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1 The Relationship Between Length and Spring Constant of an Elastic Bungee Cord: Static and Dynamic Cases Jenny Wang & McCauley Massie – PHYS113-01 – October 20, 2014 I. Introduction In this experiment, data is collected to investigate the relationship between the unstretched length of an elastic bungee cord and the spring constant of the bungee cord. Because the cord is elastic, it possesses certain spring-like qualities, so data is analyzed using Hooke’s Law, which is characterized by the following equation: 𝑭𝒔𝒑𝒓𝒊𝒏𝒈 = −𝒌𝒙 (1) This law describes the restoring force of a spring when it is stretched, stating that there is a linear relationship between this force (Fspring) and the displacement (x) of the spring. At equilibrium, the restoring force of the spring should also be equivalent in magnitude to the force applied on the spring by the hanging mass. The magnitude of this applied force is given by the equation for weight: 𝑾 = 𝒎! |𝒈| (2) In his equation, mh is the mass of the hanging mass in Kg, and g is acceleration due to gravity, 9.81m/s2. When the system is not at equilibrium, the spring must exert a force that tries to restore equilibrium. In this experiment, the force applied on the cord will remain constant. The relationship between the unstretched length of the cord and the stretch, or displacement, of the cord is used to investigate the relationship between the spring constant (k), which is related to the force in equation 1, and the length of unstretched cord. It is hypothesized that the k constant will remain constant as the length of unstretched elastic bungee cord increases. II. Methods One end of an elastic bungee cord forms a loop so that a hanging mass of 150 g can be attached. The weight of the hanging mass, given by equation 2, will constitute the force acting on the bungee cord. The bungee cord is tied to form another small loop at a distance from the first loop. This distance is the “unstretched” bungee cord length, and it is varied throughout the experiment. The second loop is attached to a force sensor attached to the top of a L-shaped rod clamped to a table. The force sensor connects to the Capstone computer program, which records the restoring force of the bungee cord as it is stretched. Measuring tape is also suspended from the L-shaped rod adjacent to the force sensor. This is used to measure the displacement of the cord. 1 Figure 1: Experimental Set-Up. A length of bungee cord is tied such that each end has a small loop. One end is attached to a force sensor mounted on an L-shaped rod clamped to a table. A hanging mass (mh) is attached to the other end. The “unstretched” length and the displacement (x) of the bungee cord are measured using a tape measure. 2 The length of unstretched bungee cord is measured from the bottom of the loop attached to the force sensor to the top of the loop that holds the hanging mass, without the hanging mass attached. For each length of unstretched string, the static stretched bungee cord length was measured with the spring and hanging mass system at equilibrium. In this case, the force applied to the bungee is equivalent to the restoring force of the bungee cord. This measurement was taken from the bottom of the loop attached to the force sensor to the top of the loop where the hanging mass is attached. In addition, the dynamic stretch for each length of unstretched bungee cord was also measured. In this condition, the hanging mass is dropped from rest from the point where the bungee cord is just on the verge of being stretched. Because it is difficult to determine the maximum amplitude of the cord in this condition with the naked eye, the frame-by-frame video analysis function of the CoachMyVideo (CMV) app is used to determine the dynamic stretched bungee cord length. Again, this measurement was taken from the bottom of the loop attached to the force sensor to the top of the loop where the hanging mass is attached. In both the static and dynamic cases, the displacement, or stretch, of the bungee cord can be calculated by subtracting the unstretched bungee cord length from the stretched bungee cord length. This stretch, which is related to the restoring force in Equation 1, can be used to calculate the spring constant, k. Because this experiment seeks to investigate the relationship between the unstretched length of the bungee cord and the spring constant, the length of the unstretched bungee cord is varied, and the subsequent stretch of the bungee is used to determine the spring constant in both static and dynamic cases. III. Results In this experiment, the stretch of the bungee cord increased linearly with the length of the unstretched bungee cord. Consequently, the spring constant decreases as the length of the unstretched bungee cord increases. Figure 2: Table Showing Static Stretched Bungee Cord Length (in cm), Average Dynamic Stretched Bungee Cord Length (in cm), and Average Force (in N) for each Unstretched Cord Length (in cm). All of the data in this table are directly measured in the experimental design. For each unstretched cord length, three trials of dynamic stretch bungee cord length and force were conducted, and the resulting values were averaged. Unstretched Cord Length (cm, ±0.2) Static Stretched Cord Length (cm, ±0.2) 5.5 11.8 18.0 30.0 38.5 13.6 28.2 45.9 92.7 115.4 Average Dynamic Stretched Cord Length (cm, ±0.2) 38.73333 62.36667 86.33333 131.7 194.6 Average Force, Fspring (N, ±0.1) 2.37 2.55 2.46 2.36 2.52 Figure 2 provides the data obtained by direct measurement in the experiment. As the unstretched bungee cord length increases, the static stretch increases. Moreover, the average dynamic stretch increases with the unstretched bungee cord length at a greater rate. The average 3 restoring force of the bungee cord remains about constant regardless of unstretched bungee cord length. Figure 3: Table Showing the Spring Constant derived from the Static Stretch (in cm) of the Bungee Cord and the Restoring Force Exerted by the Bungee Cord. The static stretch was calculated by subtracting the unstretched bungee cord length from the static stretched cord length. The restoring force of the bungee, Fspring (in N) was calculated using Equation 2. The spring constant, k (in N/m), was then calculated using Equation 1 for each static stretch measurement obtained from each unstretched cord length condition. Fspring Spring Static (N) Constant, k Stretch, x (N/m) (cm, ±0.2) 1.47 0.181 8.1 1.47 0.0896 16.4 1.47 0.0527 27.9 1.47 0.0234 62.7 1.47 0.0191 76.9 Figure 3 shows the derived values of static stretch, or displacement, of the bungee cord for each unstretched bungee cord length. For each of these values, the restoring force applied by the spring remains constant. This figure also shows the derived spring constant for each of the displacement values. The results for this experiment indicate that the spring constant decreases as the static stretch increases. Figure 4: Table Showing the Spring Constant, k, derived from the Dynamic Stretch (in cm) of the Bungee Cord and the Restoring Force (in N) Exerted by the Bungee Cord. The dynamic stretch was calculated by subtracting the unstretched bungee cord length from the dynamic stretched cord length. The spring constant, k (in N/m), was then calculated using Equation 1 for each dynamic stretch measurement obtained from each unstretched cord length condition. Fspring Spring Dynamic (N) Constant, k Stretch, x (N/m) (cm, ±0.2) 2.37 0.0713 33.23 2.55 0.0504 50.567 2.46 0.0360 68.33 2.36 0.0232 101.7 2.52 0.0161 156.1 Figure 4 shows the derived values of dynamic stretch, or displacement, of the bungee cord for each unstretched bungee cord length. For each of these values, the restoring force applied by the spring remains fairly constant. This figure also shows the derived spring constant for each of the displacement values. The results for this experiment indicate that the spring constant decreases as the static stretch increases. In general, the spring constants in the dynamic condition are smaller than the spring constants in the static condition. 4 Figure 5: Linearized Graph Showing the Relationship between Unstretched and Static Stretched Cord Length. The equation y=2.941(±0.2)x gives the linear regression of the graph, indicating that the unstretched and static stretched cord length are positively, linearly related by a factor of ~3. Static Stretched Cord Length (cm, ±0.2) Unstretched vs. Static Stretched Cord Length 140 120 100 80 60 40 20 0 y = 2.941(±0.2)x 0 5 10 15 20 25 30 35 40 45 Unstretched Cord Length (cm, ±0.2) Figure 5 shows that the total static stretched bungee cord length is positively and linearly related to the unstretched bungee cord length. Each unstretched length of bungee cord stretches about 3 centimeters when a force of about 1.5 N, calculated using equation 2, is applied to it. Figure 6: Graph Showing the Relationship between Unstretched Cord Length (in cm) and Average Restoring Force of the Bungee Cord in the Dynamic Condition (in N). For each unstretched cord length, three trials of dynamic stretch were conducted and the resulting values for the restoring force were averaged. Average Force (N, ±0.1) Unstretched Cord Length vs. Average Force 5 4 3 2 1 0 0 5 10 15 20 25 30 Unstretched Cord Length (cm, ±0.2) 35 40 45 5 Like Figure 4, Figure 6 also shows that the average restoring force of the force in the dynamic case remains constant regardless of unstretched bungee cord length. Figure 3 shows that this is also the case for the static condition. Figure 7: Linearized Graph Showing the Relationship between Unstretched Cord Length (in cm) and 1/Spring Constant (in N/m) for the Static Condition. The equation y=1.3204(±0.1)x gives the linear regression of the graph, indicating that the unstretched cord length of the bungee cord and the spring constant, k, of the bungee cord are inversely related in the static condition. In other words, the unstretched cord length is directly related to 1/k. 1/Spring Constant (N/m) Unstretched Cord Length vs. Static Stretch 1/Spring Constant 60 50 40 30 20 10 0 y = 1.3204(±0.1)x 0 5 10 15 20 25 30 35 40 45 Unstretched Cord Length (cm, ±0.2) Figure 8: Linearized Graph Showing the Relationship between Unstretched Cord Length (in cm) and 1/Spring Constant (in N/m) for the Dynamic Condition. The equation y=1.5609(±0.1)x gives the linear regression of the graph, indicating that the unstretched cord length of the bungee cord and the spring constant, k, of the bungee cord are inversely related in the dynamic condition. The unstretched cord length is directly related to 1/k. 1/Spring Constant (N/m) Unstretched Cord Length vs. Dynamic Stretch 1/Spring Constant 80 60 y = 1.5609(±0.1)x 40 20 0 0 5 10 15 20 25 30 Unstretched Cord Length (cm, ±0.2) 35 40 45 6 Figures 7 and 8 show that, in both the static and dynamic cases, the spring constant is inversely related to the unstretched length of the cord, meaning the spring constant decreases as the length of unstretched bungee cord increases. IV. Discussion The results of this experiment do not confirm the hypothesis that the spring constant will remain constant as the length of unstretched elastic bungee cord increases. In fact, the spring constant decreases as the unstretched length of the bungee cord increases, so they are inversely related. Even though these results do not support the hypothesis, they are still consistent with the theory behind this experiment. Hooke’s law, characterized by equation 1, implies that the displacement and the spring constant and inversely related. Since the force remains about constant throughout each of the trials, it is absolutely reasonable to state that the spring constant decreases as the displacement increases. Furthermore, the results imply that it is possible to model the behavior of the elastic bungee cord using Hooke’s law even though the spring constant changes depending on the length of the bungee cord. The changes in the spring constant merely indicate that the bungee cord behaves differently at different lengths. This is no different from the idea of connecting two springs that have the same properties. The new length of spring has different qualities because it is twice as long. Unfortunately, the percent error of the results cannot be calculated because there is no predicated value of the spring constant. Therefore, it is not possible to state whether or not the results of the experiment are accurate to within the uncertainty of the experiment. The results in this experiment could, however, be used as the predicted values for future experiments. Similarly, the results of future experiments can be used to determine the accuracy of the results of this experiment. Since the spring constant varies depending on the length of the elastic bungee, future experimentation should seek to reproduce the value of the slope when 1/spring constant is plotted against the unstretched bungee cord length. For example, in the static stretch condition, a linearized graph, like Figure 7, should yield the equation y=1.3204x, where 1.3204 is the slope relating 1/spring constant to the unstretched bungee cord length. This value would be used as the predicted value for a future experiment to determine the accuracy of the results of that experiment. If the percent error between the predicted value and the experimental value is less than the percent uncertainty, the results are accurate. Still, the percent uncertainties for the results of this experiment are very small, indicating that the results of this experiment are precise. For example, the percent uncertainty for the slope quantifying the relationship between the static stretched and the unstretched length of the bungee cord, depicted in Figure 5, is 0.07%. The percent uncertainty for the slope quantifying the relationship between 1/spring constant and the unstretched bungee cord length in the static condition, depicted in Figure 7, is 0.08%, and the percent uncertainty for the slope quantifying the relationship between 1/spring constant and the unstretched bungee cord length in the dynamic condition, depicted in Figure 8, is 0.06%. There are many sources of uncertainty in this experiment. For the measurements of length, there is uncertainty due to the change in angles of the camera when measuring the length of stretch using the CoachMyVideo App. Uncertainty in the measure for restoring force of the bungee cord can be attributed to the force sensor’s least count and the amount of bungee in the loops at either end of the spring. Even though these loops were tied very small, they still stretch to some degree, and they will behave differently from the single stranded length of bungee that the experiment focuses on. In the dynamic stretch condition, there is come uncertainty that can 7 be attributed to the difficulty of eyeballing just exactly where the line between slack and stretched bungee is. V. Conclusion The purpose of this experiment was to investigate the relationship between the unstretched length of an elastic bungee cord and the spring constant of the bungee cord, assuming that the behavior of the bungee cord can be modeled using Hooke’s law. The results of this experiment state that the restoring force of the bungee cord is constant regardless of the length of the unstretched bungee cord length as long as the force applied to the bungee cord is constant. Consequently, as the displacement of the bungee cord increases in both the static and dynamic stretch conditions, the spring constant of bungee cord decreases. These results not only state this relationship, but they also indicate that the behavior of the bungee cord can be successfully modeled using Hooke’s law. In addition, the static stretched bungee cord length could be determined using the equation given by the linear regression in Figure 5 if the unstretched bungee cord length is known. Similarly, the spring constant could be determined using the equations given by the linear regression in Figures 7 and 8. Since an important application of bungee cords is their use bungee jumping, it is important to understand how the restoring force of the bungee cord is related to the force applied to the bungee cord, such as the weight of the mass attached to it. If the restoring force of the bungee cord is too great, the mass could incur damage do to a very quick deceleration. In addition, the mass attached to the bungee cord could incur damage if the bungee cord stretches to far, and the mass hits the ground. This experiment was able to describe the relationship between the restoring force, the spring constant, and the displacement of the bungee cord. It was also able to determine the relationship between static stretch and unstretched bungee cord length and between dynamic stretch and unstretched bungee cord length when the mass is dropped from rest at the point where the bungee cord is just about to be stretched. However, it does not describe the relationship between dynamic stretch and unstretched bungee cord length when the mass is dropped from rest at the maximum height, where the bungee attaches to the force sensor. This information would be important to gather in order to predict the success of a bungee jump. VI. Appendix 8