At the poles of earth, a stretched wire of a given length vibrates in unison with a tuning fork. At the equator of earth, for same setting, to produce resonance with same fork, the vibrating length of wire

For a string,
Frequency of vibration \(n=\frac{1}{2 l} \sqrt{\frac{\mathrm{~T}}{\mathrm{~m}}}\)
If \(\mathrm{T}=\mathrm{Mg}\), then \(\mathrm{n}=\frac{1}{2 l} \sqrt{\frac{\mathrm{Mg}}{\mathrm{m}}}\)
If \(l_1\) and \(g_1\) are the length and acceleration due to gravity at the poles and \(l_2\) and \(g_2\) at the equator respectively, then if the frequency is the same,
\(\begin{array}{ll}
& \mathrm{n}=\frac{1}{2 l_1} \sqrt{\frac{\mathrm{Mg}_1}{\mathrm{~m}}}=\frac{1}{2 l_2} \sqrt{\frac{\mathrm{Mg}_2}{\mathrm{~m}}} \\
\therefore \quad & \frac{\sqrt{\mathrm{~g}_1}}{l_1}=\frac{\sqrt{\mathrm{g}_2}}{l_2} \text { or } \frac{l_2}{l_1}=\sqrt{\frac{\mathrm{g}_2}{\mathrm{~g}_1}} \\
\therefore \quad & \mathrm{~g}_2 \lt \mathrm{g}_1 \\
\therefore \quad & l_2 \lt l_1
\end{array}\)