(The "Smart Pulley" is "smart" because it is equipped with a photocell which you will use to determine the velocity and acceleration of the system.) Clamp the ring stand to the edge of your lab table so that the masses will be able to reach the floor.
2. Attach one end of your string to a 100 gram mass. Thread the other end through the pulley, and attach a second 100 gram mass to the other end. IMPORTANT!! The pulley is a delicate (and expensive) device. You want to allow the masses to move a reasonable distance, but be sure that you have a long enough string so that the rising mass cannot strike the pulley as the other mass falls to the floor! Have your instructor check you setup before you proceed! (Notice that if M1 = M2, the system will remain at rest at any position, or move at constant velocity if you start it moving - Newton's First Law strikes again!)
3. Plug the "Smart Pulley"'s jack into the DG1 port of the computer interface. Be sure that the interface is properly attached to the computer, and that the power supply is plugged in. Switch the interface on - the green light on the front should be on.

### The Timing Software:

1. Load the "ULI Timer" software. The computer screen should look like the diagram at left. If the pulley is attached correctly, you should see the icon labeled "1" (in the upper-left corner of the screen) change from dark to light to dark as the photocell is covered and uncovered by a spoke of the wheel.
2. In the Timing menu, select "Trigger on Gate" (it should get a check mark). This means that the timer will start when the pulley starts turning instead of immediately when you push the "Start" button.
3. Now, construct a data table for your experiment. In the Window menu, select "New Table". This opens a new window labeled (Surprise!) "Table". Click in the top box of the first column to the right of the "Row" column, then go to the Data menu and select "Dt1". This will place the interval times in your data table. Now click the next row, and select "Mid Time" from the Data menu. Your data-table window should look like the one shown at right.

### A "Run":

1. Add the smallest mass available (10 grams) to one of the masses on your apparatus. Notice that when you release the masses, the system accelerates - even with a very small mass difference.
2. Raise the larger mass to its starting position. Press the "Start" button on the ULI Timer. The timer will not start until you let go of the system. To avoid a lot of meaningless data, be ready to click the "Stop" button just before the mass hits the floor. (Practice! Teamwork!) Timing data for this trial should be displayed in your data table.

### Analyzing the Data:

The ULI Timer software will hold only one data set at a time, so something has to be done with the data you have just collected. The ULI Timer has the capability to graph the data, but instead of learning how to use it, it might be more efficient to transfer the data to the "Graphical Analysis" software you already know how to use.

1. Use the mouse to select both data columns from your data table. Copy the data to the clipboard.
2. Open the "Graphical Analysis" program, and paste your data into its data table.
3. Label the column that the ULI Timer called "Dt1" as "dt" (units are seconds), and label the "Mid Time" column "t" (units are seconds).
4. The pulley (and also each mass, since they are all connected) moves 0.015 meter during each time interval dt. Therefore, its average velocity during each time interval is 0.015/dt (in meters/second). To add this to your data table, select "New Column" from the Data menu. For its label, type "v = 0.015/dt" (units are m/s).
5. Click on the axes labels of the graph window to change the horizontal quantity to time, t, and the vertical quantity to velocity, v. You have a velocity vs. time graph for your motion.
6. Using a "Linear" analysis model, you can find the slope of the velocity vs. time graph.

### What now?

Be sure to record the masses used for each trial.

Go back to the ULI Timer software and make another "run".

Copy your data to the clipboard, and paste it into a "New Data Set" in the Graphical Analysis program. Notice that you do not need to retype the column headings for the new data set - nice feature!

Continue to add mass until the acceleration becomes too large - above all, protect the pulley!

## Results:

You can calculate the "theoretical" acceleration for each trial using equation 1 shown above. Be sure to show a sample calculation, and display your results in table form. Of course, what about the uncertainties of both the "theoretical" and "experimental" accelerations?

## Conclusions:

How closely do the "theoretical" and "experimental" accelerations agree? What do you think would do the most to improve the precision of this experiment?

last update January 16, 2008 by Jerry L. Stanbrough