Newton’s Laws and Modified Atwood’s Machine
Overview
The purpose of this investigation is to validate Newton’s Second Law of Motion and explore properties of friction and Atwood’s machine. Force and acceleration data will be measured by a GoDirect Force and Acceleration (GDFA) Bluetooth device. Both sets of electronic data will be collected and analyzed using Graphical Analysis running on a laptop computer. Mass data will be determined simply with a triple beam balance.
In this lab, an object of constant mass is pulled across a level surface by a weight hanging on a string that passes over a pulley. By changing the weight at the end of the string, different amounts of force act on the object. It is the tension in the string that is causing the object to accelerate. This tension will be measured by the GDFA, which also serves as the test object.
1. Adjust the feet of the track so that its surface is level.
2. Attach a piece of felt to the bottom of the GDFA to protect the device and promote a smooth and steady slide across the track. Place the GDFA face up on the track.
3. Attach the pulley to the end of the track and adjust the height of the pulley to match that of the hook of the force sensor.
4. You must connect (i.e. “pair” via Bluetooth) and calibrate the force sensor. Turn on the GoDirect Force and Acceleration (GDFA). Then on the laptop run Graphical Analysis and choose Sensor Data Collection. You should see the ID number of your sensor that is on the label. For this experiment you will use only channels for the Force sensor and the x-axis Accelerometer – do not enable the other two acceleration sensors or gyroscope channels.
5. Once it is paired, click on the Force sensor button at the bottom of the window. Choose Calibrate and two-point option. Keep the hook and x-axis in a horizontal orientation. Place the GDFA face up on the level track. Then enter zero for Known Value 1 with nothing touching the force sensor’s hook. Then apply a known force to the sensor by attaching a string to the hook, passing it over the pulley, and hanging a 500 gram mass on the end – note: you must hold the GDFA in place to prevent it from moving with this mass on the end of the string. Enter the correct force value for Known Value 2. Remove the 500 gram mass after this process!
6. The two-point calibration process is not available for the accelerometer. To zero the sensor, hold the GDFA in a horizontal orientation and then click on the x-Acceleration sensor button and click Zero. Check the calibration by holding the GDFA at rest either horizontal (should show 0.00 m/s2) or with the positive x-axis pointing straight up (show show 9.80 m/s2). (This works because the movable part within the accelerometer bends the same amount when pulled down by gravity as it would if it accelerated 9.8 m/s2 across the track in the positive x-direction.). If you choose to calibrate you are only prompted for one value – hold at rest with x-axis pointing upward and enter 9.80 m/s2 as the known value.
7. Adjust the data collection parameters and make the duration of the experiment 2 seconds, collecting 100 samples per second. You can tweak these parameters if you want to.
8. Now place a mass on the end of the string, pass it over the pulley, and connect the other end to the hook on the GDFA force sensor. Try the 50 g first. Do not use more than 200 grams!
9. Pull the GDFA away from the pulley and release it such that it slides freely away from the pulley for a period and then reverses direction and slides freely toward the pulley. Collect data for this motion. The data in the table should show the force and acceleration in one of these two conditions with the GDFA sliding under the influence of friction and the string with weight hanging at the end.
10. To complete the table use the graphs of acceleration vs. time and force vs. time. On each graph you should be able to identify a “spike” that represents the tug of the person followed by a “plateau” that represents the GDFA sliding away from the pulley and/or a “plateau” that represents the GDFA sliding toward the pulley. If you do not have a suitable plateau then repeat the experiment until you do!
11. On the force graph find the “plateau” that represents the GDFA sliding away from the pulley. Tap and drag to select only this portion of the graph – do not include sections of the graph where the GDFA was tugged by the person or where it slid toward the pulley. Then go to the Graph Tools menu and choose View Statistics to get the mean force and the mean acceleration. Record these mean values as tension and acceleration in the table.
12. Change the selected part of the graph to determine mean force and mean acceleration for the GDFA sliding toward the pulley. If there is not a suitable selection of data for motion toward the pulley then do another trial – for motion toward the pulley you can simply release the object from rest. Note: depending on the amount of weight at the end of the string you may be able to simply release the GDFA or you may have to actually give the mass on the string a tug downward to get it started moving and use data that shows it sliding to a stop.
13. Change the mass on the end of the string to the values shown in the table and repeat the experiment to collect further data. Note you should also measure the acceleration of the GDFA sliding in either direction with zero mass – i.e. disconnect the string (first row in the table). And choose a value of mass for one final trial (last row of the table). Remember do not exceed 200 grams. It can be something in between the other values shown.
14. Adjust the appearance of the graphs to your liking and then print ONE representative graph of each type including the analysis showing how the values in the table were determined. Do not print graphs for every trial. The goal here is to have a printed record showing one example of the manner by which the tension and acceleration values were determined.
15. Measure the total mass of the sliding object (the GDFA and anything attached to it). And measure the mass of the string.
Analyses
1. Use the results to produce a tension vs. acceleration graph. Plot the independent variable (tension) on the y-axis. Use different symbols and/or colors for the GDFA sliding away and for the GDFA sliding toward the pulley and include a key or legend. Determine an appropriate line or curve of best fit and its equation for each of the two sets of data.
2. The mass at the end of string had the same amount of acceleration as that measured for the cart. Use the acceleration and the mass of the hanging weight to calculate the tension at the “hanging end” of the string. Create a table and a corresponding graph of calculated tension (at one end of the string) versus the measured tension (at the other end) for the two sets of data – away and toward the pulley. Again use different symbols/colors and determine a best fit for each set of data.
Modified Atwood’s Machine
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Total mass of sliding object: |
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Mass of the string: |
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Sliding away from the pulley |
Sliding toward the pulley |
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Mass hanging on string (g) |
Tension |
Acceleration |
Tension |
Acceleration |
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0.0 |
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20.0 |
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50.0 |
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70.0 |
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100.0 |
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120.0 |
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150.0 |
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Do not exceed a mass of 200.0 grams hanging from the string!
1. (a) Make a free body diagram for the GDFA sliding away from the pulley. Write the equation of motion and solve for the tension symbolically. (b) Make a free body diagram for the GDFA sliding toward the pulley. Write the equation of motion and solve for the tension symbolically.
2. Consider the lines of best fit for the graph of Tension vs. Acceleration. (a) What do the slopes represent? (i.e. should equal what?) Explain your answer. (b) Assuming the values on your data sheet are accurate, calculate the percent error in the two slope values. Show your work.
3. Again consider the lines of best fit for the graph of Tension vs. Acceleration. (a) What do the y-intercepts represent (i.e. should equal what?) Explain your answer. (b) Ideally how should the two y-intercepts compare to one another? And if this is not the case how can the discrepancy be explained?
4. (a) Show one example of how you calculated the tension in the string based on the hanging mass and its acceleration. (b) Discuss the significance of the results shown in the calculated tension vs. measured tension graph.
5. Discuss error in this lab. (Things to discuss: indications and signs of error – random and/or systematic, the probable and significant cause(s) of the error that is apparent in the results. The goal of discussing error is to explain satisfactorily why the results of your lab are not quite exactly what was expected. Be as specific as possible. You will almost always have unexpected results in an experiment. Your task is to write a discussion that is intelligent, thoughtful, and insightful!)
A complete report (50 pts): (5 or 6 pages in this order)
q Completed data/results table. (10)
q Example graphs of Force vs. Time and Acceleration vs. Time, including analyses. (10)
q Tension vs. Acceleration graph with lines of best fit. (10)
q Calculated Tension vs. Measured Tension table and graph with lines of best fit. (10)
q Responses to questions (10)