Background Information: From the measured diameter and height, calculate the vol

Question

Background Information: From the measured diameter and height, calculate the volume (V) of each cylinder. The volume of the cylinder is given by: V = [?*D^2]*h/4. When you use the average value of the diameter (D) and height (h) in your calculation you will get the average value of the volume. Below we show you how to get the error in the volume leading from the errors ?D and ?h from the measurements of the diameter and height respectively. In general, if one has a quantity W proportional to a^n * b^m * c^p , where a, b, and c are variables, then the fractional error in W leading from the errors ?a, ?b an ?c from the measurement of a, b, and c is given by this error analysis equation:

Error Analysis Equation: ?W/W = ( ((n?a)/a)^2) + ((m?b)/b)^2) + ((p?c)/c)^2 )^1/2

Question : Based on the formula given for error analysis (stated below), calculate the ratio of the error in the volume to the mean volume (?V/V) in terms of the corresponding errors in the measurement of the diameter (?D) and the height (?h).

For the particular case of the volume, W corresponds to the average volume V, and ?W corresponds to the error in the volume ?V. The quantities a corresponds to the diameter with n=2, b corresponds to height with m=1. Also p=c=1 and ?c= 0. Note that in case of formulas with division with exponents n, m, p might have negative values. Also note that constants do not have errors.

Explanation / Answer

so ?V/V = ( ((2?D/D)^2) + ((?h/h)^2) )^0.5 and you work out deltaD and deltaH from your lab results : D and h are the mean values (expectations) - systematic errors are not detected.

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