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Virginia Tech Department of Physics Laboratory Report Physics 2215 Lab 2 Measurement & Uncertainty Nam¥ Lab section: {Day/hour and Instructor) = A T IS 2 W

!l. SAMPLE CALCULATIONS FOR A SET OF MEASUREMENTS J The objective of the first part of this lab, portions is to practice calculating certain values. You are given a set of measurements related to the width of a standard 8.5" X 11" paper (this is the data in the second column labeled ). The ten measurements of the width of the same paper were obtained by using a ruler with centimeter divisions. The smallest division on the ruler used was a centimeter, and thus the measurer visually estimated and recorded the width to a millimeter. Using the ten values of W, given in Table 2-1, complete the remaining entries in the Table 2-1. : W, (em) Wi — W (om) W= W)* em) : 215 —0.0| 0- 000] > 213 ~02| 0.oytt| 5 214 0] 0.0{7\ i 214 -0.\\ 0.0l L 5 21.7 0.9 O, o3¢ L - , G Bl 0. 000\ ik 215 —o.0l g | L 8 21.7 ©.19 0.03¢ | e 215 — 0.0l g.0001 10 21.6 0.09 0. 008 o G - - 0,139 T fl o ~o0.0152 lov =011 Table 2-1 Question 1-1: Calculate the mean width of the paper * First, calculate the sum of your measurements. Enter your result in Table 2-1, * Divide the sum with the number of measurements N to obtain the mean /. (In the present case, N=10.) Enter your result in Table 2-1. * Keep four significant figures in 17/, Lab 2: Measurement and 1 Incertainty

? Question 1-3: What is the statistical error of W? Keep one significant figure A v : | Question 1-4: 1 inch is exactly equal to 2.54 cm by definition. Do your measurements suggest that the width of paper you measured is larger than 8.5", or smaller than 8.5", or equal to 8.5" within errors? Or do you lack enough c information to conclude? Explain. The wdth of dhe fafeC in cm 'S L"S"\U‘\. Gl Y Meatulerents 4vgest 4eC the mogt ol t O (maller gha | gt

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i . i it with First. make sure your stopwatch is running correctly by comparing : : h. If you the stopwatches of the other groups and also with your phone or wrist walc A g ; ' VO suspect that your stopwatch is not running correctly, ask your instructor ho':u,1 Calrl - - Q calibrate it. Also, compare the length of your two-meter-stick with those of the othe groups. Question 2-1: What type of error are you trying to avoid by doing this? 9¥§kemeahC Erfot Question 2-2: s there anything more that could be done? Discuss. 1 1 i 1 § { : (\‘?;\_?.‘n'.\'.‘»;: %Y,."': {h' {r\‘ﬂé‘zt {"u(t ang Merer s:ici{ %y 1\”’ t JAT, ook M, ‘DC Ou./(wk Before you start recording data, practice synchronizing the starting of the stopwatch with the release of the ball (or the release of the ball with the starting of the stopwatch), and the stopping of the stopwatch with the ball hitting the floor. Take twms so that both members of your group get a chance to practice both tasks. When both of you are comfortable that you can do a good job, start recording the data, - Lab 2: Measurement and Uncertainty N RIS

Drop the ball so it falls through a height of 2.00 meters and measure the time of fall to two significant figures. Make five measurements and enter your results in Table 2-2 below. (If you cannot reach 2 meters, pick a height that you can reach.) Question 2-3: Calculate the mean time of fall for your five measurements using Table 2-2. Keep four significant figures. Record your values in Table 2-2. Question 2-4: Calculate the standard deviation of the measurements using Table 2-2. Keep two significant figures. Show your work and record your values in Table 2-2. Table 2-2 { t; (sec) t; — € (sec) (t; — £)? (sec)? 1 14 Q.77 (‘LOHL}{Gl 2 |.o0 e e T 0.073984 3 0. %9 0. (3R 0. o\%Y 4 0.40 T lite 0. 06384 S 0.4 ey 0404475211 . 3. e e S T : Ji= 0.72¢ 87 = 0. 050u5 Now that you know how to calculate the mean and standard deviation yourselves if the need arises, you are entitled to use a spreadsheet program to do the calculation for you. Open the "Class Notes" folder on the desktop. Under the "2215 Setup Files\Lab2" folder, open "Fall Time.xIsx". This spreadsheet will automatically calculate the mean and standard deviation when you enter your measurements in the column labeled t. To check this, enter your data from Table 2-2 into the spreadsheet and see if it calculates a mean and standard deviation in agreement with your calculation. The mean and standard deviation are at the bottom of Table 2-2 of numbers. If the two calculations disagree, check your calculations in Table 2-2 orask your instructor for assistance. +Lab2: Measurenient and Uneestainty: oo 0 eie o on 0 gy 0

Copy your previous five measurements from Table 2-2 into Table 2-3. & Question 2-5: Take 35 more data points for a (otal of 40 measurements. Input your results into the spreadsheet on your computer and into Table 2-3. Make sure you take turns in dropping the ball and using the stopwatch. Record the mean and the standard deviation that the spreadsheet computes in Table 2-3. L t; (sec) i t; (sec) ‘ 0.04 21 0. 7L 2 | 22 0. 74 3 0.5 1 23 a2l 3%2 4 0 ¢ 24 0.5l : i, 6 0.4k 26 s b \ . 7 | 27 Q .\,\(o : 0.5 % 0.4 - 5@ ? 0. 5% % 0. %o - . 10 0.4 30 05+ 11 0. 56 31 0.7 : 12 0, Ue 32 0.7 13 0, U7 33 A & 0, k7 34 o .63 . T 2 O, Bt 35 0. ¢ 16 > . » 0,66 - 0.6 . . 18 0, h4 o goog 19 0, 39 T = e - L YS Sfs ) \3 ) \\ . Table 2-3 g ,’5 . Lab 2: Measurement and Uncertainty 3 2-14

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Question 2-6: Calculate the mean plus the standard deviation, ¢ + 0,and the mean minus the standard deviation, F— O¢ . Enter these values, using three significant figures. into Table 2-4. Count the number of your data points that lie within these two values and enter this value in Table 24 Calculate the fraction of data points lying within these two values and enter this in Table 2-4. Calculate the percentage of data points lying within these two values and enter this value in Table 2-4. t + o, (sec) 0.7366| t — o, (sec) 0. L[(,OSXC( Number of data points within £ + g, 3 Y Total number of data points 40 Fraction of data points within £ + [os gbf/q'o Percentage of data points within £ + o, 2570 Table 2-4 Unless large systematic errors have skewed your measurements, you should have found that about two thirds of the data points fall withint + o,. Since the correct value of ¢ should lie in the vicinity of f, this means that the probability for each individual measurement to fall within +0; of the correct result is about two thirds. It is in this probabilistic sense that the standard deviation quantifies the size of the statistical error.

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