The ratio of the frequencies of two simple pendulums is 4 : 3 at the same place. The ratio of their respective lengths is

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The frequency of a simple pendulum is given by $n = \frac{1}{2 π} \frac{g}{l}$ .

Since the pendulums are at the same place, $g$ is constant. Thus, $n \propto \frac{1}{l}$ , which implies $l \propto \frac{1}{n ^{2}}$ .

The ratio of their lengths is $\frac{l _{1}}{l _{2}} = (\frac{n _{2}}{n _{1}})^{2}$ .

Given $\frac{n _{1}}{n _{2}} = \frac{4}{3}$ , we get $\frac{l _{1}}{l _{2}} = (\frac{3}{4})^{2} = \frac{9}{16}$ .

Answer: $9 : 16$

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