1.4 Mongolian Steps<br />EXPERIMENTER 1 DID Tyler Cash assembled steps and recorded data<br />Dates work done: 4/1/09<br /...
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1.4 Modeling And Analysis, The Mongolian Steppe Steps

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Scientific Paper on the Modeling of Mongolian Steppe steps.

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1.4 Modeling And Analysis, The Mongolian Steppe Steps

  1. 1. 1.4 Mongolian Steps<br />EXPERIMENTER 1 DID Tyler Cash assembled steps and recorded data<br />Dates work done: 4/1/09<br />OVERVIEW<br />The goal of this experiment is to find a model that predicts the maximum allowed overhang of a step. I started out using 5 steps. I repeated the experiment 5 times with 4 steps. Then I did the same process, except with 8 steps. I then derived the theoretical behavior of steps, and then applied it to my data.<br />MATERIALS AND METHODS<br />I used 8 plastic rulers as steps. Each ruler was 30 centimeters long. In order to balance the steps, I started at the top, balancing one step at a time until I reached the bottom step. To find the overhang of each step, I looked at the amount the ruler underneath was left exposed. I recorded the measurements, and then disassembled the steps and started over. I did this 5 times each for a set of 5 rulers and a set of 8 rulers.<br />The uncertainty in the data will be accounted for by using the standard deviation of the mean. The equation for standard deviation of the mean is SDOM=σ/√(N). Using this method for uncertainty should account for any random error in the experiment. <br />Theory<br />In order to derive the length of overhang of each step, I calculated the position of the center of mass for each step. It is worth noting that for this analysis to be correct, each ruler must be uniform, have the same density, thickness, and width.<br />We can approach this problem thinking of it as a system rotating about a fixed point. If there is ever more mass on one side of the system than the other, then the system will start rotating. In this experiment, should the ruler extend too far over the edge, and thus creating too much mass on one side of the system, the ruler will rotate about the edge and fall to the table.<br />In order for the first (top) step to be “balanced,” it cannot extend more than halfway over the edge. L/2 is also the center of mass of the step.<br />When balancing the second step, we must take in account the mass of the step on top of it. First, we find the combined center of mass of these 2 steps. We then place this center of mass at the edge of the third step.<br />For the third step, we repeat the process. We find the center of mass of the system of the top 3 steps, and then move it to the edge of the fourth step.<br />The center of mass location for 7 steps of Length L is:<br />StepCM<br /><ul><li>1/2 L
  2. 2. 3/4 L
  3. 3. 5/6 L
  4. 4. 7/8 L
  5. 5. 9/10 L
  6. 6. 11/12 L
  7. 7. 13/24</li></ul>This means our rulers extend a distance of 1-CM for each step.<br />Using this piece of information, we can guess that the formula for the overhang of a step is Overhang=L21# of steps.<br />RESULTS (DATA and OBSERVATIONS)<br />4 Steps7 StepsAverageSt Dev of MeanAverageSt Dev of Mean15.080.0215.020.0489897957.540.0244948976.920.3367491654.920.03741657450.0547722563.580.1067707833.680.20346989930.1095445122.460.041.980.037416574<br />It is important to note that the rulers were not flat or uniform. This caused them to be unstable. At high numbers of steps, the rulers would often topple over. This is a very likely source of error.<br />DISCUSSION (ANALYSIS RESULTS)<br />Using the theory developed in the theory section, I knew the data should fit a function of the form Y=AxB. Using the program WAPP+, I fit the data to a power function.<br />For the fit for 4 steps, WAPP+ returned a fit of <br />A = 15.08 ± .20E-01<br />B = -1.008 ± .42E-02,<br />with a reduced Chi squared value of 3.8. This indicates that the data fit the function reasonably well. This values of A and B were also very close to the predicted values of A=15 and B=-1.<br />For the fit for 7 steps, WAPP+ returned a fit of<br />A = 15.03 ± 0.49E-01<br />B = -1.017 ± 0.57E-02,<br />with a reduced Chi squared value of 2.5. This indicates that the data fit the function reasonably well. This values of A and B were also within uncertainty of the predicted values of A=15 and B=-1.<br />CONCLUSIONS<br />In this lab, I derived the relationship between overhang number and overhang of steps. I then used experimental data to confirm that relationship. The overhang of a step was found to be y=12Lx-1. My results for four steps were just outside of the expected values. My results for seven steps agreed with expected values. The results of the experiment were skewed by the fact that the rulers were not flat. I had no way of accounting for this in my measurements, so it is not accounted for in the uncertainty. If we take this into account, both fit reports may have agreed with the expected values.<br />REFERENCES<br /> Taylor, John R. 1997. 2nd ed. Sausalito, California: University of Science Books, 102.<br />

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