Simple Harmonic Motion II

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1 Simple Harmonic Motion II Objectives In this lab you will investigate the relationship between the kinetic energy and elastic potential energy of a mass attached to a Hooke s law spring in simple harmonic motion. test the Law of Mechanical Energy Conservation. measure the position, velocity, kinetic energy, potential energy, and total mechanical energy using the Vernier Motion detector. plot your data and analyze it using the Vernier Logger Pro software. Equipment Vernier Motion Detector, Vernier LabPro system (includes computer and Logger Pro ), support bracket (clamped to a small stand), a set of slotted masses, mass hanger, tape, string, rubber bands, and a meterstick. Theory Suppose an object of mass m is attached to a vertical Hooke s law spring of constant k. The mass hangs from the spring at rest after stretching it a distance x. At this point the spring force (given by Hooke s law kx) exactly matches the weight mg (see Figure 1). stretch x kx Figure 1 As m is varied, the spring force kx increases. If we set y = mg then we expect our data to follow a straight line of the form y = kx where k represents the slope of the line. Now displace the mass by a small amount and release it will execute simple harmonic motion (SHM) with a total mechanical energy E given by mg Equation 1 In the absence of friction, E is conserved (Law of Mechanical Energy Conservation). Page 1 of 7

2 Set-Up and Procedure I. Measuring the Spring Constant 1) Refer to lab Simple Harmonic Motion I, Spring Data and Analysis Steps 1 and 2. Place only a 50-gram slotted mass on the hanger and secure it with a rubber band (making a total mass of kg). 2) Open the experiment file 17b Energy in SHM. Logger Pro is now set up to plot the applied weight vs. position. Click the Collect button to begin data collection. Let the mass to hang motionless. Click on the KEEP button and enter 0.98, the weight of the mass in newtons (N). Press ENTER to complete the entry. 3) Repeat Step 2 with 150, 250, 350 grams on the hanger (making total masses of kg, kg, and kg, respectively). Record the stretch distance d in meters and enter the weights mg in newtons. When you are done, click STOP to end data collection. 4) Click the LINEAR FIT button to fit a straight line to your data. The absolute value of the slope equals the spring constant k in N/m (see Graph 1). Record this value in the first question of the data sheet. Note: If any of your graphs are too noisy (especially your velocity graphs), click the Data Collection button and enter a collection rate of 30 (per second) instead of 50. II. Measuring Kinetic and Potential Energies 1) Open Experiment file 17c Energy in SHM. In addition to plotting position and velocity, three new data columns will appear (kinetic energy, elastic potential energy, and the total mechanical energy). Select Column Options/Kinetic Energy from the Data menu and click on the Column Definition tab. Make sure that the mass is set to 0.20 kg, then click DONE. Now change the spring constant to your value (from Step 3 above) in the potential energy column. 2) Place 150 grams of mass on the hanger, making a total 200 grams including the mass of the hanger. When the spring/mass system becomes motionless, click the ZERO button to calibrate the motion detector. From now on, all distances will be measured relative to this position. Note that the position will be reported negative whenever the mass moves toward the detector. 3) Tape your meterstick to the lab table close to the spring/mass system (see Photo 3 in lab Simple Harmonic Motion I). 4) Displace the mass about 5 cm upward, then release in order to set up SHM. Click Collect to gather position, velocity, and energy data. If your position graph does not resemble Graph 2 (e.g. it is not symmetric about the time axis), repeat Steps 1 and 2 above to re-calibrate the system. If your problem persists, ask your instructor for help. Print a symmetric Position-Time and Velocity-Time graph (use the Print Graph command). Page 2 of 7

3 5) Now click on the title of the first graph (on the y axis - Position) and select Kinetic Energy. Click on the title of the second graph (on the y axis - Velocity) and select Potential Energy. You should see a display similar to Graph 2. You may have to adjust the scales of the axes in each graph in order to best view the data. Use the X button to answer Questions 2-6 on your data sheet. 6) Click on the title of the first graph and choose Position (or Velocity). Click on the title of the second graph and choose Potential (or Kinetic) Energy. You should see a Position-Time plot and a Potential Energy-Time plot (or Velocity-Time and Kinetic Energy-Time) as displayed in Graph 3. Answer questions 7 and 8 on your data sheet. 7) Click on the title of the first graph and choose Position. Click on the second graph then choose Option from the toolbar. Pick Graph Option and click on Axis Option. Check Potential, Kinetic and Total Energies. Now you should have all 3 energies on the second graph (see Graph 4). Print these two graphs and answer question 9 on your data sheet using the Print Graph command. Each student is required to submit a completed data sheet in order to receive full credit. Your lab group needs to submit only one copy of each graph. These copies are to be stapled to the data sheet of one of your lab partners - Each lab partner does not need to submit his/her own graphs. Graph 1 Page 3 of 7

4 Graph 2 Page 4 of 7

5 Graph 3 Page 5 of 7

6 Graph 4 Page 6 of 7

7 Data Sheet - Simple Harmonic Motion II Name Partners Names Date 1) Record the absolute value of the slope of Force-Position graph: This is the spring constant k. 2) What is the maximum kinetic energy? What is the minimum kinetic energy? 3) What is the maximum potential energy? What is the minimum potential energy? 4) Why are the kinetic and potential energies always positive even when the displacement and velocity are negative? (Hint: Consider Equation 1.) 5) Select one of the peaks in the kinetic energy graph. Now go to the potential energy graph and find the energy at the same time value. Is the potential energy at its maximum or minimum? 6) Select one of the peaks in the potential energy graph. Now go to the kinetic energy graph and find the energy at the same time value. Is the kinetic energy at its maximum or minimum? 7) Select one of the peaks in the potential energy graph. Now go to the position graph and record the displacement at the same time value:. Select the next peak in the potential energy graph. Now go to the position graph and record the displacement at the same time value:. Are these two values the maximum displacements of the object from its equilibrium position (i.e. x = ±A, where A is the amplitude)? (Yes/No) Are these two values equal to the equilibrium position (i.e. x = 0? (Yes/No) 8) Select one of the peaks in the kinetic energy graph. Now go to the velocity graph and record the velocity at the same time value:. Select the next peak in the kinetic energy graph. Now go to the velocity graph and record the velocity at the same time value:. Do these two values correspond to the object s greatest speeds? (Yes/No) 9) From the total energy can you say that the sum of kinetic and potential energies was constant at all times? (Yes/No) When you study Equation 1, do you expect to get a flat line for your total energy graph? (Yes/No) Page 7 of 7

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