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Repeated, but independent, measurements of a quantity allow you to determine the standard 5 deviation, and this is often a good measurement of your random uncertainty forthose individual measurements. Not surprisingly, the average (mean) of all such measurements is typically a better estimate of the experimental result than any of the individual measurements, and mathematically this uncertainty is quantified as the “standard deviation of the mean”. This can be calculated as: X = (3) VN S 2. Finding an unknown force In the table below, you find the force diagram and a typical data set that could have been measured in the second part of the lab. The objective is to determine the magnitude of the unknown force Ē. From the repeated measurements, find the average values and uncertainties (standard deviations) of the angles 04 and 02. Then use eqs.1, 2 of the write-up to calculate the components Fix, Fly, F2y, F2y, and their uncertainties. As above, you can do this by hand, or you may find it easier to have Excel do it for you (although if you do the latter, remember that Excel likes to have angels in radians, not degrees! The conversion factor is 1=3.14159 rad/180°). Calculate the magnitude of the unknown force, F=-(F1x+F2x). Compare the result with the actual value of 236.7 g*. Do the two agree to within your uncertainty? Also check if Fly+F2y=0, as we expect. Balancing an Unknown Force 250g Trial Notes 1 2 3 4 5 Fun O₂ 벌 293° 288 291 290 291 47° 49 47 47 47 02 F2 200g Based on the results of this tabulated calculation, how would you express the best estimate of the component forces, and the uncertainty on those estimates (Hint: take another careful look at APP. C, equation (3)). Fx= +/- Fy= +/- Assuming that the angle is in fact 0=50.0°, what is the discrepancy between the expected results and those you found from the data? Are these two measurements consistent with the expected results? How would your answers to the above change if the angle were, in fact, =50.4°? Keep in mind, in experiments it is as important to be careful in recording/setting your controlled variable as it is to carefully measure the thing you really are interested in. 2. Finding an unknown force In the table below, you find the force diagram and a typical data set that could have been measured in the second part of the lab. The objective is to determine the magnitude of the unknown force Ē. From the repeated measurements, find the average values and uncertainties (standard deviations) of the angles 04 and 02. Then use eqs.1, 2 of the write-up to calculate the components Fix, Fly, F2y, Fay, and their uncertainties. As above, you can do this by hand, or you may find it easier to have Excel do it for you (although if you do the latter, remember that Excel likes to have angels in radians, not degrees! The conversion factor is 1=3.14159 rad/180°). Calculate the magnitude of the unknown force, F=-(F1x+F2x). Compare the result with the actual value of 236.7 g*. Do the two agree to within your uncertainty? Also check if Fly+F2y=0, as we expect. Trial (i) Exi (g*) Fr – Fxi (Ēx – Fxi)? Fyi (g*) F, – Fy (F, – Fy:) 1 155 197 2 157 192 3 155 195 4 162 192 5 165 193 6 165 194 Ē Fy OF OF, * If the mass hanging from the string is m, the magnitude of the force acting on the string is mög, where g=9.80 m/s2 is the acceleration due to gravity. The unit of force is 1N (Newton). Thus, we would have to multiply all measured masses by 9.80 m/s2. Instead we save ourselves this trouble by defining (just for the purpose of this lab) the force unit 1g* as the weight of a mass of 1g, or the force by which a lg mass is attracted in the gravitational field of the earth. PRE-LAB EXERCISE: VECTOR ADDITION OF FORCES This Pre-Lab gives you practice with the method of estimating the uncertainty of repeated measurements and uncertainties of a function. Read the experiment first, then come back and do this Pre-Lab exercise. 1. Analyzing vector addition data In the table below, there are some typical data from the first part of the experiment. The repeated measurements were obtained with a 250.0 g* “force” hung from the Force Table at an angle of 0=50°. The table below walks you through the calculation for determining the standard deviation for a set of measurements of the two component forces (as you will do in this lab). Copy the two sets of force numbers into a spreadsheet and use the sheet to complete the table following the guidelines provided in App. D. (if you’d prefer, you can do this by hand, but you’ll be much better off if you can get used to doing such things with Excel or Googlesheets). Use equation (2) of App. C to then determine the standard deviations of the two distributions (for Fx and Fy). Finally, use the “STDEV” function in the spreadsheet program to confirm that it gives you the same results for each distribution as the full “by hand” calculation. Trial (i) Fxi (g*) (Ēx – Fxi)? Fyi (g*) (Fy – Fyi)? Fx – Fxi F E – Fyi 1 155 197 2 157 192 3 155 195 4 162 192 5 165 193 6 165 194

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