Conservation of Energy as a spring oscillates

You might find it useful to review the manner in which we simplify today's calculation of potential energy.

The goals

Setting up the experiment

  1. Set the slide switch on the force probe to 10 N. Place the motion sensor on the floor. Attach the force sensor to CH 1, and the motion sensor to Dig/Sonic 1.
  2. Grab this file from the course web site:
    If that fails, try instead grabbing a file from a shared folder: Go to "Student Shares" -> "University Physics" -> "Team Physics 311" -> "Lab Pro", and drag the file Conservation_of_Energy (note the underscores) onto your local desktop.

    Double-click on the file on your Desktop to start LabPro.

  3. Calibrate the force sensor, by
  4. Choose one of the tiny springs. Attach the aluminum mass hanger (50 grams) plus an extra 50-gram weight, making the total mass 100 grams. The mass should be about 15 or 20 cm below the bench top at equilibrium. Zero both sensors with the block at rest.
  5. Check your set-up by lifting the mass straight up about 8 cm and releasing it. Avoid side-to-side motion. What do you expect to see for position as a function of time? Collect data; do you see what you expected? If not, adjust the sensors until you get what you expect.
  6. You should see the position oscillate around 0 meters (to within about +/- 0.001 m), and the force oscillate around 0 Newtons (to within about +/- 0.0003 N). The amplitude of the motion should be around 7 or 8 cm. Look carefully; if the centers of oscillation aren't zero, you may have to adjust the readings manually to force them to zero.
    Adjust to zero manually: Return the system to equilibrium and zero all sensors. Collect data with the system at rest. Use Analyze:Statistics to determine the average position of the mass. Click on the oval Labpro icon to bring up the Sensors Window, and click on the motion sensor. On the pop-up menu, click Set Offset. A box will pop up with an initial value of the offset. Subtract the average position from the offset and enter this difference as the new offset, then click OK. (E.g. if the average position is 0.0123 and the initial offset is -0.612345, the new offset is -0.624645.) Collect data again and the position should have an average very close to zero. Close the Sensors window. Each time you zero the motion detector you will need to repeat this.

Look at the highest and lowest displacement the mass reaches. If these two extremes are not the same size, make appropriate adjustments and take new data. If sometimes helps to wait one minute after starting the block in motion before you collect data.

Be absolutely certain that you have the equipment properly zeroed or your results will be bogus!

Collecting a first dataset

Start the mass moving and let it oscillate for about twenty or thirty seconds; higher order vibrations will damp out during this time, leaving very smooth, simple, harmonic motion. Press Collect to gather data. You should see nice smooth sine wave shapes for position and velocity. Before you go any further, ask an instructor to verify that your data is good enough.

Analysis over a few cycles

You can (and should) print out several graphs to illustrate your report. If you want to plot more than one measurement on the vertical axis, click the label on the vertical axis, click More on the pop up menu, and check the boxes for the additional quantities you wish to plot.

  1. Draw on a sheet of graph paper diagram with three panels, in which the panels show the mass On your panels, use arrows to show the displacement from equilibrium, velocity, acceleration, and net force on the mass.
  2. Look at the graph of Position versus Time. It looks sort of like a sine curve, doesn't it? Suppose that we represent the mass' height above equilibrium by the equation
              y(t) = A sin (ω t + φ)  
  3. Compare the Force and Position graphs, and give a verbal description of their relation. Are force and displacement "exactly in phase", "exactly out of phase", or something else? Explain, using your diagram from question 1, and your equations from question 2.

  4. Compare the Force and Acceleration graphs, and give a verbal description of their relation. Are force and acceleration "exactly in phase", "exactly out of phase", or something else? Explain, using your diagram from question 1, and your equations from question 2.

  5. Compare the Force and Velocity graphs, and give a verbal description of their relation. Are force and velocity "exactly in phase", "exactly out of phase", or something else? Explain, using your diagram from question 1, and your equations from question 2.

  6. Find the spring constant k by making a graph of one particular quantity versus another. Print the graph. On your printed graph, make a linear fit to the graph and determine the slope and its uncertainty.
    (Hint 1: the left-hand side should be "Force", in Newtons).
    (Hint 2: the force constant is a positive value)

  7. Find the "total mass" m of the moving system by making a graph of one particular quantity versus another. Again print the graph and make a linear fit.
    (Hint the left-hand side should again be "Force", in Newtons).

    How does this "total mass" compare to the mass of the block plus the hanger? If they are not the same, explain how big the difference is, and explain where the difference ("extra" mass or "missing" mass) originates.

  8. Modify the formulae within LabPro for kinetic energy (KE) and potential energy (U).

    To modify the column for kinetic energy, go to Data: Column Options and choose the name of the column you want to modify: Kinetic Energy. In the Equation box, you should see the formula for kinetic energy, which will look something like this:

               0.5 * 0.111 * "velocity"^2
    You should know the effective mass from part 5. Replace the incorrect mass value of 0.111 with your derived mass from part 5.

  9. In a similar fashion, replace the bogus spring constant in the equation for the Potential Energy column with your actual spring constant. the total mechanical energy, E = KE + U.

  10. Make graphs that allow you to discuss conservation of energy. Examples of graphs might be any one of the energies versus time (or position, or velocity, or acceleration), one energy versus another energy, etc. You need to try different graphs and decide what they tell you about energy conservation.

  11. Is energy conserved during your measurements? There are two ways to answer this question:
    1. what is the "peak-to-peak" change in the total energy, expressed as a fraction of the average total energy? You should be able to state something like "total energy bounces around its mean by around +/- 2 percent."
    2. what is the secular change in the average total energy during the period you measured? You might compare the average energy during the first second of data to the average energy during the final second of measurements.

Analysis over a very long period

You ought to find that total mechanical energy is almost exactly the same over a few cycles of oscillation. But what if we monitor the system over many, many cycles? Change the Experiment->Data Collection parameters so that you measure 10 samples per second over a period of 180 seconds (yes, that's about 3 minutes).

  1. Reset the system to equilibrium and re-zero the sensors
  2. Pull the weight down about 10 cm and release it carefully. Wait about 30 seconds to let the wiggles dissipate
  3. Collect measurements over the long period

    While you are waiting for this to finish, you might perform calculations for the previous portions of the experiment, or start writing your report.

  4. Look at (and print) a graph of Total Energy versus Time. Is energy conserved over this long period?
  5. What is the "half-life" of energy? That is, how long does it take for the total energy to decay to half its original value?
  6. What is the "half-life" of the displacement's amplitude? That is, how long does it take for the maximum displacement during each cycle to decay to half its original value? Is this the same as the "half-life" of energy? Should it be?
  7. How much energy would be left if the spring continued to oscillate for twenty minutes?

Bonus section

Some of the energy lost by the spring and hanger is due to work performed by the force of air resistance. Recall that this force depends on several parameters, one of which is the cross-section area A of the moving object.

  1. What is the cross-section area of your moving object? Express the area in square centimeters.
  2. Create a "sail" by cutting a piece of paper so that it has a larger area -- twice the original A of your object. Tape this sail onto the bottom of the hanger so that the moving object now has double its original area. Repeat the Analysis over a long period procedure (steps 10 to 17 above). Determine the half-life for this modified object.
  3. (If you have time) Now make the "sail" three times the original area A. Determine the half-life again.
  4. (If you have time) Do it again for a sail four times the original area.
  5. In theory, how should the amount of energy lost to air resistance depend on the area of the object?
  6. In practice, how does the amount of energy lost to air resistance depend on the area of the object?
  7. Discuss.

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