Question

Force F has units of kg m/s2 . The electric force between an object of
charge Q and a second object of charge q separated by a distance r is given by F= k Qq/ r2 , where k is
a constant. In order to determine the charge of q, the electric force between the two charges is
measured to accuracy ÄF, Q is known to a accuracy of ÄQ, and r is known to an accuracy Är.
Assuming that all these errors are uncorrelated, what is the fractional error in the determination of q
(Fractional error = is Äq/q ) . Start with q= F r2/ (kQ)

Answer 1

I find it convenient to put the expression in logarithm form first.

lnq = lnF - lnk -lnQ +2*lnr
Take the differential due to small errors in F, Q and r:
dq/q = dF/F -dQ/Q +2 dr/r
(There is no appreciable uncertainty in k, so dk = 0)

When comparing the effects of uncorrelated separate error sources, you can ignore the + or - sign, since the error of each type goes randomly both ways.

The fractional error is dq/q is the square root of the sum of
(dF/F)^2 + (dQ/Q) + 2(dr/r)^2^2
That is the way random uncorrelated errors add.

I am not able to read the error numbers you have provided; just use the formula above.