= Mass ratio from spring oscillation frequencies
{title2=$m_1/m_2=(\nu_2/\nu_1)^2$}
For the same massless <spring> obeying <Hooke's law>, the <frequency> is $\nu=(2\pi)^{-1}\sqrt{k/m}$. The displayed <mass> ratio follows. For small <independent> frequency errors with common fractional <standard uncertainty> $\epsilon$, <monomial uncertainty propagation> gives $\sigma_R/R\simeq\sqrt8\epsilon$. If $\epsilon$ instead describes individual measurements and each <sample mean> uses $n$ independent trials, its <standard error> is smaller by $\sqrt n$, giving $\sigma_R/R\simeq\sqrt{8/n}\epsilon$. Four times as many trials halve this statistical uncertainty; a common calibration error needs its <covariance> included.
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