Physics 211 Experiment #4 Newton’s Second Law – Atwood’s Machine

Newton’s second law (FNET = ma) can be experimentally tested with an apparatus known as an “Atwood’s Machine” (See Figure 1.) Two weights of unequal mass, connected by a thread, are draped over a pulley, as shown in the figure. When released, the larger mass accelerates downward and the smaller one accelerates upward.

Figure (1a): The Atwood’s Machine, showing the pulley and the two masses after a run. Figure (1b): A close up of the pulley and one of the

masses after a run.

Figure (2a): A schematic diagram of the apparatus shown in Figure 1.

Figure (2b): The free body diagram for M1.

Figure (2c): The free body diagram for M2.

Newton’s Second Law applied to the larger mass, M2, implies (see Figure 2c):

FNET(on M2) = M2a

T – M2g = – M2a (1)

Newton’s Second Law applied to the smaller mass, M1, implies (see Figure 2b):

FNET(on M1) = M1a

T– M1g = M1a (2)

The tension T can be eliminated from equations (1) and (2) obtain:

M2g – M1g = M2a + M1a (3)

The magnitude of the acceleration, a, of the system is then:

(4)

The numerator, (M2 – M1)g, is the net force causing the system to accelerate. The denominator, (M2 + M1), is the total mass being accelerated. Equation (4) can be written:

(5)

Where FNET = (M2 - M1) g

And MTOTAL = (M2 + M1)

Apparatus

Pasco 750 Interface10 spoke smart pulley (Pasco ME 9387) with stereo phone plugTable clamp for pulleyTwo 5 gram mass holdersMasses: 2 x 100gm, 3 x 50gm, 2 x 20gm, 2 x 10gm, 2 x 5gm Thread

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Detailed Procedure and Analysis for Atwood’s Machine Experiment

I. Set-up of computer and interface

1. Turn on the PASCO 750 interface first. Verify that the indicator light is on.2. Turn on the computer, and login.3. Follow the instructions on the "Data Studio Information Sheet" to set up the computer for

the experiment.4. When you must choose the sensor, choose smart pulley. 5. Under measurement, choose velocity, Ch1 (m/s); make sure this is the only option

checked.6. Note, there is no calibration for this experiment.7. Now, you are ready to set up and run the experiment; the velocity as a function of time

will be recorded by the computer, and the computer will display the results graphically for you to analyze.

II. Setting up and Running the Experiment

1. Clamp the smart pulley to the lab table as shown in the Figure 1, and plug its stereo phone plug into channel 1 of the interface.

2. Set up masses M1 = 95 grams and M2 = 105 grams as shown. Use the brass masses supplied. When adding to obtain the total mass be sure to include the 5 grams for each weight holder.

3. Hold the system steady with M2 far from the ground, and allow the system to come into equilibrium (i.e. make sure the masses are not swinging).

4. To have the computer record the data, click the start button on the computer screen, and release the system. After M2 has fallen to the ground, click the stop button (which will be in the same location as the "start" button).

III). Determination of the Acceleration of the Masses

1. The computer will have plotted velocity vs. time for you. The slope of the best fitting line to the velocity as a function of time is the experimentally determined acceleration. Determine the slope of the best fitting line, and record it into an excel spreadsheet. This spreadsheet should also include the values of M1 and M2.

2. To obtain a reliable fit, only data points recorded while the masses are in "free fall" should be included in the fit. This part of the data can be selected by clicking down on the mouse from an initial position on the graph, dragging the mouse to the final position,

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and releasing. To obtain a close up view of this part of the data, click on the scale to fit icon, which is located in the top left hand corner of the graph window.

IV). Measuring acceleration for M1 + M2 constant, with different values of M1 and M2

1. To obtain the value for the acceleration of another system with the same total mass, but with different values of M1 and M2, increase the mass of M2, and decrease the mass of M1

by a fixed amount so the mass difference is 20, 30, 40, and 50 g; the total mass should remain at 200g (i.e. the mass of M2 runs from 105 to 125 g in increments of 5 g, and the mass of M1 runs from 95 to 75 g in increments of 5 g). For each run, the total mass of the system, M1 + M2 should be equal to 200 g.

2. Follow the instructions given above for each run, and record the values of the masses and the experimentally determined acceleration into the excel spreadsheet. Print out one or two typical graphs to include in your laboratory report.

V). Measuring acceleration for M2 - M1 constant, with different values of M1 and M2

1. Choose initial values of M2 and M1 of 45 and 35 g respectively. Run the experiment, analyze the velocity vs. time graph, and determine the acceleration. Record these results into a second excel spreadsheet, along with the values of M1 and M2.

2. To obtain a total of five runs, increase each mass by 40 g, rerun the experiment, and record the results. The mass difference should remain the same. Thus, M1 will take on values of 35, 75, 115, 155, and 195 g, while M2 should have the values of 45, 85, 125, 165, and 205 g.

VI). Analysis of the Results

1. Theory predicts that the acceleration is given by the net force divided by the total mass (see equations 4 and 5). Now, you should compare your experimentally determined acceleration with the theoretical prediction. To determine the theoretical prediction, create three new columns in the excel spreadsheets: one for the net accelerating force (M2 –M1)g, one for the total mass (M1 + M2), and one for the theoretically predicted acceleration:

Acceleration = (Net accelerating force)/total mass

2.Create another column for the percent difference between the experimental and theoretical values of acceleration.

3. Another way to compare experimental and theoretical results is to plot the net force FNET vs. the experimental acceleration. Equation (5) indicates that this should be a straight line with slope equal to the total mass. For the results obtained in part IV, with the total mass constant, plot FNET versus experimental acceleration, and fit the graph with a straight line.

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Compare the slope of the line with the actual total mass (.200 kg). What is the percent error? Print out this graph and include it in your laboratory report.

4.Theory predicts that when the net force is constant, the acceleration will vary inversely with total mass (see equation 5). For the data obtained in part V, with the mass difference held constant, plot the experimental acceleration versus mass. Approximate the data with a power law fit (y = c x n). Record the best fit values of c and n, and compare them with theoretical predictions based on equation 5.

VII). Include the Answers to These Questions in Your Laboratory Report

1. How does the acceleration depend on the net accelerating force when the total mass is constant?

2.How does the acceleration depend on the total mass when the net force is constant?

3.What sources of experimental error most likely caused the differences you found between aEXP and aTHEOR?

VIII) Clean up the area around you; put away the equipment and shut down the computer.

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