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Uncertainty of the centripetal force

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2 Answers

Let δn, δm, and δr be the uncertainties of n, m, and r, respectively. Then the error propagation law tells us that the relative uncertainty of the product F=4Π2n2mr is
 
δF/F = 2 δn/n + δm/m + δr/r, 
 
from which you can find δF.
Note: a constant factor (4Π2) does not contribute to the relative uncertainty, whereas the exponent 2 in n² becomes a factor of 2 in the relative uncertainty.
 
Alternatively, you can compute the total differential of F,
 
dF = (∂F/∂n) dn + (∂F/∂m) dm + (∂F/∂r) dr
= (8Π2nmr) dn + (4Π2n2r) dm + (4Π2n2m) dr.
Wouldn't you just convert to differentials?
 
dF = 4pi^2 (dn)^2 (dm) (dr)
 
where each differential would be replaced with the uncertainty of that value?

Comments

You can use the differential dF as the uncertainty, but you need to use the product rule to compute it, which gives you 3 terms.