M61 1 M61.1 PC COMPUTER ASSISTED DETERMINATION OF ANGULAR ACCELERATION USING TORQUE AND MOMENT OF INERTIA

Transcription

1 M61 1 M61.1 PC COMPUTER ASSISTED DETERMINATION OF ANGULAR ACCELERATION USING TORQUE AND MOMENT OF INERTIA PRELAB: Before coming to the lab, you must write the Object and Theory sections of your lab report plus include a Data Table. Include the derivations of equations 3 and 4 in the theory section of your lab report. You should also determine the error formula for equation 6 (page 7). SAFETY: There are no known safety concerns with the equipment for this experiment. Fire Exit: If the fire alarm sounds, immediately exit Room 131 and turn LEFT and then exit the building into the Bowl. OBJECT: The purpose of this experiment is to investigate the effect of torque and moment of inertia on the angular acceleration of a disk. APPARATUS: The lab equipment consists of 2 spinning discs connected to either a small or large diameter pulley. A thread is is attached to a hanging weight and looped over a cylindrical air bearing (not shown) and wrapped around the pulley. As the falling weight at the end of the thread falls, its torque causes the disc to rotate with some angular acceleration. Fig. 1. Side View of Lab Apparatus.

2 M61 2 M61.2 THEORY: For an object influenced by a torque (τ) τ = Iα. (1) In this equation α is the angular acceleration found by dividing the linear acceleration of the circumference of the disk (a) by its radius (r) α = a/r (2) Also in equation (1), I is called the moment of inertia since it is the rotational analogue to inertial mass. However, one cannot specify the moment of inertia of an object without specifying the axes of rotation because the moment of inertia is different (generally) about different axes. Note that equation (1) has the same form as the equation of motion of a particle moving in a straight line (F = ma). Thus, the equations of motion for a rotating body have the same form as those for a particle moving in a straight line, a property used in the analysis below. A torque τ = rf can be applied to the disc by wrapping a string around the small pulley, passing it over the cylinder bearing and attaching to it a mass M. By neglecting both the friction of the pulley and its rotational inertia, draw a free-body diagram of the apparatus and derive in the theory section of your report that the torque about the axis of rotation is given by τ = rf = Mr(g-a) (3) Where a is the linear acceleration of the falling mass, F is the tension in the string, and g is the acceleration of free fall (9.81 m/sec 2 ). By using equations (1) and (3) discussed in the theory section, derive in the theory section of your report the following theoretical angular acceleration expression: α th = M r g / (I + M r 2 ) (4) where, M = mass of falling weight r = radius of pulley I = moment of inertia of disc As the disc has a hole in the centre, its moment of inertia is given by I = 1/2 * M d (R R 2 2 ) (5) where M d = mass of rotating disc, R 1 = inner radius of disc, and R 2 = outer radius of disc.

3 M61 3 M61.3 PROCEDURE: Do NOT rotate discs against each other, or rotate the cylinder air bearing unless compressed air is turned on. 1. Check apparatus for trial 1. Check that the thread washer is attached underneath the small pulley that sits on top of the steel discs. The thread should pass through the slot in the small pulley, hang over the cylinder air bearing and be attached to a 25 g falling mass without extra weights. 2. Level apparatus. Place the bubble level on the disc and adjust the levelling screws (legs) until the unit is as level as possible. 3. Connect jack. Connect the jack with the yellow band from under the apparatus to the Input 1 plug on the PASCO Xplorer GLX s Digital Adapter. Ensure that the Xplorer GLX is turned OFF. To turn it off, depress the power button at the bottom right of the unit for about 5 seconds. 4. Open Air Supply. SLOWLY open compressed air supply until gauge reads 8 psi. 5. Check that BOTH steel discs are attached to apparatus with a solid black cap screw and that only the top disc rotates freely. (If both rotate, open hose clamp below apparatus.)

4 M61 4 M Open DataStudio. Turn on the computer and double click the DataStudio icon to open that program. Ensure that there is a USB cable connecting computer to Xplorer GLS unit. 7. Create Experiment. Click Create Experiment when the Welcome window asks How would you like to Use DataStudio?. (If this message does not appear, click on New Activity under the File Menu.) Turn on the GLX unit. 8. If the XplorerGLX File Manager window ever opens, click the Done button to close it. 9. Select General Counting. Select General Counting when the Choose sensor or instrument window appears and click the OK button. 10. Set Count Time Interval. Click the Setup button and then click the Constants tab in the bottom portion of the Experiment Setup window and set the Count Time Interval to 0.2 seconds. Click the Red X button at the top right corner of this window to close it. 11. Wind up thread. Turn the top disc to wind the thread around the pulley until the falling weight is almost raised to the level of the cylindrical air bearing. 12. Start Data Run. Release the disc and then immediately press the Start button on DataStudio (or the Start button on the Xplorer GLX). The falling mass should accelerate the top disc. 13. Stop Data Run. Press DataStudio s Stop button (or the Start button on the Xplorer GLX) when the string is unwound and the falling mass has reached its lowest position. 14. Export Data. A Pulse Count table of elapsed time values and Pulses will appear on the screen. Click on Export Data under DataStudio s File Menu. When prompted, select the appropriate Run and click the OK button on the Export Data window. Navigate to the folder for your class and save the table as a text file using a filename that begins with the surnames of the members of your group followed by today s date and your trial number. For example, the first set of data collected by Shadick and Sander on Jan 26, 2011 should be saved under the filename: shadicksander2011jan26trial Obtain Trial 2 data. Add a 10 g weight to the falling mass. Repeat steps Obtain Trial 3 data. Remove the 10 gram weight. Replace the small pulley atop the disc with a large diameter pulley provided. Repeat steps Obtain Trial 4 data. Replace the large pulley with the original smaller pulley. Replace the solid black cap screw with the cap screw with a hollow center. Also, close the hose clamp below the apparatus. Both discs should now rotate together. Repeat steps Record the mass of the discs, and measure their inner and outer radii with callipers. Also, record the radii of the two pulleys used.

5 M61 5 M61.5 ANALYSIS: 1. Open Excel. Open M.S. Excel by clicking on its icon or by clicking on Start, All Programs, Microsoft Office, Microsoft Office Excel. 2. Open Data file. Using Excel, open the text file with the data from your first trial. When the Text Import Wizard Step 1 of 3 window opens, click the Finish button. 3. Examine Data File. You should now have an excel file with 2 columns of data: Elapsed Time values in column A; Pulse Count values in column B. The file title should be in cell A1 in the top row. The column headings should be in row 2 in cells A2 and B2. If the pulse values increase to a maximum and then decrease, it means the computer was still collecting data after the falling mass reached the end of its fall and had started to rise causing a deceleration. Should this occur, you should delete all rows of data following the row with the maximum Pulse Count value. 4. Insert Pulse Frequency Column. Select a blank column following the Time column by clicking on column B, then click the Insert Menu and then click on Columns. In the new cell in the row with the column headings, type a label: Pulse Frequency (bars/s) In cell B3 below the column heading, insert the equation: =C3/0.2. This equation should calculate the pulse frequency by dividing the measured pulse count by the count time interval that you set in step 10 of the Procedure. Copy this formula into the remaining cells of this column by dragging the tiny black square at the bottom right corner of this cell down to the cell in the last row of data. 5. Add names and trial number. Insert your surname and your partner s surname into cell C1. Insert Trial followed by your trial number into cell D1. 6. Adjust Column Widths. Adjust the column widths so that the entire column headers are visible by double clicking on the column separator at the top of the chart between columns B and C and also between columns C and D. 7. Save As Excel File. Click on Excel s File Menu and then click on Save As. Click on the Save as type: drop down menu and move the slider to select the top entry Microsoft Office Excel Workbook Click the Save button. 8. Set up Linear Regression Table Headings. In cell E1, type the title Linear Regression. In cell E2, type Acceleration (bars/s/s). In cell F2, type Y-intercept (bars/s). In cell D3, type Value:. In cell D4, type Standard Error:. 9. Adjust Column Widths. Adjust the column widths so that the entire column headers are visible by double clicking on the column separator at the top of the chart between columns D and E and also between columns E and F and between columns F and G.

6 M61 6 M Enter LINEST expression. In cell E3, type =LINEST(. Below this cell, known_y s should now be highlighted. Select the cells with the Pulse Frequency Data (for example: B3:B36) and then type a comma,. Below cell E3, known_x s should now be highlighted. Select the cells with the Time Data (for example: A3:A36) and then type a comma,. Below cell E3, [const] should now be highlighted. Type TRUE and then type a comma,. Below cell E3, [stats] should now be highlighted. Type TRUE and then type a right bracket ). 11. Set up Linear Regression table. Select the range of cells E3:F4. Press the F2 key and then press CTRL+SHIFT+ENTER. The values of the slope (acceleration) and y-intercept for your data should now appear in cells E3 and F3, respectively. Their corresponding statistical standard errors should now appear in cells E4 and F4, respectively. 12. Add borders. Select the range of cells that should have borders from your table. Add borders by clicking on the Format menu and then cells and then the border tab and then choose the appropriate border style. 13. Using the number tab in the Format Cells window, adjust the cell format so that all columns always display an appropriate number of significant figures. 14. Print data table. Click on the File Menu and then click on Print and OK button to print your data table. Print a second copy for your lab partner. Tape these data tables into the appropriate part of the Data section of your lab report. The output will appear on the printer next to the blackboard at the south end of Room 131. The lab number on your computer should appear on all pages to identify your output. 15. Plot Graph. Use the chart wizard to create an XY (Scatter) of your Elapsed Time and Pulse Frequency data. Add an appropriate title to the graph that includes your surname and trial number (e.g. Trial 1 Acceleration by Shadick and Sander.) Add appropriate axis labels with units to the graph. Ensure that the axes values are displayed to a consistent number of decimal places. Save your graph as a new sheet in your excel file. Right click on a data point on the graph and add a linear trendline. 16. Print Graph. Use the File menu to print a copy of your graph.

7 M61 7 M The frequency column in the data tables is the frequency that the black bars, on the circumference of the disc, pass by the optical reader. Note that this frequency could easily be converted to a rotational tangential velocity of the rim of the disc by multiplying the frequency by the S = 0.20 cm ± 0.02 cm spacing between bars. (You need not calculate that!) The slope of your acceleration graph is expressed in units of optical bars per second 2. Convert this value to a more meaningful angular acceleration in radians per second 2 units by using the following equation: α exp = (A * S) / R 2 (6) where R 2 = (outer) radius of the disc in cm, and * means multiplication. Record this experimental value and its calculated error in your table of results in the Analysis section of your report. 18. Repeat the above steps for each set of trials. 19. By using equations 4 and 5, tabulate the theoretical angular acceleration values for each of your trials in your table of results. 20. Determine whether the theoretical angular acceleration agrees with the experimental value within its calculated error range. To do this, calculate the absolute value of the difference of the angular accelerations and test whether this difference is less than the error value. 21. Compare the experimental and theoretical values of angular acceleration by calculating their percentage difference for each trial and tabulate your results as outlined in the following table. TRIAL FALLING PULLEY DISC MOMENT MASS RADIUS OF INERTIA M (g) R (cm) I (g cm 2 ) ANGULAR ACCELERATION α exp (rad/s 2 ) ± α th (rad/s 2 ) PERCENTAGE DIFFERENCE OF α 1 2 ± 3 ± 4 ± 22. The moment of inertia of the cylinder air bearing was neglected in your analysis. This bearing has a mass of 26 g and has inner and outer radii of 0.77 cm and 1.25 cm respectively. Calculate its moment of inertia and compare with that of the discs used.

8 M61 8 M61.8 CONCLUSION: 1. Discuss whether experimental angular acceleration values agree with the theoretical values within the experimental error limits. If not, test whether their percentage differences imply that they agree within a reasonable lab error of 20%. This latter calculation includes errors in the theoretical values. 2. Discuss whether your experimental accelerations have verified the expected effects of mass, radius and moment of inertia in this experiment. The mass effect is verified if the experimental and theoretical angular accelerations agree within a reasonable lab error in BOTH trials 1 and 2. The radius effect is verified if the experimental and theoretical angular accelerations agree within a reasonable lab error in BOTH trials 1 and 3. The moment of inertia effect is verified if the experimental and theoretical angular accelerations agree within a reasonable lab error in BOTH trials 1 and 4. SOURCES OF ERROR: 1. The moment of inertia of the cylinder air bearing was neglected in your analysis. Is this error significant? Refer to your calculation in the final step of the analysis. 2. Briefly discuss any other sources of error and their qualitative effects on the results.