Saimon Islam, Yoko Koyoma. Professor Cumming Physics 113- Thursday morning November 15, 015. Dynamic Relationship between Bungee Cord s Length and Tension Force Introduction: The purpose of this experiment is to figure out the relationship between the length of a bungee cord and the tension force it experiences in a dynamic stage. A bungee cord shows elastic spring-like behavior. Using the CWE theorem, a spring constant k can be derived, from which a relationship between the tension force and the length of the cord can be established. The CWE theorem states that in a conservative system (one that neither loses nor gains energy from outside the system) the total energy sum of potential and kinetic energies is constant. In the bungee system, before the jump, all of the energy is gravitational potential energy. At the bottom of the jump, when the mass stops going downward, the velocity and the height are zero. Therefore, all of the energy is converted to elastic potential energy. Ideally, considering the height = h Mass = m Spring constant= k Stretch= x mgh=.5kx (1) In the experiment, h= l+x max X= x max Therefore, mg(l+x max ) =.5kx max () The total acting forces in the system are the gravitational weight and the elastic tension force of the cord. Methods: Initially the cord is evenly stretched out so that the elasticity is roughly even at all parts of the cord. Then one end of the cord is tied to the top of the stand with a tight loop. After that, a mass of.00 kg is tied to the cord 6.5 cm away from the top. All the heights are measured to the top of the hanging mass. Additionally, a tape measure is hanged from the stand close to the cord. Then the mass is slowly dropped to hang freely. When the mass is still, a measurement of the stretched cord is taken. Next, the mass is taken to the top of the stand and dropped to fall freely. The maximum height travelled by the mass is video recorded using an iphone camera. From the recording, the height is recorded. The total acting force is measured using the force sensor connected to the hook of the stand. The process is repeated 5 more times with the initial cord length being 9 cm, 37.5 cm, 41.5 cm, 47.5 cm and 5.5 cm.
Force sensor Measuring Tape Unstretched Cord length l Cord stretched with hanging mass in static motion. - static equilibrium length x max h=l+x max.00 kg Mass. Cord stretched with mass in dynamic motion Figure 1.1: Static and dynamic motion of an elastic cord with a hanging mass of.00 kg. In Figure 1,1, three states of the cord are shown. First the unstretched hanging cord. Secondly the cord stretched because of the hanging mass in a static equilibrium and thirdly the cord stretched to the maximum in a dynamic motion. The force sensor is connected to the cord to measure the net acting force and the measuring tape is connected adjacent to the cord so that the maximum height can be measured in a dynamic motion. Since the purpose is to find the relationship in dynamic motion, the static equilibrium length is not included in the results section.
Results: A relationship is found from the collected data. From equation, in dynamic motion, K= mg(l + x max ) x max N/m (3) mg= 1.96 N l ±.001 m x max ±.01 m h= l+x max ±.01 m F±.001 N k (dynamic)±.01 N/m.65.8 1.08 1.73 6.9.90.89 1.18 1.63 5.84.375 1.04 1.41 1.55 5.11.415 1.5 1.66 1.44 4.17.475 1.39 1.86 1.41 3.77.55 1.5.04 1.33 3.48 Table 1.1: Measured lengths, maximum stretches, maximum heights, net acting forces and spring constants in a dynamic experiment with a.00 kg mass. The lengths and the maximum heights are measured using the measuring tape as described in the methods section. The x max is derived by subtracting the l values from the h values for each of the lengths. The net acting force F is measured from the force sensor for each corresponding length. And the k values are calculated from equation 3.
Force (net) (N) h=l+x_max (m) Graph 1: l vs h.5 y = 3.945x R² = 0.988 1.5 1 0.5 0 0 0.1 0. 0.3 0.4 0.5 0.6 l (m) Graph 1: Relationship between l and maximum height h in a dynamic experiment with mass of.00 kg. Graph 1.1 shows the relationship between initial unstretched length l and the maximum height h= l+x max. The relationship between them is linear. From the graph, h=3.94l ±.01 m (4) This relationship is important because with this, we can predict the maximum height an object with.00 kg mass will travel using the elastic cord. Graph : F vs h 1.8 1.6 1.4 1. 1 0.8 0.6 0.4 0. 0 y = 1.769x -0.385 0 0.5 1 1.5.5 h (m) Graph : The relationship between height and acting net force in a dynamic experiment with a mass of.00 kg.
There is an inverse power relationship between h and F. From Graph, we find the relationship of F net = 1.763h.385 Using equation 4, F net = 1.04l.385 (5) Since the only forces working on the system are gravitational forces and tension force of the cord, Therefore, of F net = F T -mg Using equation 5, F T = 1.04l.385 +mg (6) This equation leads to the connection between the tension force and the length of the cord. In the bungee jump, we can measure the length of the cord and calculate the net acting force and the tension force from it. The measurement of the tension force is necessary because one of the key factors of a good bungee jump experience is not to experience too high a tension force. If the net force is over 3 mg, meaning if the tension force is over 4g, then the object experiences too much force and gets damaged. Therefore, the tension force must not exceed 4mg. Discussion: From the results section, we see that, F T = 1.04l.385 +mg. The results are in agreement with the original assumption. The original assumption was that the longer the cord will be, the lesser tension force the object will experience. In the experiment, the net force and the length are in inverse relationship. This means the longer portion of the cord is used, the less is the tension force acting in the system. Which is understandable since the spring constant value is also decreasing. There are several factors in this experiment which needs to be discussed. Firstly, this is not enough information to conduct the bungee jump. Because the relationship between length and tension has been established, but only for a constant mass. The effect of changing mass in the tension force is yet to be found. The equilibrium length and k value were not used in this experiment since we only focused on the dynamic section here. Most of the uncertainties in this experiment were acceptable. That being said, there were a few uncertainties that can be improved. Firstly, the elastic stretch of the circumference of the loop was not accounted for. Often tying a tight knot was difficult and therefore the stretch was larger than usual. This can be solved stretching the cord while tying the knot. Secondly, the mass, due to its shape, often had a rotating motion with a spin which made measuring the exact length of the x max problematic. Also, the spin caused another force in the system we did not account for. From Table 1.1, the measured height and the derived x max and F has a higher uncertainty. The rotating motion of the hanging mass makes is troublesome to accurately measure the height in a dynamic motion. This problem can be solved by using a mass with a different shape where this rotating motion would not occur. Additionally, the slow motion
feature of the recording device is not slow enough to find a specific time frame where the height is the maximum. Using upgraded equipment can solve this problem. Conclusion: The tension force of the cord and the length of the cord are inversely related. An equation was derived from the relationship. The equation is numerically proven from the slope of graph and modified in equation 6. The purpose of the lab is fulfilled by this equation. This experiment can be further extended by checking the effects of varying the mass on the results. A good bungee jump has three components. The mass, the height and the force the object experiences. This experiment deals with the height and the force. If we can connect this relationship with the mass, then we can successfully predict a bungee jump. Acknowledgement: 1. Bungee journal. The Bungee Jump: Potential Energy at Work, AiS Challenge, Summer Teacher s Institute, Richard Allen 3. http://www.bungee.com/bzapp/press/pt.html On my honor, I did not take any unacknowledged aid.