Bungee Constant per Unit Length & Bungees in Parallel. Skipping school to bungee jump will get you suspended.

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1 Name: Johanna Goergen Section: 05 Date: 10/28/14 Partner: Lydia Barit Introduction: Bungee Constant per Unit Length & Bungees in Parallel Skipping school to bungee jump will get you suspended. The purpose of this lab was to familiarize ourselves with the bungee we ll be using to propel our egg safely down from the fourth floor of the science center to a point just above the ground of the second floor and back up to safety in lab 10! We were trying to answer two different questions in this lab: Can we treat the bungee like a spring with a spring constant k that allows us to use Hooke s Law? If so, what is the relationship between the length of our bungee on this spring constant of the bungee? From now on, we will call the bungee s equivalent of a spring constant a bungee constant. Does putting bungees in series or in parallel work the same way as putting springs in series and in parallel in terms of the change in bungee constant? Equation 1: F = kx For F = force, k = spring constant, and x = displacement. Equation 2: k tot = k 1 +k 2 + +k n (for n springs) For k tot = effective spring constant of n springs in parallel, k i = spring constant for single spring i for i in [1, n]. Methods: Our experiment was set up as follows: For the first part, determining the effect of bungee cord length on bungee constant, we tied our bungee cord to the elevated hook with a certain length, x, of bungee hanging straight downward. We measured and recorded this length x. We then tied weights with total mass 0.05 kg to this hanging end of the bungee cord. We measured the new length from the knot at the top of the bungee cord to the knot at the hanging end attached to the weights. We recorded this new value for length. We repeated this procedure ten times, each time varying the initial x value so that the length of the bungee cord before attaching mass 0.05 kg was not the same in any two trials.

2 x i x f Figure #1: Before the Egg Drops. The length of our hanging bungee cord is x I before adding mass to the end Figure #2: After the Egg Drops. The length of our hanging bungee cord is x f after our is added to the end of the cord For the second part, determining the effect of placing bungee cords in parallel, we first tied our bungee cord to the elevated hook with a certain length, 1.12 m, of the bungee hanging straight downward. We then tied weights with total mass m to this hanging end of the bungee and measured the new length of the hanging portion of the cord. We recorded this new length. Then, we left the length, 1.12 m, the same but chose a new total mass m for the weights and attached this new mass to the hanging end of the cord. We recorded the new length of the hanging portion of the cord with this weight attached. We did this with a different value for m 10 times. Then, we modeled two bungee cords in parallel by folding our bungee cord in half and tying the cord to the elevated hook with the same length, 1.12 m, of the bungee hanging straight downward, but this time the bungee was doubled up so that we were able to tie our weights to the equivalent of two bungee cords at once. We then performed ten trials in the same manner as above with 10 different values for m, just twice the bungee cord. Then we repeated these 10 trials one last time with the bungee cord tripled up but still the same initial length 1.12 m, recording the length of our stretched bungee cord at every mass. Figure # 3: Bungees in Parallel. We hung three bungees in parallel in the second part of our experiment.

3 Bungee Constant (N/m) Results: As a result of the first half of our experiment, we were able to determine a bungee constant per unit length, i.e. a bungee constant per meter, for our single bungee cord. We collected the following data by hanging the same mass,.05 kg, on our bungee at 10 different lengths (as described above) to come up with this result: Trial # x initial (m) ± 0.01 x final (m) ± 0.01 Force (N) k (N/m) ± Figure #4: Data from part 1. We calculated Force using F=mg = (0.05kg)(9.81m/s 2 ) and k using F=kx where x = (x final)-(x initial). The values for x final and x initial were our experimental values. After calculating the bungee constant for each different length of bungee cord as described in the caption above, we were able to plot length vs. bungee constant in the following graph: 6 5 Bungee Constant vs. Length of Cord 4 k = x Length of Bungee Cord (m) Figure #5: Bungee vs. Length of Cord. The slope of this plot is the bungee constant per unit length of our bungee cord.

4 The above graph tells us that our bungee cord has a bungee constant that decreases by a factor of per meter. As a result of the second half of our experiment, we were able to determine the behavior of our bungee cord when put in parallel with another bungee cord of the same length (and same material properties). We collected the following data by hanging 10 different masses on our bungee first, then two bungees of the same length in parallel, and then three bungees of the same length in parallel: SINGLE STRING Trial # Mass (kg) x final (m) ± 0.01 Displacement (m) ± 0.01 Weight (N) Figure #6: Single Bungee Cord. These are the results of hanging10 different masses on our single bungee cord with initial length constant at 1.12 m. DOUBLE STRING Trial # Mass(kg) x final (m) ± 0.01 Displacement(m)± 0.01 Weight (N) Figure #7: Two Bungee Cords in Parallel. These are the results of hanging10 different masses on our two bungee cords in parallel with initial length constant at 1.12 m.

5 Weight (N) TRIPLE STRING Trial # Mass(kg) x final (m) ± 0.01 Displacement(m)± 0.01 Weight (N) Figure #8: Three Bungee Cords in Parallel. These are the results of hanging10 different masses on our three bungee cords in parallel with initial length constant at 1.12 m. The above results allowed us to come up with a plot of weight vs. displacement. Graphing weight v. displacement allowed us to come up with a trend line for each of the three sets of bungees. Since we found that our trend lines were very linear, we were able to interpret them like Hooke s Law, F = -kx (but since we treated F as a positive value even though it would really be negative due to gravity s direction, we ended up getting equations that looked like F = kx). Thus, the slopes of each of our trend lines become our experimental value for one single bungee, two bungees in parallel, and three bungees in parallel Weight vs. Displacement F = x F = x F = x Displacement (m) Single String Double String Triple String Figure #9: Weight vs. Displacement. This plot contains all of our data points from the three different sets of trials. For each of the three strings, we found the linear trendline obtained by plotting weight against

6 Bungee Constant k (Nm) displacement. This allowed us to model the Hooke s Law equation F= kx since F = weight and x = displacement. Thus, the slope of each of our lines is our value for k. After using the trend lines in the above graph to come up with the value of k in a single bungee of 1.12 m, two bungees of 1.12 m in parallel, and three bungees of 1.12 m in parallel, we were able to plot those k values to determine the relationship between number of bungees in parallel and effective bungee constant. 6 Bungee Constants in Parallel 5 4 y = x Number of Additional Bungees Added in Parallel Figure #10: Bungee Constant v. Number of Additional Bungees Added in Parallel. We plotted our three bungee constants (which we obtained using the above graph) against the number of additional bungees added to our one original bungee cord. So, for our three cords we plotted against 2 because we had added two additional bungee cords of the same length to obtain three bungee cords in parallel. The above plot tells us that for each bungee added in parallel, we will increase our bungee constant k by *(number of bungees of same length added in parallel). So, for example if we were to have four bungees of the same length in parallel, each with bungee constant k at that length, using our equation we could predict that the bungee constant of the four bungees together would be k *3. Discussion: The experiments conducted in this lab allowed us to determine two very important characteristics of our bungee cord, the bungee constant per unit length as well as the way the bungee constant changes by placing two or more bungees of the same length in parallel.

7 Our value for bungee constant per unit length yielded an uncertainty of ±1.57 N, which is a percent uncertainty of 39 %. This is a terribly high value for uncertainty, but upon observing the graph and noting the non-linear nature of our data points it is not surprising that we came out with such a high uncertainty. We wanted to come up with a bungee constant per meter, which would require our data points in the bungee constant vs. length graph to be very linear (thus allowing us to use Hooke s Law and treat the slope of this linear trend line as the bungee constant per unit length). However, in observing the graph it is clear that the bungee cord does not behave as similarly to a spring as we had initially hoped. The bungee constant vs. length graph would do better to have a polynomial trend line, but then this would not give us the bungee constant per unit length that we desired. In the second half of our experiment, we obtained an uncertainty value of ±0.28 for the slope of our final trend line (1.9935) which we use in our experimentally found formula for finding the bungee constant of bungees in parallel, k tot = k 1 + (1.9935)*n where k 1 is the bungee constant of any one bungee in parallel and n is the number of bungees added in parallel (total # bungees 1). This is a 14% uncertainty value. This is relatively low since, unlike the first half of the experiment, all the data sets we plotted were very linear and thus were better approximated by the linear trend lines we applied to them to obtain bungee constants. The percent error of each of our bungee constants were: 1 single bungee cord: ±0.03 Nm or 3% uncertainty 2 bungee cords in parallel: ±0.3 Nm or 11% uncertainty 3 bungee cords in parallel: ±0.7 Nm or 14% uncertainty These were used in calculating our final percent error (however it is just a coincidence that the final percent error and the percent error for 3 bungee cords in parallel happen to be the same). The primary source of uncertainty in the first half of our experiment, determining bungee constant per unit length, is the fact that our bungee cord is not an ideal spring. Because it is not an ideal spring, it does not perfectly obey Hooke s Law or the law that bungee constant is exactly inversely proportional to length of the bungee. Although it models Hooke s Law closely, we found experimentally that the relationship between bungee constant and length is not nearly as linear as we had hoped. The primary source of error in the second half of our experiment, determining relationship between bungee constant and number of bungees in parallel, is the low number of data sets we used (only three, due to time constraints). If we had had enough time to put even up to 5 or 6 bungee cords in parallel at the same length and take ten trials for each system, this would have allowed the trend line for our final graph to take more data points into account and thus give us a more accurate fit. However, considering that we were only able to put up to three cords in parallel, we did end up with a pretty low percent uncertainty. Conclusion: Through empirical experimentation we were able to successfully determine an equation for bungee constant of n bungees in parallel given a bungee constant k for any one of these cords (all must have uniform length, thus uniform k). As we hypothesized, the behavior of bungees in parallel pretty

8 closely resembled the behavior of springs in parallel. The equation we found was not exactly of the same form as equation #2, but the idea of a spring constant that increases linearly with the addition of more bungees in parallel is the same. Equation #2 would be k tot = n(k) for n springs in parallel each with the same spring constant k while our equation is k tot = n + k. We were also able to fairly successfully approximate the relationship between length of a bungee cord and its bungee constant. In order to create a more accurate approximation of this, we will revisit our bungee constant v. length of bungee cord graph and apply a different type of trend line. We will determine how we can apply data from this trend line so that given a bungee of some length x we can more precisely approximate its bungee constant. The two properties of our bungee cord that we obtained in this lab will help us choose the length of our bungee in our final egg drop as well as the number of bungees, if any, we will choose to put in parallel. Knowing the relationship between length of the bungee cord and its bungee constant will allow us to choose a length based on how far we want our egg to displace without having to find this length experimentally. For clear reasons, this will benefit us in our final quest for immortality in Lab 10!