Spring Energy Lab Ritvik Rao, Section A 1
Introduction Purpose: To design and perform an experiment which analyzes the conservation of energy in a spring-based system. Research Question: How will increasing the amount of coils of a slinky held affect how far down the bottom of a slinky will stretch? Hypothesis: As the amount of coils of a slinky held increased, the distance the bottom of a slinky stretched will increase. Coils held (x) -1 2
Procedure Before collecting measurements, the k value needed to be calculated. The slinky was attached to a force sensor and was stretched to various distances. The resulting values were plotted on a force vs. distance graph, and the k constant was the slope of the graph. The next step was to collect the data. For each setting, a number of coils was held at a table level above the ground. Then, the spring was allowed to drop. When the slinky settled, the resulting height was measured, from the top of the slinky. This process was repeated for each trial. After the measurements were collected, theoretical height was calculated, followed by the calculation of the initial and final energy. 3
Diagram g L h 4
Constants and Equations k constant: 0.5287 Gravity (g): 9.8 m/s 2 Length of un-stretched spring (l): 0.065 m Number of coils in spring (N): 33 Mass of spring (M): 0.0643 kg Mass per coil (m): 0.001948 kg h=distance between center of gravity of coiled portion and hanging portion Equation for theoretical L (total length of slinky): M g n2 2 k N 2 (where n=number of dangling coils) 5
Equations (cont.) E i =E f ; therefore, PE gi =PE s PE g =mgh; PE s =(1/2)*k*h 2 mgh=(1/2)*k*h 2 mgh: height refers to the center of mass of the slinky, which is the point where ½ of the mass is above the point and ½ is below (1/2)*k*h 2 : k is the individual spring constant of each coil. As there are 33 coils, k for each coil is 33*.5287=17.4471 mgh=(0.0643)*(9.8)*(16(16+1)/2*(.0643*9.8)/n*1/(n*k) (1/2)*k*h 2 =(1/2)*17.4471*(h 2 ) 6
Potential Energy of the slinky, PEs, J K calculation x PE s 0.6 Potential Energy vs. distance 0.235 0.116 0.335 0.168 0.5 y = 0.5287x - 0.0107 0.435 0.214 0.4 0.535 0.266 0.3 0.635 0.336 0.2 0.735 0.376 0.835 0.427 0.1 0.935 0.486 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Distance of stretched string, x, (m) 7
Table of data # of rings Avg x STDEV %RSD Theo. x % error E i /PE gi E f /PE s % change held (m) (m) of Avg (m) from Avg (J) (J) E i to E f 3 0.582 0.005 0.801 0.498 16.7 0.100 0.218 118.94 5 0.528 0.005 0.922 0.439 20.4 0.100 0.218 118.94 6 0.499 0.005 0.940 0.411 21.5 0.100 0.218 118.94 8 0.444 0.004 0.791 0.358 24.0 0.100 0.218 118.94 10 0.388 0.002 0.534 0.309 25.6 0.100 0.218 118.94 12 0.342 0.004 1.215 0.265 29.2 0.100 0.218 118.94 15 0.274 0.001 0.447 0.207 32.5 0.100 0.218 118.94 18 0.213 0.001 0.534 0.159 34.6 0.123 0.334 170.89 20 0.179 0.002 1.283 0.132 36.0 0.101 0.222 121.09 24 0.122 0.001 1.070 0.092 33.0 0.062 0.084 35.90 Avg 0.854 Avg 27.3 8
Distance of stretch, from top of slinky Graph of measured vs. theoretical height 0.700 Distance vs. held rings 0.600 0.500 y = 0.0004x 2-0.0334x + 0.6824 R² = 0.9998 0.400 0.300 0.200 0.100 y = 0.0005x 2-0.0341x + 0.5959 R² = 1 Average length Expected Length Poly. (Average length) Poly. (Expected Length) 0.000 0 5 10 15 20 25 30 Number of held rings 9
Analysis The measurements of the height of the slinky when dropped were higher than the expected values found using the measured k value, while the form of the regression functions were similar. This shows that repeated stretching of the spring may have caused a change in the k-value, resulting in the difference between the measured and theoretical values. The R 2 values for both the theoretical and measured data were equal to or close to 1, suggesting that the lines of regression were a good fit to their respective data. Because there is a clear line of regression, there is also a strong correlation between the number of held rings and the distance the slinky stretches. The change in energy was above 100 percent for all values except for the 24-ring setting, which was at 35%. 10
Conclusion The measured data supports the hypothesis that the slinky will stretch less if the number of held rings increased. Based off of the comparison of the theoretical and measured slinky lengths, it can also be concluded that the k value of the slinky decreased as a result of stretching during experimentation. The high percent increases in the final energy can be attributed to the fact that the original measured k-value was used with the measured heights, and that the k-value was not adjusted for the energy calculations. If the theoretical data was used to calculate energy, there would have been a lower percent change. Future extensions include calculating energy in a falling slinky or testing Slinkys of different material. 11
Appendix Derivation of height with Riemann sums: Based on Hooke s law, F=k x, and therefore x=f/k. This must be calculated coil by coil The k-value for each coil is the same, no matter how many coils are hanging. This is simply the number of coils multiplied by the k-value. The force on the bottom coil can be written as F 1, and the force on each subsequent coil is af 1, where a is the order of coil. For example, the second coil has a force 2F 1, because it is supporting the first coil. This sum can be written as: (F 1 /nk)+ +(nf 1 /nk), which simplifies to (n(n+1)/2)*(0.0643*(n/33)*9.8)*(1/nk) (F 1 =mass of hanging portion of spring*gravity) n(n+1)/2 can be changed based on how many rings should be included. For example, substitute n=16 for finding the height of the first 16 rings (but only in this part of the equation) 12
Pictures Above: The measurement of the k constant; To the right: An example of the setup of the apparatus 13
Rings Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average STDEV 3 0.581 0.584 0.585 0.574 0.585 0.582 0.005 5 0.520 0.533 0.530 0.529 0.529 0.528 0.005 6 0.495 0.495 0.497 0.503 0.505 0.499 0.005 8 0.443 0.442 0.439 0.446 0.448 0.444 0.004 10 0.388 0.386 0.387 0.391 0.390 0.388 0.002 12 0.342 0.340 0.337 0.346 0.347 0.342 0.004 15 0.275 0.274 0.272 0.275 0.274 0.274 0.001 18 0.212 0.214 0.213 0.213 0.215 0.213 0.001 20 0.180 0.181 0.182 0.177 0.177 0.179 0.002 24 0.122 0.123 0.120 0.123 0.121 0.122 0.001 # of rings Avg x STDEV %RSD Theo. x % error E i /PE gi E f /PE s held (m) (m) of Avg (m) from Avg (J) (J) 3 0.582 0.005 0.801 0.498 16.7 0.100 0.218 0.163731 5 0.528 0.005 0.922 0.439 20.4 0.100 0.218 0.175426 6 0.499 0.005 0.940 0.411 21.5 0.100 0.218 0.181924 8 0.444 0.004 0.791 0.358 24.0 0.100 0.218 0.196477 10 0.388 0.002 0.534 0.309 25.6 0.100 0.218 0.213562 12 0.342 0.004 1.215 0.265 29.2 0.100 0.218 0.233902 15 0.274 0.001 0.447 0.207 32.5 0.100 0.218 0.272885 18 0.213 0.001 0.534 0.159 34.6 0.123 0.334 0.327462 20 0.179 0.002 1.283 0.132 36.0 0.101 0.222 0.377841 24 0.122 0.001 1.070 0.092 33.0 0.062 0.084 0.545771 Avg 0.854 Avg 27.3
Distance of stretch, from top of slinky # of rings Avg x STDEV %RSD Theo. x % error E i /PE gi E f /PE s % change held (m) (m) of Avg (m) from Avg (J) (J) E i to E f 3 0.582 0.005 0.801 0.498 16.7 0.100 0.218 118.94 5 0.528 0.005 0.922 0.439 20.4 0.100 0.218 118.94 6 0.499 0.005 0.940 0.411 21.5 0.100 0.218 118.94 8 0.444 0.004 0.791 0.358 24.0 0.100 0.218 118.94 10 0.388 0.002 0.534 0.309 25.6 0.100 0.218 118.94 12 0.342 0.004 1.215 0.265 29.2 0.100 0.218 118.94 15 0.274 0.001 0.447 0.207 32.5 0.100 0.218 118.94 18 0.213 0.001 0.534 0.159 34.6 0.123 0.334 170.89 20 0.179 0.002 1.283 0.132 36.0 0.101 0.222 121.09 24 0.122 0.001 1.070 0.092 33.0 0.062 0.084 35.90 Avg 0.854 Avg 27.3 0.005909 0.576 0.184198 0.148847 0.158149 0.009848 0.518 0.197355 0.148847 0.158149 0.011818 0.487 0.204664 0.148847 0.158149 0.015758 0.428 0.221037 0.148847 0.158149 0.019697 0.369 0.240258 0.148847 0.158149 0.023636 0.319 0.263139 0.148847 0.158149 0.029545 0.244 0.306996 0.148847 0.158149 0.035455 0.178 0.195673 0.195673 0.039394 0.140 0.159703 0.159703 0.047273 0.075 0.098164 0.098164 D 0.700 0.600 0.500 0.400 0.300 y = 0.0005x 0.200 2-0.0341x + 0.59 R² = 1 0.100 0.000 0 5 10 Num
Distance vs. held rings y = 0.0004x 2-0.0334x + 0.6824 R² = 0.9998 Average length Expected Length 959 Poly. (Average length) Poly. (Expected Length) 0 15 20 25 30 mber of held rings