A block suspended from a vertical spring is in equilibrium,...

Let the mass of the block be $m$ and the spring constant be $k$.

In equilibrium, the spring force balances the weight of the block:

$kx = mg \Rightarrow \dfrac{m}{k} = \dfrac{x}{g}$

The time period of oscillation for the block-spring system is given by:

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

Substituting the value of $\dfrac{m}{k}$, we get:

$T = 2\pi \sqrt{\dfrac{x}{g}}$

The time period of a simple pendulum of length $L$ is given by:

$T = 2\pi \sqrt{\dfrac{L}{g}}$

Since the frequencies (and hence time periods) are equal, we equate the two expressions:

$2\pi \sqrt{\dfrac{x}{g}} = 2\pi \sqrt{\dfrac{L}{g}}$

Squaring both sides, we obtain:

$L = x$