1 Bungee Lab 1 11/6/2014 Section 6 Is Hooke s Law an Appropriate Model for a Bungee Cord? Introduction At the end of this project we hope to have designed a quality bungee experience for an egg. To get to that point, we have to first figure out exactly how the bungee cord behaves. In this preliminary experiment, we decided to see how Hooke s Law applied to the cord, thus attempting to model it as a spring. Hooke s Law states that (1) F = - kx, where the negative sign indicates that it is a restoring force, F is the spring force, k is the spring constant, and x is displacement. This experiment aims to reveal how varying mass affects cord stretch on a constant length of elastic cord. Then, converting mass to weight and comparing that to displacement can reveal the behavior of the elastic cord at various forces. If Hooke s Law were a good fit for modeling the bungee cord, the relationship between force, which in this case is weight, and displacement is expected to be linear. Methods Before doing anything else, the cord needs to be stretched out in order to avoid inaccuracies that could arise during the experiment from the cord being stretched out by a hanging mass and then not returning to the same equilibrium position. For the setup, the hanging apparatus is clamped to a table above the floor. The apparatus is made of a vertical rod with a horizontal attachment that is able to move up and down to control height above the floor. This
2 attachment has a small metal component to hang things from. A slipknot is tied at the end of the cord; this will be where the hanging mass is attached. A second slipknot is tied at some place in between the two ends of the cord; this will be where the cord is attached to the hanging apparatus. The excess cord is wrapped around the top of the hanging apparatus so it is out of the way and does not affect the experiment. Figure 1. Setup Diagram. An elastic cord with a hanging mass at the bottom is attached to the hanging apparatus. The equilibrium position of the cord is measured from the bottom of knot 2 to the top of knot 1, as labeled in Figure 1. This initial measurement will be subtracted from each length that results from each hanging mass to find the respective cord displacements. After establishing the equilibrium length of the cord as 0.629 ± 0.001 m, a hanging mass of 0.025 kg was added, and the new length from the bottom of knot 2 to the top of knot 1 was measured to be 0.699 ± 0.001 m. The equilibrium length is subtracted from this new length to get x = 0.070 ± 0.001 m for Equation 1. This procedure was repeated for six more trials up to a hanging mass of 0.175 kg. Measurements do not proceed past this mass because you do not want the cord to stretch past three times the original length or it could get damaged. In order to see if
3 Hooke s Law models this data, you need values for F (Equation 1). These are determined by converting mass to weight by multiplying each mass by the force of gravity, g. Results Our data did not show a linear relationship between F and x. Figure 2. Data Collection. Original cord length is 0.629 ± 0.001 m, weight found by F=mg. Trial Mass F x (kg, ± 0.001 kg) (N, ± 0.001 N) (m, ± 0.001 m) 1 0.000 0.000 0.000 2 0.025 2.406 0.070 3 0.050 4.812 0.158 4 0.075 7.218 0.291 5 0.100 9.624 0.466 6 0.125 12.030 0.675 7 0.150 14.435 0.898 8 0.175 16.841 1.108 Figure 2 displays our raw data and calculated data for each trial. As stated before, Hooke s Law suggests that the relationship between F and x should be linear. Since converting from mass to weight only requires multiplication by a constant, that relationship should also be linear.
4 Figure 3. Displacement vs. Mass. Adaptation of Hooke s Law. 1.200 Cord Stretch with Varying Masses 1.000 y = 6.5x - 0.11 Displacement (m, ± 0.001 m) 0.800 0.600 0.400 0.200 0.000 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200-0.200 Mass (kg, ± 0.001 kg) In Figure 3, the uncertainty of slope is ± 0.5 m/kg and the uncertainty of the y- intercept is ± 0.05 m. It is important to notice that a linear regression does not fit our data well (Figure 3). However, we did notice that it appeared as though there were two different linear regressions if you split the data points up into groups with the first group having points from Trials 1, 2, and 3, and the second group consisting of the points from Trials 4, 5, 6, 7, and 8 (Figure 2).
5 Figure 4. Weight vs. Displacement. Data split into two separate models of Hooke s Law. 2.000 Weight vs. Displacement (Long Cord) 1.800 1.600 F = 1.19x + 0.41 Weight (N, ± 0.001 N) 1.400 1.200 1.000 0.800 0.600 0.400 0.200 F = 3.1x 0.000 0.000 0.250 0.500 0.750 1.000 1.250 Displacement (m, ± 0.001 m) We split the data into groups and looked at F instead of mass so the slope of each line would represent k (Equation 1). For the first segment where F = 3.1x, the uncertainty of the slope is ± 0.2 N/m, and the y- intercept is not significant because it is less than the uncertainty. For the second segment where F = 1.19x + 0.41, the uncertainty of the slope is ± 0.03 N/m, and the uncertainty of the y- intercept is ± 0.02 N. These uncertainties were found using regression analysis on Excel. This graph made us wonder if the k of the bungee cord changed at an exact point described by where the two lines intersect. For Figure 4, this point is (0.207,0.657). This could suggest that k changes once F = 0.657 ± 0.001 N and at a displacement 32.9 ± 0.1% of equilibrium length. This uncertainty was found from percent uncertainty of displacement. In order to test this theory, we repeated the exact procedure described above but with equilibrium
6 lengths of 0.429 ± 0.001 m and 0.196 ± 0.001 m to see if k behaved similarly. After recording mass and displacement, we noticed trends similar to that of Figure 3 for both the medium and the short cord lengths. We then modeled the data in the same way used to produce Figure 4. Figure 5. Weight vs. Displacement of Medium and Short Cord Lengths. Slope is k of Equation 1 measured in N/m. 2.000 Weight vs. Displacement Weight (N, ± 0.001 N) 1.800 1.600 1.400 1.200 1.000 0.800 0.600 0.400 0.200 Short: F = 9.4x Medium: F = 4.5x Short: F = 3.5x + 0.42 Medium: F = 1.71x + 0.42 0.000 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 Displacement (m, ± 0.001 m) For the first segment of the medium cord where F = 4.5x, the uncertainty of the slope is ± 0.4 N/m, and the y- intercept is not significant. For the second segment of the medium cord where F = 1.71x + 0.42, the uncertainty for the slope is ± 0.05 N/m, and the uncertainty for the y- intercept is ± 0.03 N. These uncertainties were found from regression analysis in Excel. For the medium cord, the point of intersection is (0.146, 0.673), or intuitively, at F = 0.673 ± 0.001 N and a
7 displacement 34.0 ± 0.2% of the equilibrium length. This uncertainty was found from percent uncertainty of displacement. For the first segment of the short cord where F = 9.4x, the uncertainty of the slope is ± 0.8 N/m, and the y- intercept is not significant. For the second segment of the short cord where F = 3.5x + 0.42, the uncertainty for the slope is ± 0.1 N/m, and the uncertainty for the y- intercept is ± 0.02 N. Again, regression analysis in Excel was used to find uncertainties. For the short cord, the point of intersection is (0.070, 0.673), or at F = 0.673 ± 0.001 N and a displacement 35.7 ± 0.5% of the equilibrium length. This uncertainty was found by the percent uncertainty of displacement. When comparing Figures 4 and 5, the average point of intersection is F = 0.667 ± 0.002 N and at a displacement 34.2 ± 0.5% of the equilibrium length. Discussion Our data did not model a linear relationship between F and x as we expected. Our percent uncertainty for F is 0.3% and our average percent difference is 2%. Our percent uncertainty for percent of equilibrium length is 1% and our average percent difference is 5%. Because percent uncertainty is less than our average percent difference in both components, our results are not accurate. Therefore, these values are not in agreement with choosing Hooke s Law as a model. All of the first line segments of the respective lengths of elastic cord have high percent uncertainties for the k value. This could be because there are only three data points for each line. Having more data points could lower those uncertainties. Also, we used the same end of the cord to make each of the three lengths for the experiment. This could imply that that end could have gotten overworked and stretched out to where it affected the cord s ability to return completely to the equilibrium position. This systematic error seems probable because the uncertainties in the displacements are larger than other uncertainties of the experiment, and the average percent
8 difference is much larger than the percent uncertainty, making the value very inaccurate. The affect of an outstretched cord is impossible to quantify because you cannot pinpoint when the outstretching occurred, or if it was a gradual process. Even if you could, you would not be able to provide a quantitative value to it, thus the only way to evaluate the affects of this error is to eliminate the chance of it occurring. If the experiment were repeated, after initially stretching out the cord, I would pick three separate sections of the cord to test instead of all on one end, therefore hopefully eliminating any inaccuracies in displacement that exist in our current data. Although it is clear that Hooke s Law is not an accurate way to model the behavior of the bungee cord, our data still suggests that there is a relationship between F and x because all three lengths did produce similar overall trends. Conclusion This experiment aimed to reveal if the behavior of a bungee cord followed Hooke s Law by measuring how varying mass affects cord stretch on a constant length of elastic cord. If Hooke s Law was a good fit for modeling the bungee cord, the relationship between force, which in this case is weight, and displacement would have been linear. Hooke s Law is not a good fit to model the bungee cord at every force applied. When we split our data up to try to model Hooke s Law at different ranges of forces and evaluated the point where k changes, we still received inaccurate data. Looking back at Figure 5, instead of splitting the data points up into two linear segments of Hooke s Law, it seems possible that the data points follow a polynomial trend line in both cord lengths. Going forward, this experiment could be repeated to instead determine if fitting a polynomial trend line would produce accurate values.