The string, the spring and the pulley shown in figure (12-E9) are light. Find the time period of the mass m.

When only 'm' is hanging let the extention in the spring be 'l'.

So, $T_1=kl=mg$

When a force F is applied, let the further extension be 'x'.

$\therefore T_2=k(x+l)$

$\therefore$ Driving force = $T_2-T_1=k(x+l)-kl$

$=Kx$

$\therefore$ Acceleration = $\frac{Kl}{m}$

$T=2\pi \sqrt{\frac{\text{displacement}}{\text{Acceleration}}}$

$=2\pi \sqrt{\frac{x}{kx/m}}=2\pi \sqrt{\frac{m}{k}}$