\(v=\sqrt{\frac{N}{\mu}}\)

The tension \(N\) in the string varies as :

\(N=\pi \frac{M g}{L} \times x\) where \(x\) is length from the ground.

\(d t=\frac{d x}{v_x} \text { and } v_x=\sqrt{\frac{M g x}{L \times M / L}}=\sqrt{g x}\)

\(\int \limits_0^T d t=\int \limits_0^L \frac{d x}{\sqrt{g x}}\)

\(T=\int \limits_0^L 2 \sqrt{x} d x\)

\(T=\int \limits_0^L 2 \sqrt{L_g} \quad \dots (i)\)

If time to cover half length is \(T_2\).

\(T_2=\sqrt{2 L g}\) [By putting limits \(0\) to \(L / 2\) in equation \((i)\)]

\(\frac{T}{\sqrt{2}}=T_2\)