Monday, 17 October 2016

homework and exercises - The approximate uncertainty in r


The volume of a cylinder is given by the expression


V=πr2h


The uncertainties for V and h are as shown below



V±7%h±3%



What is the uncertainty in r?



Now, the obvious answer would be 2%, from dVV=dhh+2drr


However, rearranging to r2=Vπh gives 2drr=dVV+dhh

which gives a different answer of 5%. Thus, by simply rearranging the formula, we get different values for uncertainty in r.


How do you explain this?


(The mark scheme lists 5% as the correct answer)



Answer



You're confusing independent and dependent variables. When you propogate from uncertainties in the xi to some f(x1,x2...), the formula δf(x1...)=|fxi|δxi assumes that each of the xi is an independently measured variable and that f is a dependent variable to be calculated from the xi.



In the example you give, you have two independent measurements of V and h and are expected to calculate the uncertainty in r. Well, to use the above formula, you need to write r as a dependent variable of V and h. Therefore, it's only correct to solve for r first, and then calculate the uncertainty.


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