= Repeated-timing uncertainty in a pendulum gravity measurement
{title2=$\sigma_g^2/g^2=(\sigma_L/L)^2+4\sigma_T^2/(NT^2)$}
For <simple pendulum> <oscillation period> $T$ in the <small-angle approximation>, $g=4\pi^2L/T^2$. If the length measurement and $N$ identically precise timing measurements have <independent> errors, the timing <sample mean> has <standard error> $\sigma_T/\sqrt N$. <Propagation of uncertainty> gives the title formula. A target fractional <standard uncertainty> $\epsilon$ requires $N\ge4(\sigma_T/T)^2/[\epsilon^2-(\sigma_L/L)^2]$, rounded upward to an integer. No amount of repeated timing can beat the floor $\sigma_L/L$ without improving the length measurement; correlated errors or a timing <systematic error> require a different model.
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