Energy in Simple Harmonic Motion

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

The spring can hold elastic potential energy, or PEelastic. We calculate PEelastic by using

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 DKE + DPEelastic = 0, which we can test experimentally.

objectives

·  Examine the energies involved in simple harmonic motion.

·  Test the principle of conservation of energy.

Materials

LabPro or CBL2 interface
TI Graphing Calculator / slotted mass set with hanger, 50 g to
300g in 50-g steps
DataMate program / wire basket
Vernier Motion Detector / ring stand
spring, 10-20 N/m / 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.

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.

5. 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.

Figure 1

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 75cm 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. Connect the Vernier Motion Detector to the DIG/SONIC1 port of the LabPro or the DIG/SONIC port of the CBL2 interface. Use the black link cable to connect the interface to the calculator. Firmly press in the cable ends.

4. Set up the calculator for the Motion Detector. Start the DATAMATE program and press to reset the program.

  1. Select SETUP from the main screen.
  1. Press to select MODE and press .
  1. Select TIME GRAPH from the SELECT MODE menu to collect motion data as a function of time.
  1. Select CHANGE TIME SETTINGS from the TIME GRAPH SETTINGS menu to set the data collection rate.
  1. Enter “0.02” as the time between samples in seconds (remember to finish your entry with ). If you are using a TI-73 or TI-83, use “0.04” for the time between samples.
  1. Enter “100” as the number of samples. If you are using a TI-73 or TI-83, use “50” as the number of samples.
  1. Select OK twice to return to the main menu.

5. 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.

  1. Select ZERO from the setup menu.
  1. Select DISTANCE.
  1. With the mass hanging at rest from the spring, press to zero the Motion Detector .

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.

6. 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. Select Start to begin data collection.

7. After 2 seconds, data collection will stop. Press to view the distance graph. 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, select MAIN MENU and the select START to repeat data collection.

Analysis

1. From the position and velocity of the mass data you can calculate the spring potential energy and the kinetic energy. To do this will require manipulating the data outside of the DataMate program, so select MAIN SCREEN and then QUIT. Now you will calculate the kinetic energy KE using ½mv2, storing the data in list seven. The velocity data are stored in list L7. Where m is the mass of your bob in kg, press 0.5 m L7^2 L7. This will replace the distance data by the kinetic energy data. (On the TI-83 and 83 Plus, to enter L7 you must press [LIST], and then choose L7 from the list. It will be entered as LL7. To enter L2 or any of the first six lists, press and then the corresponding digit. On the TI-73, access lists by pressing [STAT]. You will then see a list of available lists. You will need to scroll down using the cursor keys to see all the lists. On other calculators directly enter the list name using the alphanumeric keys.)

2.   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 previous lab. The PE result is stored at each time in list six. The distance data are stored in list L6. Where the 10 is the spring constant in N/m (use your own value), press 0.5 10 L6^2 L6. Enter the list names as you did in the previous step for your particular calculator.

3. Finally, calculate the total energy KE+PE, and store the result in list eight. To do this, press L6 L7 L8. Enter the list names as you did in the previous step for your particular calculator.

4. Plot the KE, PE, and total energy on your calculator:

TI-73, TI-83, TI-83 Plus, and TI-84 Plus Calculators

  1. Press [stat plot] and select Plot 1. ([PLOT] on the TI-73.)
  1. Use the arrow keys to position the cursor on each of the following Plot1 settings. Press to select any of the settings you change: Plot1 = On, Type =, Xlist = L 1, Ylist=L7, and Mark =. This is the kinetic energy.
  1. Press [stat plot] and select Plot 2.
  1. Use the arrow keys to position the cursor on each of the following Plot2 settings. Press to select any of the settings you change: Plot2 = On, Type =, Xlist = L 1, Ylist=L6, and Mark = +. This is the potential energy.
  1. Press [stat plot] and select Plot 3.
  1. Use the arrow keys to position the cursor on each of the following Plot3 settings. Press to select any of the settings you change: Plot3 = On, Type =, Xlist = L 1, Ylist=L8, and Mark = •. This is the total energy.
  1. Press and then select ZoomStat (use cursor keys to scroll to ZoomStat) to draw a graph with the x and y ranges set to fill the screen with data.

TI-89, TI-92, and TI-92Plus

  1. Press , select Data/Matrix Editor, then Current. Press to select the Plot Setup menu.
  2. Using the cursor keys, highlight Plot 1 and press to select it.
  3. Choose Scatter for the Plot Type, and then Box for the Mark.
  4. Use the cursor keys to move to the x line. Press to enter the x-axis. Similarly enter for the y-axis. Press twice. This is the kinetic energy.
  5. Highlight Plot 2 and press to select it.
  6. Choose Scatter for the Plot Type, then Plus for the Mark.
  7. Press to enter the x-axis and to enter the y-axis. Press twice. This is the potential energy.
  8. Highlight Plot 3 and press to select it.
  9. Choose xyline for the Plot Type, then Dot for the Mark.
  10. Press to enter the x-axis and to enter the y-axis. Press twice. This is the total energy.
  11. Press [WINDOW]. Press to access the Zoom menu.
  12. Select ZoomData to fill the graph with your data.

5. Inspect your kinetic energy vs. time graph (marked with) 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.

6. Inspect your elastic potential energy vs. time graph (marked with +) 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.

7. Record the three energy graphs by printing or sketching.

8. Compare your total energy graph prediction (from the Preliminary Questions) to the experimental data for the spring-mass system.

9. What do you conclude from the total energy vs. time graph (marked with •) 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.

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.

Physics with Calculators 23 - 3