Understand the system

A block of mass $m$ is attached between two identical springs, each having spring constant $2k$, and the other ends are fixed.

When the block is displaced by a small distance $x$ to one side:

• the left spring gets compressed by $x$
• the right spring gets stretched by $x$

Both springs exert restoring forces toward the equilibrium position.

Find restoring force due to each spring

For one spring of spring constant $2k$, restoring force magnitude is $F = (2k)x$

Since there are two springs, both contribute in the same restoring direction.

Hence total restoring force is $F_{\text{net}} = -(2k)x -(2k)x = -4kx$

So the effective spring constant is $k_{\text{eff}} = 4k$

Write angular frequency

For SHM, $\omega = \sqrt{\frac{k_{\text{eff}}}{m}} = \sqrt{\frac{4k}{m}} = 2\sqrt{\frac{k}{m}}$

Find time period

$T = \frac{2\pi}{\omega} = \frac{2\pi}{2\sqrt{k/m}} = \pi \sqrt{\frac{m}{k}}$

Match with options

$T = \pi \sqrt{\frac{m}{k}}$

This matches Option B.

Verification with stored answer

Stored correct answer: B

Derived answer: B

Hence, the derived answer agrees with the stored answer.