Uncertainty of the centripetal force

Question

A lab experiment was performed to find the centripetal force on a rotating body which measured mass(m), frequency(n), and radius(r).  It was found that F_c=4Π²n²mr.  How do I write an expression for the uncertainty of the centripetal force?  Thank you 

Answer 1

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.

Answer 2

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?