Using Hooke s Law to Solve for Length of Bungee Cord Needed for Egg Drop Introduction This experiment is the second part of a three- part experiment. The first two lead up to the final in which we aim to successfully drop an egg attached to the cord without the egg breaking. The first experiment we did involved using Hooke s Law (Equation 1) to determine the cord s characteristics. Hooke s Law represents the F = - kx (1) relationship of an ideal spring where F is the restoring force, k is the spring constant, and X is the displacement from equilibrium. The restoring force, weight, was found by multiplying the attached mass by the gravitational constant. We found the spring constant, k, of our bungee cord by measuring the displacement of cords varying in length after hanging different masses on it. The goal of the current experiment was to develop an equation that can be used to determine the length of cord necessary to drop our egg with. Similar to our first experiment, we sought to determine the spring constant, k. However, we did so dynamically in order to have a more accurate idea of what the displacement will be. This experimental design also assumes that the bungee cord acts as an ideal spring. This is a conservative system; thus, no energy is gained or loss according to the Classical Work Energy (CWE) Theorem. Equation 2 is a derivation relating the CWE theorem and Hooke s Law where m is the mass, h is the total length of the drop, Mgh = ½kx 2 (2) k is the spring constant, and x is the displacement of the bungee cord from equilibrium.
Methods Overall, we measured the displacement of a doubled- up bungee cord by dropping a hanger with mass attached to it and measuring how far the bungee stretched from equilibrium. Length L (m) Mass m (kg) Displacement x (m) Step 1 Step 2 Step 3 Figure 1: Setup and procedure of the bungee cord system Figure 1 displays the order of the procedure. First, we doubled the bungee cord by folding it in half and knotting it, leaving a hoop to hang it from. We did this in order to find a more direct relationship between the spring constant and the length of the cord used per trial. Then, we attached the knotted cord to a metal bar and hung a measuring tape parallel. Each measurement off the measuring tape was taken from the center of the knot on the cord. Another knot was made at the bottom of each chosen length of cord, 0.10-0.275 m. This length is the cord s equilibrium point (Figure 1: Step 1). A hanger was attached to the loop from the bottom knot. Four different masses, 0.10-0.16 kg, were chosen to make four trials per length. The
masses were fastened with painter s tape to ensure their stability. The hanger was dropped straight down from where the bungee cord was connected to the metal bar (Figure 1: Step 2). We used an application on an ipad called Coach My Video to measure the displacement. The application allowed us to video the drop, zoom in afterwards, and play the video in small steps to see where on the measuring tape the knot hit full extension. This measurement is also known as the height, or h. Displacement, x, was calculated by subtracting l from h (Figure 1: Step 3). By deriving Equation 1, we found that the restoring force, F, was equal to the weight, or mass multiplied by gravity, of the hanger (Equation 3). Using algebra, we are able to find k (Equation 2). Graphically, a weight vs. displacement graph shows K = (mg)/x (3) that the spring constant was the slope of each linearized graph. Each k was then graphed vs. length of cord. The result of the linearized version of this graph gave us an equation of the relationship between length of cord and spring constant. This equation will be used to determine was length of cord to use when dropping our egg. Results The results indicate that as a double stranded bungee cord gets longer, the spring constant weakens (Table 1). Equation 3 provides support the results as k is inversely related to x. In Figure 2, the relationship is apparent based off the slope of each weight vs. displacement line.
Bungee Resting Length L (m, ± 0.01) Spring Constant k (uncertainty) 0.10 8.37 (±0.71) 0.16 5.45 (±0.36) 0.21 3.52 (±0.16) 0.28 2.91 (±0.03) Table 1: Spring constant per bungee resting length. Weight Vs. Displacement 1.7 Weight (N) 1.6 y = 8.3721x + 0.1179 1.5 1.4 1.3 1.2 y = 5.4494x - 0.023 y = 3.5211x + 0.1301 y = 2.9104x - 0.0257 Length = 0.10 m Length = 0.16 m Length = 0.21 m Length = 0.275 m 1.1 1 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 Displacement (X) Figure 2: Weight versus displacement for each length of bungee cord. The slope of each line represents the spring constant, k.
The bungee cord s spring constants per resting lengths were then graphed in order to derive a formula that represents a relationship between the two (Figure 3). This formula will be the formula used in determining the length of bungee needed for the egg drop. K- value 9 8 7 6 5 4 3 2 1 0 K- value Vs. Length of Bungee y = 0.6997x - 1.084 0 0.05 0.1 0.15 0.2 0.25 0.3 Length of Bungee (m) Figure 3: Spring constant versus Length. The relationship between the two variables is shown; as the cord lengthens, k drops. The linearization of Figure 3 results in the final equation (Equation 4) needed to determine the length of cord necessary to successfully drop an egg without breaking. K = 0.881L 0.3652. (4) Discussion The goal of this experiment was to determine an equation that we could use to determine the length of cord we need. On the egg drop day, we will be given the
height that the egg will drop from and mass of the egg. Using this information, we can combine Equations 2 and 4 to solve for the length of cord necessary (Equation Mgh = ½ (0.881L - 0.3652)(h L) 2 (5) 5). The height of jump, length of cord, and length of elongation all relate to each other; the length plus elongation should equal the height. So in Equation 2, elongation, x, can be substituted for length subtracted from height, or h- l. Equation 4 is solving for the spring constant, k, so 0.881L 0.3652 can be substituted in for k in Equation 4. On egg drop day, Equation 5 should result in the length of bungee cord we need in order to safely drop the egg from a certain height without it breaking. However, sources of uncertainty undoubtedly exist. First, we completed this experiment with the assumption that the bungee cord acts as an ideal spring. However, after reading the Bungee Journal, some of our peers concluded that their bungee cord did not behave like an ideal spring (Busch & Wilbur, section 5; Melkun & Towne, section 5). If their experiments generalize to our bungee cord, then we did not take other external forces acting on the system into account. This implication could result in the wrong length of cord being used; too long and the egg will crash or too short and the rebound might crack the egg. A further source of uncertainty is our bungee cord. During experimentation, the cord snapped on four different occasions. We were able to continue the experiment only by piecing together the broken pieces. We would have liked to have trials at longer lengths, but our cord did not allow for that. There are several implications of the cord snapping. First, we will be dropping the egg with a
completely different cord. Equation 5 was derived specifically from the first cord. It s a relationship based on that cord s spring constant and length of cord. A different cord could have a varying relationship, and the length of cord obtained might not translate to the new cord. We can try to avoid this by prestretching the cord as much as possible. Second, the limit of trial lengths that could be completed might have affected the result of Equation 5. The egg is going to be dropped from between 8 9m, and 0.275 m is not the best representation of the actual height. Getting a longer length might have resulted in an equation that serves as a better model than Equation 5 will. A final source of error lies in how elongation, x, was measured. In theory, the application on the ipad should have worked fine. However, the quality of the video, after zooming in enough to see where on the measuring tape the cord stretched to, was very poor. We attempted to be as accurate as possible by counting the blurry lines as best as possible. Perhaps using just the ipad camera application and slow motion feature would have worked better Conclusion In conclusion, this experiment sought to unveil an equation that could be used to find the length of bungee cord necessary to safely drop an egg from a great height. Equation 5 takes the height of the drop, length of the cord, spring constant of bungee, mass of egg, elongation of bungee, and gravity into account. On drop day, we will enter in the mass of the egg and height of the drop to obtain the necessary length.