A rod AB of length L is hung from two identical wires 1 and 2. A block of mass m is hung at point O of the rod as shown in fig. The value of x so that a tuning fork excites the fundamental mode in wire 1 and the second harmonic in wire 2 is

The correct option is C. $\frac{L}{5}$

Let $T_1$ and $T_2$ be the tensions in wires 1 and 2 respectively. Let m be the mass per unit length of each wire and l be the length of each wire.

Given

$\frac{l}{2l}\sqrt{\frac{T_1}{m}}=\frac{2}{2l}\sqrt{\frac{T_2}{m}}$

which gives $\frac{T_1}{T_2}=4$

For rotational equilibrium of the rod about O, we have

$T_1\times AO=T_2\times BO$ or $T_1\times x=T_2(L-x)$

$4=\frac{(L-x)}{x}$

which gives $x=\frac{L}{5}$.

Hence the correct choice is (a).