Banding Together: Examination of the Conservation of Energy BY: ACHILLES GATSONIS LAB PARTNER: PATRICK TAN DUE: 12/8/2017 P SECTION
Introduction The purpose of this lab was to design and perform an experiment which analyzes the conservation of energy in a spring-based system. How does decreasing the length of the bungee cord affect the final height reached by an object which is dropped from the top of the bungee cord? If the length of the bungee cord is shortened, then the final height reached by the object will increase, where the final height reached by the object is inversely proportional to the length of the bungee cord.
Methodology: Procedure A bungee cord was made by tying rubber bands together. A rubber band was tied to the end of the bungee cord and a ball was placed within the band. To secure the ball within the rubber band, multiple rubber bands were tied around the ball and the aforementioned rubber band. The rubber band at the opposite end of the bungee cord was connected to a hooked metal rod which was perpendicular to the ground and attached to a beam parallel to the ground. The length of the bungee cord was changed by connecting a different rubber band of the bungee cord to the hooked metal rod perpendicular to and above the ground. Afterwards, a meter stick was used to measure the new length.
Methodology: Procedure The ball, attached to the bungee cord, was dropped by Patrick from the height at which the bungee cord was attached to the metal rod. Patrick counted down from 3 before releasing the ball. Each time the ball was released, stickers were placed on the pole by Achilles to mark the optically observed final height reached by the ball. After the ball was dropped 10 times, a meter stick was used by Achilles to measure the heights of the stickers. Patrick recorded the heights into a table.
Methodology: Materials - Rubber bands - Hooked metal rod - Ball - Meter stick - Vernier Force Sensor - Logger Pro - Excel - Word - PowerPoint
Methodology: Diagram Hooked metal rod Parallel beam h i L PE gi g m B Δx h f PE gf + PE s Ball Bungee cord Figure 1. Diagram of the apparatus
Methodology: Constants and Equations m B = 0.090 kg h i = 2.045 m g = 9.8 m/s 2 PE g = m B gh F s = k x s PE s = 1 2 k x x = h i L h f h f = mg(mg + 2kL) mg + h i L k k
Results: Pictures Figure 2. Photograph of Patrick and Achilles during experimentation Figure 3. Photograph of Patrick holding the ball at the initial height
Results: Data Summary Table 1. Summarized data table L h favg STDEV %RSD h ft %err TE i TE f %change (m) (m) (m) of h favg (m) of h f (J) (J) of E IV1 1.255 0.130 0.048 36.681 0.121 7.456 1.808 1.770 2.082 IV2 0.964 0.578 0.043 7.524 0.435 32.994 1.808 1.373 24.056 IV3 0.675 0.992 0.016 1.582 0.840 18.137 1.808 1.418 21.552 IV4 0.449 1.265 0.037 2.956 1.188 6.444 1.808 1.618 10.509 IV5 0.260 1.597 0.022 1.351 1.533 4.203 1.808 1.663 8.013 Avg 10.019 Avg 13.847 Avg 13.243
Final Height, h favg (m) Results: Graph 1.800 1.600 1.400 1.200 1.000 0.800 0.600 0.400 0.200 Final Height vs. Length y = -1.4445x + 1.9534 R² = 0.998 y = -1.4166x + 1.8441 R² = 0.9915 0.000 0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 Length, L (m) Measured Data Theoretical Data
Analysis As L increases, the graph shows a negative trend for h favg. h ft also shows a negative trend as L increases. Linear trends could not be identified for STDEV or %RSD because their values show neither regular increase nor regular decrease as L increases. %err generally shows a negative trend as L decreases. Because h f and L cannot be negative, their values are limited to the set of positive numbers. TE f is lower than TE i because the system is not a closed system; thus, energy will escape from the system, resulting in TE f being lower than TE i. The minimum %err, 4.203%, is seen at the lowest bungee cord length, 0.260 m. The maximum %err, 32.944%, is seen at the bungee cord length of 0.964 m. The average %err, 13.847%, indicates that the data has low accuracy. The average %RSD of the data, 10.019%, indicates that the data had low precision. The R 2 value for the Measured Data trendline, 0.998, indicates that the mathematical model is a strong fit.
Conclusion The hypothesis was that shortening the length of the bungee cord would increase the final height reached by the object. The hypothesis is supported by the results of the experiment. For a follow-up to this lab, the final height reached by the ball could be measured by a sensor. By using a sensor, measured final heights would be both more precise and more accurate because error caused by misgauging the final heights would be eliminated. One source of error could have been misgauging the final height optically. By standing rather than crouching down at roughly the same height as the final height, the final height could have been measured to be higher than it was. A second source of error could have been the movement of the hooked metal rod. Because the metal rod was not fully secured to the beam parallel to the ground, it could have moved slightly, changing the initial height. This movement could have resulted in the ball reaching a height that was higher than the actual final height.
Appendix A: Full Main Data Table 2. Full main data L h f1 h f2 h f3 h f4 h f5 h f6 h f7 h f8 h f9 h f10 h favg STDEV %RSD h ft %err TE i TE f %change (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) of h favg (m) of h f (J) (J) of E IV1 1.255 0.060 0.090 0.100 0.120 0.110 0.142 0.163 0.133 0.149 0.235 0.130 0.048 36.681 0.121 7.456 1.808 1.770 2.082 IV2 0.964 0.687 0.609 0.585 0.579 0.570 0.562 0.551 0.543 0.547 0.548 0.578 0.043 7.524 0.435 32.994 1.808 1.373 24.056 IV3 0.675 1.001 0.997 1.003 1.008 1.011 0.993 0.980 0.971 0.964 0.994 0.992 0.016 1.582 0.840 18.137 1.808 1.418 21.552 IV4 0.449 1.332 1.312 1.295 1.271 1.264 1.243 1.224 1.242 1.232 1.231 1.265 0.037 2.956 1.188 6.444 1.808 1.618 10.509 IV5 0.260 1.641 1.621 1.605 1.599 1.594 1.591 1.595 1.586 1.575 1.566 1.597 0.022 1.351 1.533 4.203 1.808 1.663 8.013 Avg 10.019 Avg 13.847 Avg 13.243
Appendix B: Determination of k To determine the spring constant of the bungee cord, a test was conducted using a Vernier Force Sensor and the Logger Pro program. The rubber band from which the bungee cord was connected to the hooked metal rod was attached to the Vernier sensor. The bungee cord, with the ball still connected, was placed in a relaxed position. The ball was held in place by Patrick while Achilles pulled the sensor. The sensor was pulled to five different distances. This test was done for each bungee cord length. Afterwards, the data was graphed and the lines of best fit were found for each bungee cord. The slopes of these lines were the values for the spring constants. Table 3. Bungee cord lengths Lengths L 1 = 1.255 m L 2 = 0.964 m L 3 = 1.001 m L 4 = 1.312 m L 5 = 1.621 m
Force, F (N) Appendix B: Determination of k Force vs. Stretch Table 4. Vernier Force Sensor test table L 1 L 2 L 3 L 4 L 5 Δx s F F F F F (m) (N) (N) (N) (N) (N) IV1 0.10 0.715 0.664 1.279 1.825 2.540 IV2 0.15 1.110 1.018 1.842 2.347 3.242 IV3 0.20 1.534 1.448 2.187 2.905 4.025 IV4 0.25 1.812 1.784 2.373 3.509 4.788 IV5 0.30 2.265 1.985 2.909 3.520 5.328 6.000 5.000 4.000 3.000 y = 14.244x + 1.1358 R² = 0.9965 y = 9.104x + 1.0004 R² = 0.948 y = 7.582x + 0.6016 R² = 0.974 2.000 y = 6.816x + 0.0166 R² = 0.9863 1.000 0.000 y = 7.604x - 0.0336 R² = 0.9961 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Stretch, Δx s (m) L1 L2 L3 L4 L5
Appendix C: Derivation of h f h f was obtained through the derivation below. The quadratic formula was used in the derivation. Onl one solution was taken, as the other solution was not physically possible. mgh i = mgh f + 1 2 k(h i L h f ) 2 mgh i = mgh f + 1 2 k(h i 2 2h i h f 2h i L + 2h f L + h f 2 + L 2 mgh i = mgh f + 0.5kh i 2 2kh i h f kh i L + kh f L +.5kh f 2 +.5kL 2 0 =.5kh f 2 + mg kh i + kl h f + (.5kh i 2 kh i L +.5kL 2 mgh i ) h f = kh i mg kl h f = kh i mg kl h f = kh i mg kl m 2 g 2 +2mgkL k h f = mg mg+2kl mg+ h i L k k (mg kh i +kl) 2 4(.5k)(.5kh 2 i kh i L+.5kL 2 mgh i ) k m 2 g 2 2mgkh i +2mgkL+k 2 h 2 i 2k 2 h i L+k 2 L 2 (k 2 h 2 i 2k 2 h i L+k 2 L 2 2kmgh i ) k
Appendix D: Derivation of TE i and TE f TE i was obtained through the derivation below. TE i = PE gi TE i = m B gh i TE f was obtained through the derivation below. TE f = PE gf + PE s TE f = m B gh f + 1 2 k x TE f = m B gh f + 1 2 k(h i L h f )