Determine the time period of the mass m , assuming the string,...

Let the equilibrium extension of the spring be $x_0$.

At equilibrium, the tension in the string balances the weight of the mass $m$.

$T_0 = mg$

The tension is also equal to the spring force.

$T_0 = kx_0 \Rightarrow kx_0 = mg$

When the mass $m$ is displaced downwards by a distance $x$, the string pulls the spring upwards by the same distance $x$.

The new extension of the spring becomes $x_0 + x$.

The new tension in the string is $T = k(x_0 + x)$.

The net restoring force acting on the mass $m$ in the downward direction is:

$F = mg - T$

$F = mg - k(x_0 + x)$

Substituting $mg = kx_0$:

$F = kx_0 - kx_0 - kx = -kx$

The acceleration of the mass is $a = \dfrac{F}{m} = -\dfrac{k}{m}x$.

This is the equation of a simple harmonic motion $a = -\omega^2 x$, where $\omega^2 = \dfrac{k}{m}$.

The time period of oscillation is given by $T = \dfrac{2\pi}{\omega}$.

$T = 2\pi \sqrt{\dfrac{m}{k}}$