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Palm 17 Energy in Simple Harmonic Motion We can describe an oscillating mass in terms of its position, velocity, and acceleration as a function of time. We can also describe the system from an energy perspective. In this experiment, you will measure the position and velocity as a function of time for an oscillating mass and spring system, and from those data, plot the kinetic and potential energies of the system. Energy is present in three forms for the mass and spring system. The mass m, with velocity v, can have kinetic energy KE KE 21 mv 2 The spring can hold elastic potential energy, or PEelastic. We calculate PEelastic by using PE elastic 21 ky 2 where k is the spring constant and y is the extension or compression of the spring measured from the equilibrium position. The mass and spring system also has gravitational potential energy (PEgravitational = mgy), but we do not have to include the gravitational potential energy term if we measure the spring length from the hanging equilibrium position. We can then concentrate on the exchange of energy between kinetic energy and elastic potential energy. If there are no other forces experienced by the system, then the principle of conservation of energy tells us that the sum KE + PEelastic = 0, which we can test experimentally. OBJECTIVES Examine the energies involved in simple harmonic motion. Test the principle of conservation of energy. MATERIALS LabPro interface Palm handheld Data Pro program Vernier Motion Detector spring, 10-20 N/m slotted mass set with hanger, 50 g to 300 g in 50 g steps wire basket ring stand twist ties PRELIMINARY QUESTIONS 1. Sketch a graph of the height vs. time for the mass on the spring as it oscillates up and down through one cycle. Mark on the graph the times where the mass moves the fastest and therefore has the greatest kinetic energy. Also mark the times when it moves most slowly and has the least kinetic energy. Physics with Vernier 17 - 1 Palm 17 2. On your sketch, label the times when the spring has its greatest elastic potential energy. Then mark the times when it has the least elastic potential energy. 3. From your graph of height vs. time, sketch velocity vs. time. 4. Sketch graphs of kinetic energy vs. time and elastic potential energy vs. time. PROCEDURE 1. Attach the spring to a horizontal rod connected to the ring stand and hang the mass from the spring as shown in Figure 1. Securely fasten the 200 g mass to the spring and the spring to the rod, using twist ties so the mass cannot fall. 2. Place the Motion Detector at least 75 cm below the mass. Make sure there are no objects near the path between the detector and mass, such as a table edge. Place the wire basket over the Motion Detector to protect it. 3. Plug the Motion Detector into the DIG/SONIC 1 port of the LabPro interface. Connect the handheld to the LabPro using the interface cable. Firmly press in the cable ends. Firmly press in the cable ends. 4. Set up the handheld and interface for the Motion Detector. Turn on the handheld and start Data Pro, by tapping the Data Pro icon on the Applications screen. Choose New from the Data Pro menu or tap to reset the program. a. On the Main screen, tap . b. While still on the Setup screen, tap . c. Enter “0.02” as the time between samples in seconds using the onscreen keyboard (tap “123”) or using the Graffiti writing area. d. Enter “100” as the number of samples. e. Tap twice to return to the Main screen. Figure 1 5. Make a preliminary run to make sure things are set up correctly. Lift the mass upward about five centimeters and release. The mass should oscillate along a vertical line only. Tap to begin data collection. 6. After 2 seconds, data collection will stop and a graph of distance vs. time will be displayed. The distance graph should show a clean sinusoidal curve. If it has flat regions or spikes, reposition the Motion Detector and try again. To collect again, get the mass moving again, and then tap to repeat data collection. Once you have a clean run, do not move the motion detector or the ring stand for the rest of the experiment. 7. Sketch or print your graphs and compare to your predictions. Comment on any differences. 17 - 2 Physics with Vernier Energy in Simple Harmonic Motion 8. To calculate the spring potential energy, it is necessary to measure the spring constant k. Hooke’s law states that the spring force is proportional to its extension from equilibrium, or F = –kx. You can apply a known force to the spring, to be balanced in magnitude by the spring force, by hanging a range of weights from the spring. The Motion Detector can then be used to measure the equilibrium position. You will plot the distance vs. weight to find the spring constant k. a. When you have finished examining the graphs, tap on the Graph screen, to return to the Main screen. b. On the Main screen, tap . c. On the Setup screen, tap , then choose Events with Entry. d. Enter the Entry Label (Weight) and Unit (N). You can enter this information using the onscreen keyboard (tap “abc”), or by using the Graffiti writing area. e. Tap twice to return to the Main screen. f. Tap to get ready to collect data. In this mode, the position of the mass will only be measured when you choose. You will then enter the weight in newtons on the handheld. 9. Take your force and distance data. a. Hang a 50 g mass from the spring and allow the mass to hang motionless. b. Tap and enter “0.49” using the onscreen keyboard as the weight of the mass in newtons (N). Tap to store the data pair. c. In the same manner, hang 100, 150, 200, 250, and 300 g from the spring, recording the position and entering the weights in N. It is important that the length of the mass not change as the value of the mass is changed. d. After entering the last weight, tap , then tap to view a graph of distance vs. force. 10. The graph of distance vs. force should be linear. The magnitude of the slope is k–1, since force is on the x axis. Find the slope of the data. a. Tap , then tap . b. From the Fit Equation menu, choose Linear. The linear-regression statistics for these two data columns are displayed on the screen. c. The inverse of the slope magnitude is the spring constant k in N/m. Record the slope and its inverse in the data table. d. Tap to display the distance vs. force plot with the fitted equation. e. Tap twice to return to the Main screen. 11. Remove the 300 g mass and replace it with a 200 g mass for the remaining steps. 12. As you did before, set up the handheld for a time graph. a. On the Main screen, tap . b. While still on the Setup screen, tap . c. Enter “0.02” as the time between samples in seconds using the onscreen keyboard (tap “123”) or using the Graffiti writing area. d. Enter “100” as the number of samples. e. Tap to return to the Main screen. Physics with Vernier 17 - 3 Palm 17 13. In order to measure distances from the equilibrium position of the mass, it is necessary to zero the Motion Detector. Measuring from equilibrium allows easy calculation of the elastic potential energy, since the distance will correspond directly to spring stretch or compression. a. Tap . b. With the mass hanging at rest from the spring, tap . When the button changes back to , the process is complete, and readings for the sensor should be close to zero. c. Tap twice to return to the Main screen. Distances will now be measured from the current position of the mass, with displacement above the current position measured as positive. Displacement below the current position will be read as negative. 14. Take distance vs. time data as you did before. Lift the mass upward about five centimeters and release. The mass should oscillate along a vertical line only. Tap to begin data collection. 15. After 2 seconds, data collection will stop and a graph of distance vs. time will be displayed. If it has flat regions or spikes, reposition the Motion Detector and try again. To collect again, get the mass moving again, and then tap to repeat data collection. DATA TABLE Slope of Distance vs. Force m/N Spring constant N/m ANALYSIS 1. From the position and velocity of the mass data you can calculate the spring potential energy and the kinetic energy. First, you will calculate the kinetic energy KE using ½mv2, storing the data in a new column. a. b. c. d. e. On the Graph screen, tap , then tap . Enter the column name (KE) and unit (J). Tap and choose the formula, A * B * X^C. Choose X = DIG/SON 1:VEL(m/s) as the Column for X. For the A = value, enter “0.5” using the onscreen keyboard (tap “123”) or using the Graffiti writing area. f. For the B = value, enter the mass of your bob in kg. g. For the C = value, enter “2”. h. Tap to display the graph of KE vs. time. In a similar way you can calculate the spring’s elastic potential energy (PE) using ½kx2. Here, the stretch or compression of the spring x comes from your distance data and the spring constant k from the linear fit done in step 11. The PE result is stored at each time in a new column. i. On the Graph screen, tap . Then tap and choose New Column. j. Enter the column name (PE) and unit (J). k. Tap and choose the formula, A * B * X^C. l. Choose X = DIG/S 1:DIST(m) as the Column for X. m. For the A = value, enter “0.5”. 17 - 4 Physics with Vernier Energy in Simple Harmonic Motion n. For the B = value, enter the value you obtained for the spring constant, N/m. o. For the C = value, enter “2”. p. Tap to display the graph of PE vs. time. 2. Examine the KE and PE graphs. a. Inspect your kinetic energy vs. time graph for the motion of the spring-mass system by tapping on the y-axis label of the displayed graph and choosing KE(J). Explain its shape. Be sure you compare to a single cycle beginning at the same point in the motion as your predictions. Comment on any differences. b. Tap on the y-axis label of the displayed graph and choose PE(J). Inspect your elastic potential energy vs. time graph for the motion of the spring-mass system. Explain its shape. Be sure you compare to a single cycle beginning at the same point in the motion as your predictions. Comment on any differences. c. Record the three energy graphs by printing or sketching. 3. If mechanical energy is conserved in this system, how should the sum of the kinetic and potential energies vary with time? Sketch your prediction of this sum as a function of time. 4. Finally, calculate the total energy KE + PE, and store the result in a third new column. a. b. c. d. On the Graph screen, tap . Then tap and choose New Column. Enter the column name (TE) and unit (J). Tap and choose the formula, X1 + X2. Choose X1 = KE(J) as the Column for X1. Use the Scroll buttons on the handheld to scroll through the options for the Column for X1. e. Choose X2 = PE(J) as the Column for X2. Use the Scroll buttons on the handheld to scroll through the options for the Column for X2. f. Tap to display the graph of total energy vs. time. 5. Compare your total energy graph prediction (from the Preliminary Questions) to the experimental data for the spring-mass system.What do you conclude from the total energy vs. time graph (marked with a closed circle) about the total energy of the system? Does the total energy remain constant? Should the total energy remain constant? Why? If it does not, what sources of extra energy are there or where could the missing energy have gone? EXTENSIONS 1. In the introduction, we claimed that the gravitational potential energy could be ignored if the displacement used in the elastic potential energy was measured from the hanging equilibrium position. First write the total mechanical energy (kinetic, gravitational potential, and elastic potential energy) in terms of a coordinate system, distance measured upward and labeled y, whose origin is located at the bottom of the relaxed spring of constant k (no force applied). Then determine the equilibrium position s when a mass m is suspended from the spring. This will be the new origin for a coordinate system with distance labeled h. Write a new expression for total energy in terms of h. Show that when the energy is written in terms of h rather than y, the gravitational potential energy term cancels out. 2. If a non-conservative force such as air resistance becomes important, the graph of total energy vs. time will change. Predict how the graph would look, then tape an index card to the bottom of your hanging mass. Take energy data again and compare to your prediction. Physics with Vernier 17 - 5 Palm 17 3. The energies involved in a swinging pendulum can be investigated in a similar manner to a mass on a spring. From the lateral position of the pendulum bob, find the bob’s gravitational potential energy. Perform the experiment, measuring the horizontal position of the bob with a Motion Detector. 4. Set up a laboratory cart or a glider on an air track so it oscillates back and forth horizontally between two springs. Record its position as a function of time with a Motion Detector. Investigate the conservation of energy in this system. Be sure you consider the elastic potential energy in both springs. 17 - 6 Physics with Vernier