A Case Study of Hooke s Law and the Variation of Spring Constant James Freeman PHYS 113; Professor Khalifa
Freeman 2 Introduction Bungee jumping is more than an extreme sport when viewed through the lenses of physics: it is a dynamic process that explores the balance between gravity, acceleration, and the spring-like qualities of an elastic cord. By utilizing an egg as a jumper attached to an elastic cord, or bungee rope, we hope to successfully analyze these relationships. The purpose of this experiment is to replicate an effective bungee jump scenario, with success being measured in two ways. First, we aim to obtain the closest proximity to the ground as possible without actually touching it. Second, the jumper is not to be damaged by too rapid of a deceleration incurred by the elastic properties of the cord. The main scope of this experiment focused on Hooke s Law and bungee cord characteristics, which create an interesting interplay between stretch, or displacement, and force due to the deformation of rubbery, long-chain molecules. We used a variety of procedures, both static and dynamic, in order to model this relationship and in turn help us to create a successful bungee jump. As stated, our experiments relied on Hooke s Law, which aptly describes the elastic quality of a spring, or bungee cord: 1) F = -kx with F being the force, k being the spring constant which characterizes the stiffness or strength of a spring or cord, and x being displacement. In our experiments, we used individual masses to play the role of the force by hanging them from the cord and then measuring the subsequent displacement. This allowed us to find the spring constant in both dynamic and static situations, which we believe to be a crucial aspect in our final experiment. We hypothesized that there would be a linear relationship between force and displacement, but the procedures we utilized were ultimately to collect enough data to assess the boundaries and conditions relating to different spring constants. Methods The first method in this experiment involved measuring the displacement of the cord after hanging a mass from it. First, we assembled the setup, as reproduced in Figure 1. In our set-up, we had an elastic bungee cord attached to a stand. A knot was tied in the cord where it met the top of the stand. We also tied a small knot in the bottom of the cord, in order to create a loop from which we could hang our masses. By doing this, we were able to measure the displacement from a constant position on the spring: from the knot in the top of the cord to the knot in the bottom of the cord. We measured and recorded the starting length in Microsoft Excel, and then began to add mass. Several different values of mass were added to a constant starting cord length. After we weighed increasing values of displacement from the same starting length, we removed mass and lengthened the starting cord length. We then restarted the process, gradually adding mass and measuring the displacement. This was accomplished for four different initial cord lengths, with values of 56.1 cm, 83.3 cm, 115.5 cm, and 137.5 cm. This concluded the static portion of our experiment.
Freeman 3 Figure 1: A reproduction of the setup of the static part our experiment. A hanging mass was hung from an initial cord length. The new length in the cord was recorded as displacement, or x. After completing this, we moved onto the dynamic portion of our experiment. This portion involved keeping the mass constant, but varying the length of the cord. The setup is reproduced in Figure 2. We attached the mass to the end of the cord, and one partner held it level with the top of the stand, adjacent to where the cord itself was tied to the stand. A tape measure was also hung from the top of the stand, going to the floor. This enabled us to see the magnitude of displacement during the dynamic process. The mass was then dropped. Because the process is extremely quick and it is difficult to see exactly to what extent the mass drops next to the tape measure with the naked eye, we filmed the drop with an ipad. With the aid of the ipad, we were able to film the drop and then watch it back in slow-motion in order to see the precise value of displacement. After dropping the mass at a certain length and recording it in Excel, the starting cord length was then altered. As stated before, in this portion of the experiment the mass was kept constant. Four different cord lengths were recorded.
Freeman 4 Figure 2: A reproduction of the dynamic portion of this experiment. The attached mass was hung level with the top of the stand, and dropped. The ensuing displacement as the mass plummeted to the ground was measured. Results Through this process, we were successfully able to find the spring constant, k, based off the relationship between force and displacement in the static trials. An example of the data from one trial in the static experiment can be seen in Figure 3. In order for the mass to act as our force, it was converted into SI units and multiplied by the gravitational constant, 9.81 m/s 2. The displacement was found by subtracting the initial length from the new length. This was then converted to meters, and graphed along with force, as seen in Figure 4. Hanging Mass (grams) Hanging Mass (kg) Starting Length (cm): 56.1 Force New Length (mg) (cm) Total Displacement (cm) Displacement (m) 20 0.02 0.1962 56.9 0.8 0.008 50 0.05 0.4905 64.7 8.6 0.086 70 0.07 0.6867 66.8 10.7 0.107 90 0.09 0.8829 75.5 19.4 0.194 110 0.11 1.0791 83.9 27.8 0.278 130 0.13 1.2753 91.7 35.6 0.356 150 0.15 1.4715 101 44.9 0.449 170 0.17 1.6677 111.9 55.8 0.558 190 0.19 1.8639 123.3 67.2 0.672 210 0.21 2.0601 134.1 78 0.78 230 0.23 2.2563 145.4 89.3 0.893
Freeman 5 250 0.25 2.4525 155.2 99.1 0.991 Figure 3: Data of Trial 1 in the static portion, with a cord starting length of 56.1cm. Figure 4: Graphs showing the relationship between force and displacement. The slope of each graph is the spring constant for that length of cord. From these graphs, it is clear that there is a linear relationship between force and displacement. The slope of each graph is the spring constant for each length of cord. Because we have multiple spring constants, we wanted to look at the relationship between starting length and spring constant. This relationship is seen in Figure 5, and shows that there is an exponential relationship between the two, which will be further considered in the Discussion section.
K Freeman 6 K vs Length 2.5 2 1.5 y = 3.381e -0.01x R² = 0.9804 1 0.5 0 0 20 40 60 80 100 120 140 160 Starting Length (m) Figure 5: Relationship between Spring Constant k and initial starting length, showing an exponential trend. The results of the dynamic section are less in depth, for we just wanted a brief overview to get an idea of the relationship, as seen in Figure 6. From this data and the graph seen in Figure 7, we see that there is also a linear relationship between starting length and displacement. This concept will be acknowledged further in the Discussion section. Mass: 110 g Starting Length (cm) New Length (cm) Displacement (cm) 30 84.5 54.5 54.7 150.4 95.7 75.7 210.75 135.05 92.7 267 174.3 Figure 6: Data table from dynamic portion. Starting length was subtracted from displacement in order to find displacement.
Starting Length (cm) Freeman 7 Starting Length vs Displacement 200 150 y = 1.8958x - 5.0711 R² = 0.9949 100 50 0 0 20 40 60 80 100 Displacement (cm) Figure 7: Graph portraying a linear relationship between starting length and displacement after a constant mass is dropped. Discussion As stated previously, there is a clear relationship between force and displacement, which is the spring constant. In essence, our static experiment was perfectly described by Hooke s Law. As mass increases, there is more force which in turn creates more displacement. Spring constant also increases with higher masses and higher values of displacement. Our hopes are that this will be useful in the final bungee jump, for knowing the spring constant for different lengths of cord and with different masses is a valuable asset. We also have a visible relationship between k and displacement, as seen in Figure 5. This association is slightly more complex than Hooke s Law, for it is exponential rather than linear. There are lower spring constants with longer cord lengths, which means that the shorter the string, the stiffer or stronger it is. With more experimentation, we might be able to create an equation allowing us to know the spring constant based off of length. We could then find the ensuing displacement once the force is known, hopefully giving us a successful jump scenario. When viewing these results, uncertainty must also be kept in mind. In each aspect of this experiment there are uncertainties, ranging from inaccurate mass values to precision in measurement. Our cord also could have been permanently deformed as more mass was added and left for small periods of time during the measurement process. When thinking about uncertainty, it must be kept in mind that the R 2 values in each slope are extremely close to a value of 1, signifying an extremely high level of certainty in each trial, or almost a perfect fit. Therefore, from this we know that uncertainty is minimal in our experiment. As there are no true values to compare our answers to, overall uncertainty and error cannot be computed. If this experiment were to be conducted once more, we might make several alterations to the procedure in order to maximize precision and accuracy, and minimize error. First of all, we
Freeman 8 could weigh the individual masses that were added to the cord. Even though their masses are written on their side, years of use could alter their true mass. Therefore, we could weigh them and solve this. Also, we could measure the cord before and after masses were hung, to see if there was any permanent deformation, which could alter our results. Conclusion This experiment touched upon the important relationship described in Hooke s Law, and gave us confirmation of this concept. By running both static and dynamic tests, we were able to see that there is a linear relationship between force, displacement, and the spring constant. We also see that the spring constant can be compared with the length of the cord, and has an exponential relationship. Lastly, we found a linear relationship between the starting length of a cord and the ensuing displacement after a constant mass is dropped. These results were valuable additions to our final bungee project. To continue this work, we could further explore the concept touched on our dynamic portion of the experiment, and vary masses. We could also see the relationship between displacement and force when a constant mass attached to a constant cord length is dropped from varying heights, in order to see the effect of gravity. For now though, we have a thorough understanding of Hooke s Law, and are looking forward to continuing this work in later labs.