= Solution
For a <simple pendulum> in the <small-angle approximation>, the <oscillation period> is $T=2\pi\sqrt{L/g}$, so
$$
g=\frac{4\pi^2L}{T^2}.
$$
Interpret the quoted random timing error and length error as <standard uncertainties>, with <independent> errors and no additional <systematic error>. For $N$ <independent> measurements, the <standard error> of the <sample mean> of the period is $\sigma_{\overline T}/T=0.02/\sqrt N$. The length has relative <standard uncertainty> $\sigma_L/L=0.1/25.0=0.004$. By <propagation of uncertainty>,
$$
\left(\frac{\sigma_g}{g}\right)^2=\left(\frac{\sigma_L}{L}\right)^2+4\left(\frac{\sigma_{\overline T}}{T}\right)^2=(0.004)^2+\frac{(0.04)^2}{N}.
$$
Requiring a relative <standard uncertainty> no greater than $0.01$ gives
$$
N\ge\frac{0.04^2}{0.01^2-0.004^2}=19.0476\ldots,\qquad\boxed{N_{\min}=20\text{ measurements}.}
$$
For $N=20$, the combined relative <standard uncertainty> is $0.009798$, or $0.980\%$; $N=19$ gives $1.001\%$ and just fails. Repeating the timing does not reduce the fixed length <measurement uncertainty>. The stated “accuracy” is consequently a statistical uncertainty target, not a guarantee for every experimental outcome. This is an example of <repeated-timing uncertainty in a pendulum gravity measurement>.
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