Force and Bungee Cord Length: Will Longer Length Un-Stretched of Cord result in a Greater or Lesser Force than a Shorter Length of Un-Stretched Cord Introduction: Answering the request of quite the adventurous egg, we ve been tasked to figure the appropriate length of nylon cord so as to supply an exciting bungee jump experience. The height will be between 8 and 9 meters and will feature pterodactyl skeleton, a flight of stairs, and a wide array of tables and chairs as scenery. To ensure that this jump is as thrilling as possible without being too much so, it is our job to determine the length of nylon cord that will allow the jumper to come as close to the ground as physically possible without experiencing an overwhelming restoring force more than four times the jumper s weight. Previous experiments (see: Examining the relationship between the force on a bungee cord and how long the cord stretches) have found the absence of a constant k-value as a property of the given cord which was explored using Hooke s Law for an ideal spring F = -kx (1) Zooming out from a focus on k-values, this experiment focuses on the direct correlation between length of cord, un-stretched, and the resulting force exerted on the jumper. Though, as previously mentioned, this cord does not possess a characteristic k-value, it can be thought of as possessing an average k-value over its entire length. This cord is quite easily stretched. In comparison to a string of yarn that yields considerably less stretch, our nylon cord s average k- value value should be relatively smaller for, as described by R. Reese, a [spring] with a large [spring] constant k is a stiff [spring] (284). As a result, we expect the force on the mass to increase with increased cord length. Methods: We examined the relationship between un-stretched cord length and force exerted on the hanging mass by holding the hanging mass at constant mass and varying only the length of unstretched cord. The magnitude of the hanging mass was held constant as we will be given a specific egg with a specific mass all its own come B-Day (Bungee Day). One end of the bungee cord was knotted about the hook of the force sensor, which was connected to our computer, ready to record data in the Capstone program. The loop of the aforementioned knot was tightened as much as possible so as to avoid unaccounted for stretch (Figure 1). Our hanging mass was held constant at 150 grams as the mass of our dare-devil-egg (not to be confused with deviled) will be between 100 and 200 grams. To avoid any issue with our hanging mass, we also taped the 100 gram weight to the 50 gram hanging weight. For each trial, the un-stretched cord length varied, meaning that the hanging mass began at a different distance from the force sensor each time. On the opposite end of the point of attachment to the force sensor, we tied a loop-knot at random and attached the hanging mass per trial. Each trial consisted of holding the bottom of the hanging mass even with the bungee cord s point of attachment to the force sensor and simply letting it fall from rest. Un-stretched cord lengths were recorded in an Excel spreadsheet in centimeters as was force, which was collected from the Capstone program, in newtons.
Force sensor Support pole Bungee cord Table Hanging mass Figure 1. Experimental set-up Results: Figure 2 displays our experimental data Trial Cord Length Force (± 0.1 cm) (N) 1 16.3 4.45 2 22.3 4.16 3 27.4 4.08 4 37.0 3.75 5 33.5 3.86 6 19.5 4.26 Figure 2. Experimental results. Cord length measured with an experimental uncertainty of ± 0.1 cm. Force values taken from Capstone
To better visualize this relationship, we graphed force on the y-axis and cord length on the x-axis (Figure 3). 4.5 Force v. Cord Length 4.4 4.3 4.2 y = 7.6492x -0.195 Force (N) 4.1 4 3.9 3.8 3.7 0 5 10 15 20 25 30 35 40 Cord Length (cm) Figure 3. Scatter plot of the force exerted by the 150 g mass versus the varied un-stretched cord length with standard errors for the slope and y-intercept 0.06 and 0.002, respectively found via linear regression. When graphed, the data showed that as cord length increased, force decreased. Yet, this did not quite fit a linear relationship but instead, a relationship that showed force to be approximately proportional to cord length to the negative 0.195 power had a better fit. In response, we took each of our measured cord lengths to the negative 0.2 power in an effort to produce a linear relationship (Figure 4) Trial Cord Length -0.2 Force (cm) (N) 1 0.572 4.45 2 0.537 4.16 3 0.516 4.08 4 0.486 3.75 5 0.495 3.86 6 0.552 4.26 Figure 4. Table of experimental data with measured cord lengths raised to a negative 0.2 power.
and replaced cord length on the x-axis with these cord length to a power values (Figure 5). 4.5 4.4 4.3 Force v. Cord Length to a Power y = 7.5486x + 0.1178 4.2 Force (N) 4.1 4 3.9 3.8 3.7 0.480 0.490 0.500 0.510 0.520 0.530 0.540 0.550 0.560 0.570 0.580 Cord Length to a Power (cm) Figure 5. Scatter plot of the force exerted by the 150 g mass versus the varied un-stretched cord length to a negative 0.2 power with standard errors for the slope and y-intercept 0.30 and 0.56, respectively found via linear regression. We again found that as cord length decreased, force increased, though it is actually less intuitively seen in Figure 5. To emphasize this pattern, examine the minimum and maximum lengths in Figure 6, both original and to a power. The minimum cord length is marked with a yellow star in Figure 5 and the maximum with a purple star. Cord Length Cord Length -0.2 Force (± 0.1 cm) (cm) (N) 16.3 0.572 4.445 37.0 0.486 3.75 Figure 6. Minimum and maximum cord length values only to be used to see the relationship in Figure 5 Discussion: The forced inverse linear relationship found between cord length and force exerted on the mass suggests that the two are proportional. Such a relationship allows us to calculate the amount of force we could expect to be exerted on our egg given a length of our cord using equation y = 7.5486x + 0.1178 where force is y and cord length is x from Figure 5. However, due to the standard error of the slope of the linearized graph, 0.56, the certainty of this relationship is questionable. This high standard error could be due to the possibility that the
inverse relationship is simply insignificant. This theory has merit as the same weight is being exerted on the cord trial after trial, resulting in a relatively constant force over each length of cord with the small variance being due to the length of cord. Given a constant mass, the length of the cord may not be the driving force when the jumper is to be restored or, yanked up, for a better term, after free-fall. It would be difficult to disprove this significance or lack thereof for the range of cord lengths is quite limited to about 10 to 40 centimeters in which beyond about 40 centimeters with the given support pole height, the mass is unable to avoid a collision with the floor, skewing the measured force. These questionable results could be helped with going back to Hooke s Law, equation (1). Instead of examining a relationship between cord length and force, examining a known relationship, such as that between F and x or, force and displacement, could have given more intuitive as well as helpful results in terms of future calculations. Examining the displacement has the potential to provide a function that will allow for the calculation of the cord s average k- value. By itself, this would again appear fruitless, but if used in conjunction with the CWE theorem, further information could be extrapolated when assuming that the top of the jump consists only of potential energy while the bottom of the jump consists only of kinetic energy in which mgh = ½ kx 2 (2) Such an experiment would promote calculation for h in equation (2), h being the height of the cord at maximum displacement during stretching. This experiment would have served better to have considered the displacement of the cord when in dynamic equilibrium for the above reasons and to have truly provided us with the means to propel forward and determine the requirements for an egg s successful jump. Conclusion: According to our linearized data, force and cord length have an unexpected inverse relationship in which as cord length increases, force decreases. However, this relationship may be insignificant, but may only be emphatically determined with repeated experimentation that also involved record of the cord s dynamic displacement. References Cao, A. (2014). Examining the relationship between the force on a bungee cord and how long the cord stretches. Retrieved from http://bungeejournal.academic.wlu.edu/files/2014/11/ Lab-report-2.pdf Reese, R. L. (2000). University Physics (Vol. 1). Brooks/Cole Publishing Company.