LABORATORY MANUAL Relative uncertainty method for multiplication dividing The re

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LABORATORY MANUAL Relative uncertainty method for multiplication dividing The relative uncertainty method involves three steps 1. First, one converts all of the uncertainties to relative uncertainties 2. Second, the relative uncertainties are added together 3. Third, the relative uncertainty sum is converted back to a numerical uncertainty Question #21. Calculate the relative uncertainty associated with each measurement the rst one is done for you; divide the uncertainty by the measurement value): 0.0 13034 (i.e., 1.303496) Distance (30.69±0.4 m): Time (4.52 ± 0.02 s); a. 0.4 m / 30.69 m b. Again, notice that the relative uncertainties have no units. The units cancel when you divide numerical uncertainty (which has a unit) with the measurement value (which has the same unit). Question #22. Calculate the sum of the relative uncertainties: 0.013034 + Question #23. Show how the range 6.789823 m/s ± 0.118539 m/s can be obtained as follows. Divide the distance by the time (30.69 m, 4.52 s) to get the mid value (6.789823 m/s) and then multiply that mid value by the relative uncertainty (from Question #22) to get the uncertainty (± 0.1 18539 m/s): The answer to Question #23 (6.789823 m/s ± 0.118539 m/s) is not exactly the answer to Question #20 (6.790348 m/s ± 0.1 18541 m s) because the short-cut (relative uncertainty method) is an approximation to the max/min method. However, it is pretty good, and the smaller the relative uncertainties, the better the agreement. 2017.6 Page 8

Explanation / Answer

#24. given

x = 15.4 +- 0.3

y = 17.3 +- 0.2

z = 10.1 +- 0.1

A = xy/z^2

now, Amax = (xmax)*(ymax)/(zmin)^2 = (15.4+0.3)(17.3+0.2)/(10.1-0.1)^2 = 15.7*17.5/10^2 = 2.7475

Amin = (xmin)*(ymin)/(zmax)^2 = (15.4-0.3)(17.3-0.2)/(10.1+0.1)^2 = 15.1*17.1/10.2^2 = 2.48183

Aav = (Amx + Amin)/2 = 2.614667

error = (Amax - Amin)/2 = 0.132833

hence

A = 2.614667 +- 0.132883 m^2/s^2

#25. x = 15.4 +- 0.3

y = 17.3 +- 0.2

z = 10.1 +- 0.1

A = xy/z^2

A = 15.4*17.3/10.1^2 = 2.611705

dA/A = dx/x + dy/y + 2dz/z = 0.3/15.4 + 0.2/17.3 + 2*0.1/10.1

dA = 0.132787

hence A = 2.611705+-0.132787 m^2/s^2

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