Figure shows spring + block + pulley system which are light. The...

Let when $m$ is hanging, the extension in the spring is $l$, then tension in the string is, $T_1=k l=m g$ when force is applied, let the further extension is x . The tension in the spring is $T_2=k(x+l)$$\begin{aligned} & \therefore \text{ Driving force }=T_2-T_1=k(x+l)-k l=k x \\ & \therefore \text{ Acceleration }=\frac{k l}{m}\end{aligned}$$\begin{aligned} \Rightarrow \quad T & =2 \pi \sqrt{\frac{\text{ displacement }}{\text{ acceleration }}} \\ & =2 \pi \sqrt{\frac{x}{\frac{k x}{m}}}=2 \pi \sqrt{\frac{m}{k}}\end{aligned}$$$