SPRING ENERGY LAB Sean Flanagan Section A December 9, 2016 Lab Partners: M. Chava, J. Barnhart
Introduction The purpose of this experiment was to design an apparatus that used spring energy to test the Law of Conservation of Energy. How will increasing the distance a spring is compressed (Δx) affect the max height (h f ) of an object attached to a pendulum when the spring is released and contacts the mass? It was hypothesized that an increase in the distance that the spring is compressed will result in an increased maximum height of the object where Δx h f.
Methodology Setup The nut was tied to a pipe on the ceiling of Mr. Ellis room to create a pendulum. The metal dowel was held in place on top of a table with duct tape. The string was made long enough so that the nut could slide onto the dowel without slack and the string be perpendicular to the dowel. Markings were made on the end of the dowel 1 cm apart for 5 intervals (starting at 2 cm) using a permanent marker. One small binder clip was attached to the dowel in a manner that allowed the spring and nut to slide onto the dowel and not extend farther than the length of the dowel. A second binder clip was attached to the dowel to compress the nut against the spring (against the first clip).
Methodology (Cont.) Testing The front binder clip was pushed back along the dowel to compress the spring to the marked intervals (2-6 cm). At each interval, the front binder clip was removed and the spring decompressed to launch the nut off the dowel and into an upward swing. The final height of the nut was measured using a tape measure. 10 trials were conducted at each setting and the final heights were recorded in a data table.
Diagram a g h f PE gf PE s m n v i h i PE gi
Constants and Equations Constants m n = 31.96 g h i = 74.15 cm g = 9.8 m/s 2 Equations h TT = 1 2 kk xx2 + mmmmh ii mmmm k = 235.53 N/m (See Appendix B for full derivation of equation) (See Appendix A)
Data Summary Δx h avg STDEV %RSD h t %err TE i TE f Change % E (m) (m) of h avg of t avg (m) of h (J) (J) (J) IV1 IV2 IV3 IV4 IV5 0.02 0.900 0.00992 1.10213 0.892 0.94198 0.279 0.282 0.933 0.03 1.046 0.01338 1.27955 1.080 3.15739 0.338 0.328-3.260 0.04 1.344 0.03654 2.71846 1.343 0.08236 0.421 0.421 0.082 0.05 1.489 0.05698 3.82734 1.681 11.4654 0.527 0.466-12.950 0.06 1.714 0.0279 1.6281 2.095 18.2038 0.656 0.537-22.255 Avg 2.11 Avg 6.77
Graph Final Height vs. Compression (Δx) 2.2 Final height (h f, m) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 h T = 17.077Δx 0.7705 R² = 0.965 h f = 9.0233Δx 0.5975 R² = 0.9797 Observed Theoretical 0.6 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Compression (Δx, m)
Images Figure 1 The spring launch being loaded and set to 4 cm. Figure 2 The final height (h f ) being measured with a tape measure.
Analysis The strength of the mathematical model produced was high with an R 2 value of 0.98. The precision of the experiment was also high with an average RSD value of 2.11%. The accuracy of the experiment was moderate with an average percent error value of 6.77. A power trend-line was used to fit the data because of the data seemed to increase, more or less, at a constant rate. The data collected was lower than the theoretical data on average. This translated into final energy values that were lower than the initial energy values. This seems to contradict the Law of Conservation of Energy, however, these discrepancies can be accounted for with the sources of error.
Conclusions The hypothesis was supported as the final heights did increase as the compression distance increased. However, the observed values were lower than the theoretical values. The Law of Conservation of Energy states that Energy Initial ± Work = Energy Total. In the experiment, a few different types of work and energy were not accounted for. The first air resistance, or the work of Drag would have lowered the observed values. Next, thermal energy produced a result of motion also would have contributed to the lower values. The nut was also on the dowel which means that the work of friction was also not accounted for. Lastly, the spring that was used to launch the pendulum also launched off of the dowel. This action takes energy and also would have removed some from the system. In total, the factors described removed energy from the system and the resulting values, being lower than theoretical, support this.
Future Extensions In the future, this lab could be improved with a number of adjustments. The spring used could be fixed to the dowel to eliminate the energy used. The heights could be found using video analysis to heighten the precision of the experiment. The force of friction could be measured on the dowel to account for another type of work. Lastly, the whole experiment could be adapted to work in an environment where thermal energy would be a minimal factor.
Appendix A (Spring Constant) Spring Constant: Raw Data Δx Force (m) (N) 0.01 2.219 0.02 4.181 0.03 7.13 0.04 9.312 0.05 11.43 Force (N) 12 10 8 6 4 2 0 Force vs. Displacement (Spring Constant) F = 235.53Δx - 0.2115 R² = 0.9962 0 0.01 0.02 0.03 0.04 0.05 0.06 Displacement (Δx, m)
Appendix B (Equation Derivation) EE ii = EE ff PPEE ss + PPEE gggg = PPEE gggg 1 2 kk xx2 + mmmmh ii = mmmmh ff h ff = 1 2 kk xx2 + mmmmh ii mmmm
Appendix C (Full Data) Δx h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h 10 h avg STDEV %RSD h t %err TE i TE f % E Change (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) of h avg of t avg (m) of h (J) (J) (J) IV1 0.02 0.905 0.897 0.887 0.904 0.913 0.885 0.916 0.899 0.901 0.896 0.900 0.010 1.102 0.892 0.942 0.279 0.282 0.933 IV2 0.03 1.032 1.043 1.039 1.024 1.036 1.063 1.048 1.064 1.057 1.052 1.046 0.013 1.280 1.080 3.157 0.338 0.328-3.260 IV3 0.04 1.326 1.386 1.383 1.265 1.339 1.342 1.337 1.324 1.357 1.383 1.344 0.037 2.718 1.343 0.082 0.421 0.421 0.082 IV4 0.05 1.571 1.459 1.536 1.476 1.443 1.567 1.433 1.511 1.403 1.488 1.489 0.057 3.827 1.681 11.465 0.527 0.466-12.950 IV5 0.06 1.734 1.681 1.719 1.697 1.679 1.763 1.746 1.702 1.695 1.721 1.714 0.028 1.628 2.095 18.204 0.656 0.537-22.255 Avg 2.11 Avg 6.77