Work and Energy Experiments Experiment 16 When a juggler tosses a bean ball straight upward, the ball slows down until it reaches the top of its path and then speeds up on its way back down. In terms of energy, when the ball is released it has kinetic energy, KE. As it rises during its free-fall phase it slows down, loses kinetic energy, and gains gravitational potential energy, PE. As it starts down, still in free fall, the stored gravitational potential energy is converted back into kinetic energy as the object falls. If it were to bounce, the ground does work on the object to give it kinetic energy and the ball will move back up again. If there is no work done by frictional forces, the total energy will remain constant. In this experiment, we will see if this works out for the toss of a ball. Because we don't want to smash our motion detectors, we will use the time between bounces of a ball to simulate a ball being thrown upward (the ground throws the ball up.) In this lab, only consider the time between when a ball leaves the ground, and then returns. Motion Detector Table Objectives: Measure the change in the kinetic and potential energies as a ball moves in free fall. See how the total energy of the ball changes during free fall. Preliminary Questions For each question, consider the free-fall portion of the motion of a ball tossed straight upward, starting just as the ball is released (left the ground) to just before it is caught (hits the ground.) Assume that there is very little air resistance. 1. What form or forms of energy does the ball have while momentarily at rest at the top of the path? 2. What form or forms of energy does the ball have while in motion near the bottom of the path? 3. Sketch a graph of velocity vs. time for the ball. 4. Sketch a graph of kinetic energy vs. time for the ball. 5. Sketch a graph of potential energy vs. time for the ball. 6. If there are no frictional forces acting on the ball, how is the change in the ball s potential energy related to the change in kinetic energy? Procedure: 1. Measure and record the mass of the ball you plan to use in this experiment.
2. Open the file in the Experiment 16 folder of Physics with Computers. Two graphs are initially displayed on the screen. The two graphs are distance vs. time and velocity vs. time. The horizontal axis has time scaled from 0 to 3 s. 3. Connect the Motion Detector to DIG/SONIC 2 of the LabPro. Place the Motion Detector as shown above 4. Drop the ball directly below the motion detector. Verify that the distance vs. time graph corresponding to the free-fall motion is parabolic in shape, without spikes or flat regions, before you continue. This step may require some practice. If necessary, repeat the toss, until you get a good graph. When you have good data on the screen, proceed to the Analysis section. Analysis: 1. Click on the Examine tool, and move the mouse across the distance or velocity graphs of the motion of the ball to answer these questions. a. Identify the portion of each graph where the ball had just left the ground and was in free fall. Determine the height and velocity of the ball at this time. Enter your values in a data table. b. Identify the point on each graph where the ball was at the top of its path. Determine the time, height, and velocity of the ball at this point. Enter your values in a data table. c. Find a time where the ball was moving downward, but a short time before it was caught. Measure and record the height and velocity of the ball at that time. d. For each of the three points in the data table, calculate the Potential Energy (PE), Kinetic Energy (KE), and Total Energy (TE). Use the ground as the zero of your gravitational potential energy. 2. How well does this part of the experiment show conservation of energy? Explain. 3. Logger Pro can graph the ball s kinetic energy according to KE = ½ mv 2 if you supply the ball s mass. To do this, choose Modify Column Kinetic Energy from the Data menu. You will see a dialog box containing an approximate formula for calculating the KE of the ball. Edit the formula to reflect the mass of the ball. 4. Click on the top graph s vertical axis label and change the display to Kinetic Energy, KE. 5. Inspect your kinetic energy vs. time graph for the toss of the ball. Explain its shape. 6. Logger Pro can also calculate the ball s potential energy according to PE = mgh. Here m is the mass of the ball, g the free-fall acceleration, and h is the distance measured from the ground to the ball. As before, you will need to supply the mass of the ball. To do this, choose Modify Column Potential Energy from the Data menu. You should modify the equation to reflect height measured from the ground. The equation is set up as if the motion detector were on the ground. 7. Click on the bottom graph s vertical axis and change the display to the Potential Energy, PE. 8. Inspect your potential energy vs. time graph for the free-fall flight of the ball. Explain its shape. 9. Sketch the two energy graphs. 10. Compare your energy graphs predictions (from the Preliminary Questions) to the real data for the ball toss. 11. Logger Pro will also calculate Total Energy, the sum of KE and PE, for plotting. Click on a graph s
vertical axis label and display the Total Energy, TE. Sketch the graph. 12. What do you conclude from this graph about the total energy of the ball as it moved up and down in free fall? 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? Since we are only considering the time between when the ball left the ground and returned to the ground, a loss of energy due to bouncing is not an acceptable answer. Conclusion: 1. What would change in this experiment if you used a very light ball, like a beach ball? 2. What would happen to your experimental results if you entered the wrong mass for the ball in this experiment? Experiement 17 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 2 KE mv The spring can hold elastic potential energy, or PE elastic. We calculate PE elastic by using PE = 1 2 elastic = 1 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 (PE gravitational = 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 + PE elastic = 0, which we can test experimentally. Objectives: Examine the energies involved in simple harmonic motion. Test the principle of conservation of energy. Preliminary questions 1. Sketch a graph of the height vs. time a mass on a 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. ky 2
4. Sketch graphs of kinetic energy vs. time and elastic potential energy vs. time. Procedure 1. Mount a 1kg mass and spring as shown in Figure 1. Connect the Motion Detector to DIG/SONIC 2 of the LabPro. Position the Motion Detector directly below the hanging mass, taking care that no extraneous objects could send echoes back to the detector. Be extremely careful you do not drop anything on the detector The mass should be about 60 cm above the detector when it is at rest. Using amplitudes of 10 cm or less will then keep the mass outside of the 40 cm minimum distance of the Motion Detector. 2. Open the Experiment 17 folder from Physics with Computers. Then open the experiment file Exp 17a. Two graphs should be displayed on the screen. The top graph is distance vs. time, with the vertical axis scaled from 0 to +2 m. The lower graph is velocity vs. time with the vertical axis scaled from 2 to 2 m/s. The horizontal axis on both graphs has time scaled from 0 to 5 s. The data collection rate is 50 samples/s. 3. Start the mass moving up and down by pulling it about 10 cm and then releasing it. Take care that the mass is not swinging from side to side. Click collect. Sketch the graphs. How do they compare to your predictions? Figure 1 4. 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. Open the experiment file Exp 17b Spring Constant. Logger Pro is now set up to plot the applied weight vs. distance. 5. Click collect to begin data collection. Hang a mass from the spring and allow the mass to hang motionless. Enter weight of the mass in newtons (N). Press ENTER to complete the entry. Now hang 3 more masses from the spring, recording the position and entering the weights in N. 6. Click on the Regression Line tool, to fit a straight line to your data. The magnitude of the slope is the spring constant k in N/m. Record the value. 7. Use a mass of between 500g and 1kg for the below experiements. 8. Open the experiment file Exp 17c. In addition to plotting position and velocity, three new data columns have been set up in this experiment file (kinetic energy, elastic potential energy, and the sum of these two individual energies). You may need to modify the calculations for the energies. If necessary, choose Modify Column Kinetic Energy from the Data menu and substitute the mass of your hanging mass in kilograms for the value 0.20 in the definition. Similarly, change the spring
constant you determined above for the value 5.0 in the potential energy column. 9. With the mass hanging from the spring and at rest, zero the Motion Detector. From now on, all distances will be measured relative to this position. When the mass moves closer to the detector, the distance reported will be negative. 10. Start the mass oscillating in a vertical direction only, with an amplitude of about 10 cm. Gather position, velocity, and energy data. Analysis 1. Click on the y-axis label of the velocity graph to choose another column for plotting. Uncheck the velocity column and select the kinetic energy and potential energy columns. 2. Compare your two energy plots to the sketches you made earlier. Be sure you compare to a single cycle beginning at the same point in the motion as your predictions. Comment on any differences. 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. Check your prediction. Click on the y-axis label of the energy graph to choose another column for plotting. Select the total energy column in addition to the other energy columns. 5. From the shape of the total energy vs. time plot, what can you conclude about the conservation of mechanical energy in your mass and spring system? Conclusion 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. Discuss how this experiment would be performed