Propagation of uncertainty allows one to determine the error in an experiment. T

Question

Propagation of uncertainty allows one to determine the error in an experiment. This sa is important in a single trial experiment and less so in multi-trial experiments, where the data can be statistically analyzed. Propagation of uncertainty also allows one to determine which of the measurements contributes the most to the overall uncertainty. Consider the equation for the surface area of a cone: A = phirs + phir2. where r = radius of the base and s = length of the side. The uncertainty in the surface area depends on the uncertainties in the radius and side measurements. Let r = 3.56plusminus0.05 cm and s = 15.00plusminus0.05 cm. Calculate the surface area and its uncertainty. Which measurement, r or s. contributes the most to the uncertainty in the surface area? Why? Read Appendix C to solve.

Explanation / Answer

A = pi*r*s + pi*r^2

(dA/A) = (dr/r) + (ds/s) + 2(dr/r)

(dA/A) = 3*(dr/r) + (ds/s)

Given,

r = 3.56 cm

dr = 0.05 cm

s = 15 cm

ds = 0.05 cm

A = pi*r*s + pi*r^2 = pi*3.56*15 + pi*(3.56)^2 = 207.576 cm^2

(dA/A) = 3*(dr/r) + (ds/s)

dA/A = 3*(0.05/3.56) + (0.05/15) = 0.04546816

So dA = (207.576)(0.04546816) = 9.438 cm^2

So surface Area = 207.576 cm^2

So uncertainity in SA = 9.438 cm^2

So SA = 207.576 +/- 9.438 cm^2

Since (dA/A) = 3*(dr/r) + (ds/s), because of the factor '3', r contributes most to the uncertainity in SA.

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