$v=\sqrt{\frac{N}{\mu}}$

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 $(i)$]

$\frac{T}{\sqrt{2}}=T_2$