1 M62 M62.1 CONSERVATION OF ANGULAR MOMENTUM FOR AN INELASTIC COLLISION

Transcription

1 1 M62 M62.1 CONSERVATION OF ANGULAR MOMENTUM FOR AN INELASTIC COLLISION PRELAB: Before coming to the lab, you must write the Object and Theory sections of your lab report and include the Data Tables. You should also determine all error formulae used in analysis. 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 object of this experiment is to investigate conservation of angular momentum for an inelastic collision of a steel ball and a disc and test whether initial angular momentum is proportional to radius from the rotation point. APPARATUS FIGURE 2 Determination of v FIGURE 1 Determination of I d FIGURE 3 Collision Setup

2 THEORY In this experiment, a moving ball is caught by an arm attached to an initially stationary disc. The collision causes the disc to rotate with uniform angular velocity. 2 M62.2 Consider a point mass m moving with velocity v at a distance r and angle θ with respect to a reference point P as shown in Figure 4. The point mass then has an angular momentum L about point P given by: L = r p = m r v so L = mrvsinθ (1) m θ v r FIGURE 4 In this experiment, the point mass is replaced by a steel ball of mass m. The ball hits the catch arm on the disc at a distance R and angle of 90 with respect to the disc axis. Hence the initial angular momentum of the ball about the disc axis at the moment of collision is given by: L i = mrv (2) After the collision, the disc, catch arm and ball rotate with constant angular velocity ω. If the total moment of inertia is I, then the final value of angular momentum of the system will be given by: L f = Iω (3) P As the ball rotates at a distance R from the rotation axis, the Parallel Axis Theorem implies that the ball's contribution to the moment of inertia is given by: I = (4) b mr b + mr where m is the ball s mass and r b is the ball's radius.

3 PROCEDURE 3 IMPORTANT: Do NOT rotate the discs against each other, or rotate the cylinder air bearing, unless compressed air is turned on. M Check that an ALUMINUM disc is placed atop the lower steel disc. 2. Check that the thread washer is attached underneath the small pulley which sits on the TOP disc. The thread should pass through the slot in the pulley, hang over the cylinder bearing and be attached to a 25 g falling mass. 3. Position the catch arm atop the brass spacer which should rest on the small pulley on the centre of the top disc. Secure these in place with the long RED cap screw. The catch arm should be located directly above the small hole in the aluminum disc which is covered with masking tape. The tape creates extra lift which is designed to counteract the torque due to the catch arm in order to keep the disc level. 4. Level apparatus. Place the bubble level on the disc (not the table) and adjust the levelling screws (legs) until the unit is as level as possible. 5. 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. The GLX should be OFF. 6. SLOWLY open the compressed air supply until gauge reads 8 psi. 7. Check that the hose clamp below the apparatus is OPEN, so that only the TOP disc is free to rotate. Determination of Moment of Inertia of Disc (see Figure 1) 8. Open DataStudio. Turn on the computer and double click the DataStudio icon to open that program. 9. Create Experiment. Click Create Experiment when the Welcome window asks How would you like to Use DataStudio?. Switch on the GLX Digital Adapter. (If this message does not appear, click on New Activity under the File Menu.) 10. If the XplorerGLX File Manager window ever opens, click the Done button to close it. 11. Select General Counting. Select General Counting when the Choose sensor or instrument window appears and click the OK button. 12. 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. 13. 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. 14. 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.

4 4 M 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. 16. Save Experiment. Click on the File Menu and then click on Save Activity to open the Save As Dialog box. Type an appropriate name for this file including your surname and click the Save button. 17. 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, 2010 should be saved under the filename: shadicksander2010jan26trial1. GRAPHICAL ANALYIS OF ACCELERATION DATA 18. Open Excel. Open M.S. Excel by clicking on its icon or by clicking on Start, All Programs, Microsoft Office, Microsoft Office Excel. 19. 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. 20. 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. 21. Insert Pulse Frequency Column. Insert 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 12 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. 22. 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 D 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.

5 5 M 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. 25. 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:. 26. 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. 27. 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 ). 28. 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. 29. 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. 30. 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. 31. Print data table. Click on the File Menu and then click on Print and the 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. Your printouts will be labelled with the LAB number of your computer.

6 6 M 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 an appropriate 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 trend line. 33. Print Graph. Use the File menu to print a copy of your graph. 34. 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. This frequency could easily be converted to a rotational tangential velocity of the rim of the disc by multiplying the frequency by the D = 0.20 cm ± 0.02 cm spacing between bars. (You need not calculate that!) The slope of your acceleration graph (A) 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 D) / R d (5) where R d = (outer) radius of the disc in cm. Record this experimental value and its calculated error in your table of results in the Analysis section of your report. Inelastic Collision Between Ball and Disc (see Figure 3) 35. Set up for Collision. Remove the thread washer with the falling mass. Using the red cap screw with the brass spacer next to the red cap, attach the catch arm atop the small pulley above the centre of the top disc. The brass spacer should now be above the catch arm. The rubber foot on the catch arm should be positioned directly above the small hole in the aluminum disc which was covered with tape. 36. Select New Activity. Click on DataStudio s File Menu and then click on New Activity. Repeat steps 9 12 to prepare DataStudio to accept more data. 37. Start Data Run. Place the ramp on the unit so that the horizontal end is next to the 8.0 cm mark on the catch arm. The catch arm and disc should be stationary. Hold the steel ball at the top of the ramp and release it and immediately press the Start button. 38. Stop Data Run. The ball should enter the catch arm at an impact distance of 8.0 cm and cause the disc to rotate with uniform angular velocity. After at least 1 second of rotation, press the Stop button. 39. Record Pulse Count. Examine the Pulse Count column in DataStudio s data table. In the data table, record C, the second pulse count value displayed after the sudden increase in pulse counts when the collision occurred. (The first reading should be ignored as the disc may not have been moving during the entire count period. Later readings should also be ignored as they will be affected by friction.)

7 7 M Convert to frequency. Convert the Pulse Count value to a frequency by dividing the pulse count by the integration time that you input into DataStudio when you repeated step 12. Record this frequency in your data table. 41. Repeat steps 36 40, using the steel ball at an axis collision distance of 4 cm. 42. OMIT any trials using the aluminium ball. Measurement of Initial Ball Speed (see Figure 2) 43. To measure the velocity of the steel ball as it leaves the ramp, mount the ball ramp on the edge of your table such that the ball will roll off the ramp and follow a parabolic path to the floor. To determine where the ball hits the floor, tape a piece of white paper to the floor in the area where the ball will hit. Now place a piece of carbon paper, carbon side down, over the white paper. The ball should then make a mark at the point it hits the floor. Allow the ball to strike the paper six times. Always release the ball from the same point at the top of the ramp. 44. Measure the vertical distance h that the bottom of the ball moves from the bottom of the ramp to the floor. Measure the horizontal distance x that the ball travels from the bottom of the ramp to the centre of the impact scatter pattern on your paper sheet. 45. Record the mass and diameter of the ball. 46. Record the disc and pulley diameters. Also record the mass of the falling weight. 47. Set up the apparatus so that it is ready for step 8. ANALYSIS 1. In step 34 of the procedure, you should have already converted the slope of the graph to a more meaningful angular acceleration in radians per second 2 units by using the following equation: A D α = (5) exp R d where R d = (outer) radius of the disc D = 0.20 cm ± 0.02 cm spacing between bars A = linear regression value for slope of graph. Also calculate the absolute error in this quantity. 2. Calculate the linear acceleration a of the falling mass using the following equation: a = α r p (6) where r p = radius of the pulley sitting on the disc. Also calculate the absolute error in this quantity. 3. Calculate the torque τ that the falling mass exerts on the disc using the formula τ = Mr p (g a) (7) where M is the mass of the falling weight. Also calculate the absolute error in this quantity.

8 8 M Using these results, calculate the effective moment of inertia of the disc and catch arm and its error by using the formula: τ I d = (8) α 5. Calculate the additional moment of inertia due to the ball in your first collision trial using equation (4) from theory. Also calculate the absolute error in this quantity. 6. Sum the I values from steps 4 and 5 to obtain the total effective moment of inertia I. Tabulate your result for this trial. Also calculate the absolute error in this quantity. 7. Convert and tabulate the rotational frequency f, recorded in step 40, into an angular velocity of the rotating disc as well as its error using the equation: f D ω = (9) R d 8. Calculate and tabulate the final angular momentum value using equation (3) in theory. Also calculate the absolute error in this quantity. 9. Calculate and tabulate the horizontal velocity v of the ball from the data obtained in step 44 and its error by using the equation: g v = x 2 h 10. Calculate and tabulate the initial value of angular momentum using equation (2) from the theory section. Also calculate the absolute error in this quantity. 11. Calculate the difference of angular momentum values Li Lf and the sum of their errors. Also calculate and tabulate the percentage difference between the initial and final values of angular momentum. 12. Repeat Steps 5 11 of your analysis using the data for the 2 nd collision trial. 13. 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. CONCLUSION 1. Did the initial and final values of angular momentum agree within their calculated experimental error? 2. Discuss whether your results indicate that angular momentum was conserved. 3. What expressions were you able to verify? Explain. 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.

9 9 M62 M62.9 CONSERVATION OF ANGULAR MOMENTUM FOR AN INELASTIC COLLISION Data Height of ball above floor, h (cm): Horizontal distance, x (cm) Ball mass m (g) Ball diameter d b (cm) Ball radius r b (cm) Disc diameter d d (cm) Disc radius R d (cm) Pulley diameter d p (cm) Pulley radius r p (cm) Mass of falling weight, M (g) Acceleration from graph (bars/s 2 ) Inelastic Collision Results Tabulate the results of your calculations for each trial as shown. Also calculate the percentage difference between the initial and final values of angular momentum. Due to time constraints, calculus-based error calculations are required only for the first trial. Assume similar errors on angular momentum values in trial 2. v ( cm/ s) R ( c m) m ( g) L i ( g cm 2 / s) I ( g cm 2 ) C f ( bar s per s) ω ( r ad/ s) L f ( g cm 2 / s) Per cent age Di f f er ence i n L ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±