Identify the effective spring constant

The block is attached to two identical springs, each of spring constant $k = 20\ \text{N/m}.$

When the block is displaced by a distance $x$, one spring gets compressed and the other gets stretched by the same amount.

Each spring exerts a restoring force of magnitude $kx.$

So the total restoring force on the block is $F = kx + kx = 2kx.$

Hence, the effective spring constant is $k_{\text{eff}} = 2k = 2(20)=40\ \text{N/m}.$

Use the time period formula for SHM

For a mass-spring system, $T = 2\pi\sqrt{\frac{m}{k_{\text{eff}}}}.$

Here, $m=2\ \text{kg}, \qquad k_{\text{eff}}=40\ \text{N/m}.$

Therefore, $T = 2\pi\sqrt{\frac{2}{40}} = 2\pi\sqrt{\frac{1}{20}}.$

$T = \frac{2\pi}{\sqrt{20}} = \frac{\pi}{\sqrt{5}}.$

Compare with the given form

Given, $T = \frac{\pi}{\sqrt{x}}.$

Comparing with $T = \frac{\pi}{\sqrt{5}},$ we get $x=5.$

Comparison with stored answer

Stored correct answer: $5$

Our derived answer matches the stored answer.