17 Atwood’s Machine 17 – Page 1 of 5

Atwood’s Machine

Introduction

The purpose of this lab is to study an Atwood’s Machine apparatus, built with a PASCO Super Pulley, as shown in Figure 1. The Photogate is used to measure the velocity of both hanging masses as one moves up and the other moves down. The slope of the graph of velocity vs. time is the acceleration of the system. Careful measurement when there is no net force allows the student to compensate for friction.

Equipment:

Qty Items Part Number1 Photogate ME-9498A1 Super Pulley (from ME-3420) ME-9450A1 Mounting Rod SA-92421 Table Clamp ME-94721 Multi-Clamp ME-95071 Rod, 90 cm ME-87381 Mass and Hanger Set ME-89791 No-Bounce Pad SE-73471 String SE-8050

Required, but not included:1 550 Universal Interface UI-50011 PASCO Capstone Software

Written by Stuart Loucks

Figure 1: Atwood’s Machine SetupFigure 1: Atwood’s Machine SetupFigure 1: Atwood’s Machine SetupFigure 1: Atwood’s Machine SetupFigure 1: Atwood’s Machine SetupFigure 1: Atwood’s Machine SetupFigure 1: Atwood’s Machine Setup

17 Atwood’s Machine 17 – Page 2 of 5

Setup

1. Fasten the table clamp and the 90-cm rod to the edge of the table as shown in Figure 1.

2. Attach the pulley to the photogate using the mounting rod as shown in Figure 2. The pulley may need to be first removed from its own clamp.

3. Use the multi-clamp (see Figure 3) to fasten the photogate to the vertical rod.

4. Plug the photogate cord into Digital Input #1 of the 550 Universal Interface.

5. Position the yellow no-bounce pad (see Figure 1) below the photogate to protect the falling mass hanger when it reaches the table.

6. Cut a 1.2 m piece of string and tie loops on each end to hold a mass hanger. Pass the string up and over the pulley as shown in Figure 3.

7. Add a single 100-g mass and a single 20-g mass to one mass hanger (5 g) for a total mass of 125 g, which will be mass m1. To a second mass hanger add a single 100-g mass for a total of 105 g, which will be mass m2. Hang the masses on the string over the pulley.

8. Adjust the height of the pulley or length of string so that when one mass hanger is touching the no-bounce pad, the other mass hanger is a few centimeters below the pulley. Also make sure the photogate is horizontal so the string does not pull sideways at all on the pulley.

9. In Capstone, open Timer Setup in the left toolbar. Choose a pre-configured timer for the photogate in channel 1. Select a photogate with pulley as the type of timer, and select Position and Linear Speed as measurements that will be visible. The spoke arc length is 0.015 m and the spoke angle is 36 degrees.

10. In the lower toolbar, open Recording Conditions. Set a Stop condition to be measurement based for Position (in meters) above 0.30. Later during measurements, this condition will cause recording to stop automatically once the masses have moved 0.30 m. You can adjust this condition if needed to suit your experiment.

Written by Stuart Loucks

Figure 2: Pulley and PhotogateFigure 2: Pulley and PhotogateFigure 2: Pulley and PhotogateFigure 2: Pulley and PhotogateFigure 2: Pulley and PhotogateFigure 2: Pulley and PhotogateFigure 2: Pulley and Photogate

Figure 3: Rods and ClampFigure 3: Rods and ClampFigure 3: Rods and ClampFigure 3: Rods and ClampFigure 3: Rods and ClampFigure 3: Rods and ClampFigure 3: Rods and Clamp

17 Atwood’s Machine 17 – Page 3 of 5

Theory

An Atwood's Machine consists of two unequal masses connected by a single string that passes over an ideally massless and frictionless pulley as in Figure 4. When released from rest, the subsequent velocity and acceleration are both downward for the larger mass m1, while these quantities are both upward for the smaller mass m2.

The free-body diagrams (Figure 5) show the forces acting on each of the masses, and the direction of each mass’s acceleration. The tension is T throughout the string. The magnitude, a, of the acceleration is the same for each mass, even though they accelerate in opposite directions.

Using the convention that the downward direction is positive, and applying Newton’s 2nd Law to the descending mass, m1, gives:

m1g – T = m1a (1)

Applying Newton’s 2nd Law to the ascending mass, m2, gives:

m2g –T = m2(-a) (2)

1. Show that eliminating T between Equations 1 and 2 gives Equation 3 for the acceleration, a:

a=(m1−m2

m1+m2) g (3)

2. Show that Equation 3 reduces to the expected result in the limiting case where m1 = m2. Explain why this result is expected.

3. Show that Equation 3 reduces to the expected result in the limiting case where m1 = 0. Explain why this result is expected.

Written by Stuart Loucks

Figure 5: Free-Body Diagrams

Figure 4: Atwood’s MachineFigure 4: Atwood’s MachineFigure 4: Atwood’s MachineFigure 4: Atwood’s MachineFigure 4: Atwood’s MachineFigure 4: Atwood’s MachineFigure 4: Atwood’s Machine

Figure 5: Free-Body DiagramsFigure 5: Free-Body DiagramsFigure 5: Free-Body DiagramsFigure 5: Free-Body DiagramsFigure 5: Free-Body DiagramsFigure 5: Free-Body Diagrams

17 Atwood’s Machine 17 – Page 4 of 5

Procedure

1. In Capstone, create a graph of Linear Speed vs. Time.

2. Move m2 to its lowest point. Click on Record, release m2, and then Stop recording just before m1 strikes the pad.

3. The masses may touch as they pass each other. If so, repeat the run. You can delete unwanted runs using the Delete Run feature in lower toolbar. You should see a linear region on the speed graph as the masses move freely.

4. Turn on Curve Fit on the graph toolbar and select Linear fit. What is the physical meaning of the slope of the line?

5. Record the measured value of the acceleration, including units.

6. Using Equation 3, calculate the theoretical value for the acceleration.

7. Calculate the percent error for the acceleration:

% Error= Measured−TheoreticalTheoretical

∙ 100

Written by Stuart Loucks

17 Atwood’s Machine 17 – Page 5 of 5

Accounting for Friction

1. Add an additional 20-g mass to m2. The two masses should now balance. If there were no friction, and we gave m1 a push downward, it would continue at constant speed.

2. Move m1 to its highest point. Click on Record. Give m1 a gentle push downward and release it. You should see that the speed decreases.

3. Add 0.5 g to mass m1 and repeat the measurement. Does the speed still decrease? In Data Summary at the left, double click on this run and change its name to "Extra 0.5 g".

4. Using the 0.5-g, and 1-g masses, adjust the mass of m1 until the speed stays as constant as possible. For each run, change the name to indicate how much extra mass was added to m1. Use the graph toolbar to allow simultaneous viewing of multiple data sets, and select all of these runs to be visible for comparison.

5. Record this added “frictional” mass that makes the speed as constant as possible.

6. To account for friction, this mass should be subtracted from the actual value of m1 in the numerator (only) of Equation 3. This effectively adds a negative "friction force" equal to the weight of the “frictional mass”. Making this change in Equation 3, recalculate the theoretical acceleration.

7. Using this adjusted theoretical (accepted) value instead, recalculate the percent error.

8. How did the correction for friction affect your final results?

Written by Stuart Loucks