Disobeying Hooke’s Law

Disobeying Hooke’s Law Introduction The purpose of this study was to find out what happened when one stretches either one spring or multiple springs, whether they obeyed or disobeyed Hooke’s Law, and how this differs to when one stretches a rubber band. Hypothesis Springs obey Hooke’s law – the extension of a spring should be proportionate to the force applied. [1] However, a spring will only behave in this way until it has reached its elastic limit, after which it will stretch but also deform and will not return to its previous length. Abstract Each spring passed the elastic limit when more than 10N of force was applied, while two springs in parallel passed the limit when more than 22N of force were applied. Measurements were taken while loading the weights onto the weight hanger. However, rubber bands behave quite differently and not linearly, so results were taken both during loading and unloading. Equipment: Retort stand, spring(s) / rubber band, 100g and 50g weights, weight hanger, metre ruler, counterbalance. Record just the length of the spring from the top hook to the bottom hook with no force applied. 2. Set up the equipment as shown in the diagram, hooking the weight hanger onto the bottom of the spring(s) / rubber band. 3. Add weights, after each addition record the value on the ruler from the bottom of the weight hanger. 4. If using a rubber band, also record the values during unloading after removing each weight. spring(s) / rubber band 1. A counterbalance was found to be a necessary addition in order to prevent the equipment from falling over. The hanger’s length (14cm) limits the number of weights available to place. To record values incrementing by 50 grams, the observer placed a 50g weight, 100g weight hanger metre ruler value to record clamp Method retort stand counterbalance table recorded the value and then substituted it for a 100g weight to conserve space on the hanger. Disobeying Hooke’s Law 2 Two springs in series Results The observer repeated the experiment four times: hooking the weights onto just one spring, two springs in series, two springs in parallel and a rubber band. Regular spring The observer clamped the spring and then hooked the weight hanger on the bottom. A metal pointer attached to the bottom of the hanger pointed to the reading on the ruler. The readings for Total spring length do not include the length of the hanger (14cm). The readings for Total extension are equivalent to the readings for Total spring length subtract the length of the spring with no force applied. 1N ≈ 100g (on Earth) Force Total spring Total applied / N length / cm extension /cm 0.0 5.0 0.0 1.0 9.0 4.0 1.5 11.2 6.2 2.0 13.3 8.3 2.5 15.4 10.4 3.0 17.2 12.2 3.5 19.4 14.4 4.0 21.5 16.5 4.5 23.7 18.7 5.0 25.7 20.7 5.5 28.0 23.0 6.0 30.2 25.2 6.5 32.3 27.3 7.0 34.3 29.3 7.5 36.4 31.4 8.0 38.3 33.3 8.5 40.4 35.4 9.0 42.4 37.4 9.5 44.3 39.3 10.0 46.3 41.3 The observer connected one spring onto another to the connect the two in series. The springs passed the elastic limit and irreversibly deformed after 9N force, so a value for 10N was not recorded. Force Total spring Total applied / N length / cm extension / cm 0 10.2 0.0 1 17.2 7.0 2 23.9 13.7 3 31.6 21.4 4 39.6 29.4 5 46.7 36.5 6 54.7 44.5 7 62.4 52.2 8 69.6 59.4 9 79.3 69.1 Two springs in parallel The observer connected two springs in parallel with a metal wire at both top and bottom and attached them to the clamp. Force Total spring Total applied / N length / cm extension / cm 0 3.9 0.0 1 6.2 2.3 2 8.9 5.0 3 10.9 7.0 4 13.0 9.1 5 15.0 11.1 6 17.1 13.2 7 19.0 15.1 8 21.0 17.1 9 23.1 19.2 10 24.9 21.0 11 26.8 22.9 12 28.9 25.0 13 30.8 26.9 14 32.7 28.8 15 34.9 31.0 16 36.8 32.9 17 38.6 34.7 18 40.8 36.9 19 42.6 38.7 20 44.8 40.9 21 47.5 43.6 22 50.5 46.6 Disobeying Hooke’s Law 3 Rubber band (unloading) Rubber band The observer clamped the rubber band at the top and hooked the hanger onto it. Rubber bands behave differently to springs [3], so values were taken both during loading and unloading. Force Rubber Total applied band extension /N unloading / cm length 10 34.2 29.1 9 34.0 28.9 8 33.8 28.7 7 33.4 28.3 6 32.8 27.7 5 32.2 27.1 4 30.9 25.8 3 28.0 22.9 2 21.2 16.1 1 13.1 8.0 0 5.1 0.0 Rubber band (loading) Force applied /N 0 1 2 3 4 5 6 7 8 9 10 Rubber Total band extension / loading cm length 5.1 0.0 8.9 3.8 12.1 7.0 16.0 10.9 19.2 14.1 23.0 17.9 26.4 21.3 28.9 23.8 30.8 25.7 32.6 27.5 34.2 29.1 Discussion and graphs Graphs are plotted with the results of the Total extension column against the Force applied column. Graph 1 – Single spring Extension of a single spring against force applied Force applied (N) 12.0 y = 0.2405x 10.0 8.0 6.0 4.0 2.0 4.0 6.2 8.3 10.4 12.2 14.4 16.5 18.7 20.7 23.0 25.2 27.3 29.3 31.4 33.3 35.4 37.4 39.3 41.3 0.0 0.0 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.0 44.0 Total spring extension (cm) Total spring length / cm Line of best fit This graph shows that the results are linear and that one can express the line of best fit with the formula y = 0.2405x. This equation relates to Hooke’s Law – F = k x e [2], where k = 0.2405. The equation for a line is y = mx + c, however, in this case, c = 0, showing that extension is directly proportionate to the force applied. Disobeying Hooke’s Law 4 Graph 2 – Springs in parallel Extension of springs in parallel against force applied 24 22 Force applied (N) 20 18 16 14 12 10 8 6 4 5.0 2 0.0 2.3 7.0 9.1 11.1 13.2 15.1 17.1 19.2 21.0 22.9 28.8 26.9 25.0 31.0 34.7 32.9 38.7 36.9 40.9 43.6 46.6 y = 0.4818x 0 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.0 44.0 48.0 Spring length (cm) Total extension / cm Best fit (Total spring length) This graph’s line of best fit is also linear and the average spring constant here is almost precisely twice as large, due to the two springs in parallel acting in the same way as one spring that is twice as stiff would. The last recorded value when 22N of force was applied has a somewhat longer extension since it is the start of the graph’s plateau due to the springs passing their elastic limit. Graph 3 – Springs in series Force applied (N) Extension of springs in series against force applied 10 9 8 7 6 5 4 3 2 1 0 69.1 y = 0.1339x 59.4 52.2 44.5 36.5 29.4 21.4 13.7 7.0 0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 Spring length (cm) Total extension / cm Best fit (Total spring length) In this case, the spring constant is half of the regular spring, with the two springs in series extending twice as much for each extra newton of force. Disobeying Hooke’s Law 5 Graph 4 – Rubber band Extension of a rubber band and the force applied on it 11 10 9 Force applied (N) 28.9 25.7 8 28.7 23.8 7 28.3 21.3 6 27.7 17.9 5 27.1 14.1 4 25.8 10.9 3 7.0 2 1 29.1 27.5 22.9 3.8 0.0 8.0 0 0.0 16.1 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 Rubber band extension total (cm) Loading length Rubber band unloading length Best fit (Loading) Best fit (Unloading) The rubber band disobeyed Hooke’s Law, but not entirely. While loading, the rubber band almost behaved linearly. During unloading, however, the rubber band did something very different – its extension decreased very little until just over 3N of force were still being applied, when the rubber band’s pattern abruptly became linear, and the extension once again was directly proportionate to the force applied. Conclusion Overall, springs do obey Hooke’s Law, and the results support the hypothesis. On average, a single spring extended 4.2 cm for each extra Newton of force. Two springs in parallel extended roughly twice as little, at 2.0 cm extension per Newton, while two springs in series extended twice as much, extending by 7.7 cm for each added Newton of force. Rubber bands, however, are not springs, and only somewhat followed Hooke’s Law. Due to the different behaviour of rubber bands, two separate sets of results were recorded.