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UMass Amherst PHYSICS 132 - Physics 132 Lab 1

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Phys 132-99 Volume of the Library Part 1 - Experimental design 1. Methods Length and Width - To measure the length and width, we each measured the size of our feet and walked, heel to toe, along one section of the repeating cement square pattern. We then multiplied this measurement by how many repetitions of this pattern we saw along two intersecting sides of the library. Height - To find the height we measured the height of each stair and counted how many stairs were between the first and second floors. We then multiplied the number of stairs by the height of each stair. We also assumed there was about 1 meter of extra space between the floors and ceilings of each floor. Since there are 30 floors, we multiplied the distance between the first and second floor by 30. This measurement is made under the assumption that each stair is the same height and each floor is the same height. 2. Calculations Team member Length m Width m Height m Volume m3 % error Victoria 36.6 32.94 97.5 117,546.5 25.05 Zoe 33.46 33.46 93.75 104,960 11.66 Julia 37.13 33.41 93.75 116,298.1 23.72 Eliza 31.14 28.02 93 81,146.5 13.67 Part II - Statistical Analysis 1. The mean and standard deviation for each of the three measurements: Value Length m Width m Height m Mean 34.58 31.96 94.5 Standard deviation 2.43 2.28 1.762. The volume calculated from the means was 104,439 m3. We think the uncertainty in this volume is +/- 2,000 m3 because each of our measurements varies by a few meters from one another and sometimes more. Physical simulation of the Monte Carlo Method Draw # Length m Width m Height m Volume (m3.) 1 37.13 33.41 93 115,368 2 33.46 33.41 93.75 104,803 3 36.6 33.46 93.75 114,810 4 37.14 33.46 97.5 121,164 5 36.6 32.94 93.75 113,025 6 37.13 33.41 93.75 116,298 7 33.46 33.46 93 104,120 8 33.46 33.46 93.75 104,960 9 37.13 28.02 97.5 101,437 10 37.13 28.02 97.5 101,437 Mean 35.92 32.31 94.73 109,742 1. What is the average and standard deviation for the volumes you have calculated using this method? Average: 109,742 m3 Standard Deviation: 7,137 2. Calculate the percent error on each of the three dimensions. Length m Width m Height Percent error (%) 3.89 1.08 .024 3. Based on our calculated values for percent error, the length will have the greatest impact on the uncertainty of the volume, because it has the highest percent error of the three dimensions. Whenrandom values of length are chosen, the average is 3.89% greater than the expected, which was our average. This value varies the most of our values according to the random test. Computer Simulation of the Monte Carlo Method 4. The rand() or void function returns a random number in the range of 0 to RAND MAX. RAND_MAX is a constant in which the value is usually 32767. 5. The NORMINV function calculates the inverse of the Cumulative Normal Distribution Function for a supplied value of x, and a supplied distribution mean & standard deviation. Using a Computer to do the Monte Carlo Method 6. The mean of the volume is 104,249 m3, and the standard deviation of the volume is 10,418. 7. Our estimate was too low. The uncertainty or standard deviation is over 10,418. 8. The percent error of the volume calculated from the monte carlo test and the volume calculated from our average is 0.1%. This value is closer to the product of all of the percent errors rather than the sum. 9. Looking at percent errors is important because percent error gives us an idea of how accurate the random test calculates mean compared to the mean we calculated from our measurements. 10. We think the volume we calculated will be higher than the actual volume because most of us measured the length to be greater than the width and I think they may actually be equal. Also our measurements of height were based off of assumptions.11. Our measurements were not tight to the exact dimensions of the building but instead the boundaries just beyond the fence around the building. We would’ve gotten a closer measurement if our steps lined the exact edge of the walls of the building. While most of our measurements were off by a few meters, some of them were very close to the exact dimensions of the library (length and width). 12. The standard deviation from our Monte Carlo simulation was 10,564. Our average measured volume was 109,742 m3 . The actual volume of the library is 94,000 m3 . Our average volume measurement was off by 1.5 standard deviations. 13. We individually measured the length, width, and height after agreeing upon a method. It would have been more beneficial to us to communicate while we were measuring as well to ensure our values were in the same range. We definitely could have measured the dimensions differently and more effectively. For example, we could have used satellite images of the library and referenced them with our own measurements and tally of cement squares and bricks surrounding the library. Our height measurement also involved many assumptions which may have not been true. If we wanted to walk from floor to floor and measure each floor that would have been far more accurate than applying the measurements of 1 floor to all of the


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UMass Amherst PHYSICS 132 - Physics 132 Lab 1

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