The Dynamic Relation between String Length and Height with a Set Mass Introduction: The purpose of this experiment was to find the relation between string length and height while keeping the mass constant in order to complete the Bungee Challenge. The Bungee Challenge is the final test of our results from this experiment. An egg attached to a bungee string will be dropped from the fourth floor of the Great Hall to see whether it hits the ground or not. The independent variable in this experiment is string length and the dependent variable is height. Finding the appropriate string length for a given height is what needs to be found in this experiment. This experiment is empirical and the trials that were conducted were used to find a final equation that relates string length of a bungee cord to the height from which the mass is released. Equation 1: y = mx + b Equation 2: y = 15.307x + 1.13 Equation 1 models the lines on the graphs of this experiment. y represents height, x represents unstretched string length, m represents (Height/Length), and b represents the intercept which shows error. Equation 2 is the final result found in this experiment. y represents (Height/Length) since it is the slopes calculated from the other graphs in figures 3,5, and 7. X represents Mass and 15.307 and 1.13 are constants in the final equation. Methods: The length of the bungee string was varied in each trial and the height that the hanging mass stretched was recorded for each string length. Three tests were conducted each with a different mass that was kept constant within its own test. The set-up shown in figure 1 took place in the lab room. One end of the bungee string is attached to the end of the metal bar and a
loop is tied further down on the string and attached to the hanging mass. It was necessary to stretch the new string a little to wear it in and it is crucial to take the mass off of the string after each trial, so that the string does not get overstretched with time. The dynamic system is being examined in this experiment. Figure 1: Bungee String Apparatus. The bungee string was attached at the top of the pole and the hanging mass was hung on the tied knot. Once the set-up was completed, the length of the unstretched string, which is the length of the string when the hanging mass is not attached, was measured and recorded. The length of the string was measured from the top of the metal bar to the beginning of the tied loop. The measuring tape was hooked onto the metal bar to the right of where the bungee string was attached so that the numbers were visible. A mass of 0.05 kg was tested first and attached to the tied loop via the hanging mass. The hanging mass was lifted by one of the partners so that the top of the hanging mass was even with the metal bar. The other partner started recording on an ipad, using the standard camera application on video mode, once the hanging mass was ready to
be released. The hanging mass was released from that height and was caught after it stretched its full length and began moving upward. The recording of the drop was then used to find the exact point in the fall where the hanging mass reached its maximum stretched length (maximum stretched length is what this experiment refers to as height) and began to move upward. The measuring tape was seen on the video and the line on the measuring tape where the top of the hanging mass reached was recorded as the height. This same procedure was done for five different string lengths within this first test. The remaining two tests were carried out in the same way except with masses of 0.1 kg and 0.15 kg. Results: The final result of this experiment was an equation that gives the relation between string length and height (Equation 2). Mass (kg) String Length (m) Height (m) Uncertainty of the Height and String Length (m) 0.05 0.37 0.707 (+/-) 0.001 0.05 0.48 0.927 (+/-) 0.001 0.05 0.53 0.987 (+/-) 0.001 0.05 0.55 1.057 (+/-) 0.001 0.05 0.695 1.337 (+/-) 0.001 Figure 2: Test 1 Table. This table shows the recorded string length and heights for the mass of 0.05 kg. The string length was measured from the top of the metal bar to the beginning of the knot and then recorded in this table. The height was measured from the top of the metal bar to the top of the hanging mass and then recorded in this table. The uncertainty of the height and the string length are the same because it was calculated by least count of the measuring tape. The same methods were used to collect data in the remaining tables.
Figure 3: Test 1 Graph. This graph shows the relation between string length and height for the mass of 0.05 kg. The height vs. string length shows the relation between those two variables and this slope becomes the y-variable in the final slope vs. mass graph. This graph shows the linear relationship between height and string length. The same principles are used in each of the remaining graphs. Mass (kg) String Length (m) Height (m) Uncertainty of the Height and String Length (m) 0.1 0.235 0.602 (+/-) 0.001 0.1 0.395 1.047 (+/-) 0.001 0.1 0.435 1.157 (+/-) 0.001 0.1 0.54 1.407 (+/-) 0.001 0.1 0.58 1.527 (+/-) 0.001 Figure 4: Test 2 Table. This table shows the recorded string length and heights for the mass of 0.1 kg.
Figure 5: Test 2 Graph. This graph shows the relation between string length and height for the mass of 0.1 kg. Mass (kg) String Length (m) Height (m) Uncertainty of the Height and String Length (m) 0.15 0.2 0.602 (+/-) 0.001 0.15 0.27 0.917 (+/-) 0.001 0.15 0.32 1.137 (+/-) 0.001 0.15 0.35 1.237 (+/-) 0.001 0.15 0.445 1.527 (+/-) 0.001 Figure 6: Test 3 Table. This table shows the recorded string length and heights for the mass of 0.15 kg.
Figure 7: Test3 Graph. This graph shows the relation between string length and height for the mass of 0.15 kg. Mass (kg) Slope of the graphs (L/H) 0.05 1.9113 0.1 2.6289 0.15 3.442 Figure 8: Mass and Slope Table. This table shows the three tested masses along with each of the three slopes from the three graphs displayed above (Figures 3, 5, and 7). Figure 9: Slope vs. Mass Graph. This graph shows the relation between the slope of the other graphs and the masses tested. This linear graph and the equation of the trend line was the reason that these tests were conducted. Equation 2 gives the relation between length, height, and mass of the object released. The slopes of the other three graphs were used as the y-values on this graph and the three tested masses were used as x-values on this graph, respectively. The reason for choosing to graph length vs. height is because the relationship between those two variables is needed. The slope vs. mass graph was chosen because it includes all three of the tests in one graph which makes the overall equation more accurate. By taking the three
slopes and the three masses and making them points on the graph in figure 9, all of the data is represented and put together in one final equation (equation 2). Equation 2 was the base for the final result which is (H/L) = 15.307M + 1.13. The y-variable represents slope which is (H/L) and the x-variable represents mass which is M. This was the equation that was found in this empirical experiment and it shows the relation between its variables clearly. The three graphs in figures 3, 5, and 7 all had a y-intercept, but that uncertainty was so small that it can be thought of as negligible. By setting the intercept to (0, 0), it makes it easier to obtain the final equation because the small uncertainty can be disregarded. The slope is barely changed by this and therefore the final equation will be even less affected. The types of uncertainty include measurements, oscillatory motion, inconstant string stretch, and air resistance. There was uncertainty in the measurement of the initial release height due to the loop that attached the string to the metal bar. That uncertainty was (+/-) 0.001 m calculated by the least count of the measuring tape. The uncertainty in the height can also be the least count of the measuring tape at (+/-) 0.001 m. The percent uncertainty of string length was 0.23% and the percent uncertainty of height was 0.087%. Regression analysis was done on each of the graphs. The graph in figure 3 had a slope of 2.6289 with a standard error of 0.0529. The graph in figure 5 had a slope of 3.442 with a standard error of 0.243. The graph in figure 7 had a slope of 1.9113 with a standard error of 0.0715. The graph in figure 9 had a slope of 15.307 with a standard error of 0.5514 and an intercept of 1.13 with a standard error of 0.05956. The intercept in the graph of figure 9 is ideally supposed to pass through (0, 1). The extra 0.13 error in the intercept can be explained by the other uncertainties such as the tied loops and the measuring of height. Those two uncertainties are what caused the intercept to have that slight error in the end. Discussion:
The final equation is (H/L) = 15.307M + 1.13. This equation is wanted because it allows us to determine the length of the string given a set height and a set mass. This empirical experiment was designed to give us the relation between string length, mass, and height. The equation that was calculated from the tests did exactly that. We will use this equation in the bungee challenge to determine the length of the bungee string that should be used given the height and hanging mass. The substitution of variables for the final equation can be supported by the graphs in figures 3, 5, and 7. The slope of those graphs is height over string length which makes y = (H/L) in the final equation and mass is along the x-axis in figure 9 which is why x=m. The slope and intercept in the graph of figure 9 are both constants in the final equation. The uncertainty in the string length was definitely the most significant, as shown with the percent uncertainty of 0.23%, due to the variation in the size of loops tied each time when the string length was varied. Air resistance does exist, but should not be a problem because it is factored into the experiment and the testing environment will remain the same during the bungee challenge. This experiment represented the real world very well because we did not base our experiment off of any known equation and conducted the tests in real world surroundings. Something that could have been done differently is releasing the hanging mass from the bottom of the attached string to the metal bar in order to reduce that source of uncertainty. This will reduce the amount of uncertainty because the hanging mass will be dropped from the exact same height as where it is attached which will make the recorded height more accurate. Conclusion: An equation that relates the string length, height, and mass of the object released was found and can now be used to calculate string length at any given height or mass. This equation
will work only with the certain string used and will follow the calculated equation (equation 2) most closely in the same environment. This equation allows us to find the length of the string needed for a certain height and a certain hanging mass and will be used for this in preparation for the bungee challenge. Our results can be used for any height, hanging mass, or string length given as long as two out of the three variables are provided. Tying knots in the bungee string to see how that affects the results is definitely something that can be done to improve this experiment and further build on this topic because a bungee string with knots tied in it will slow the momentum of the hanging mass down faster than a string without knots.