8-Apr-2015: Conservation of energy of a mass-spring system

PURPOSE

The purpose of this experiment was to apply what we learned about energy and see if a system follows the law of conservation of energy.

PROCEDURES

Figure 1: Derivation of the gravitational potential energy of the spring

Figure 2: Derivation of the kinetic energy of the spring

Before setting up our experiment, we came up with a way to relate the gravitational potential energy and the kinetic energy of the spring to known quantities. We related the gravitational potential energy of the spring to the position of the bottom of the mass hanger shown in Figure 3 (circled in blue). Then, we related the kinetic energy of the spring to the velocity of the bottom of the mass hanger. The derivations of these relationships are shown in Figures 1 and 2 above.

Figure 3: Set-up of our experiment

We started this lab by setting up the system shown in Figure 3. There are a couple of things to note about this set-up. First, the force sensor (circled in green) was calibrated before proceeding with the measurements. Second, an index card was taped to the bottom of the mass hanger (circled in blue) so that the motion sensor (circled in red) could clearly identify its location. The motion sensor was zeroed with the spring unstretched

After constructing our set-up, we added 200 g to the mass hanger and released the system. We allowed the combined masses to oscillate for a few seconds while the motion sensor gathered data. Using the position data, we were able to find the system's elastic and gravitational potential energy. From the velocity data, we calculated the system's kinetic energy. We then found the total energy of the system from these values.

Figure 4: Energy vs time

Figure 5: Energy vs position

Next, we graphed these derived values with respect to time and then with respect to position. These graphs are shown above in Figures 4 and 5. The shapes of these graphs were what we expected to see. In the first graph, it can be seen that the times at which gravitational potential energy was at its maximum, the elastic potential energy was at its minimum. This makes sense because as the mass moves farther away from the ground, it gains more gravitational potential energy. At the same time, the spring becomes less stretched and the elastic potential energy goes down. The slope of the gravitational potential energy versus position graph was positive, while the slope of the kinetic energy versus position graph was negative for the same reasons.

CONCLUSION

In this experiment, we witnessed firsthand that energy is conserved in a system. It was interesting to see that the graphs shown in Figures 4 and 5 were so close to what we expected. However, there were still some disparities between what we expected and the experimental values. There are several possible causes for this result. First, the sample rate at which the data was collected may not have been big enough to fully capture the motion of the system since the system was oscillating at such a fast rate. Another reason why there were disparities between the expected values and the experimental ones may have been that the position of the bottom of the mass hanger may not have been recorded correctly by the motion sensor. Since there was so much movement by the mass hanger, its bottom may not have been in clear view of the motion sensor at all times.

Despite these sources of error, we were able to get results that were relatively accurate. Therefore, we concluded that our experiment was a success.