What is the algebraic expression for the uncertainty in the centripetal acceleration σac for uniform circular motion in terms of the uncertainties in the period and the radius of the motion? (Use the following as necessary: σT, σr, ac , T, and r.)

Answer 1

ac = 4 pi^2*r/T^2

ln(ac) = ln(4 pi^2) + ln r - 2 lnT
Now take the differential.
d(ac)/ac = lnr/r - 2 dT/T

For the standard deviation of the relative error, error, this leads to:

[σ(ac)/ac]^2 = [σr/r]^2 + [2*σT/T]^2

The relative error in ac, squared, equals the sum of the square of the relative error of r and the square of twice the relative error in T.

This assumes that errors in T and r are uncorrelated. Sometimes this is called the RSS (Root of the Sum of the Squares) rule.