As per given figures, two springs of spring constants $$K$$ and $$2K$$ are connected to mass $$m$$. If the period of oscillation in figure (a) is $$3 \text{ s}$$, then the period of oscillation in figure (b) will be $$\sqrt{x} \text{ s}$$. The value of $$x$$ is ______.

We are given two spring-mass configurations with springs of constants $$K$$ and $$2K$$.

Figure (a): Springs in series

When springs are connected in series, the effective spring constant is:

$$k_a = \frac{K \cdot 2K}{K + 2K} = \frac{2K^2}{3K} = \frac{2K}{3}$$

The time period is:

$$T_a = 2\pi\sqrt{\frac{m}{k_a}} = 2\pi\sqrt{\frac{3m}{2K}} = 3 \text{ s}$$

Figure (b): Springs in parallel

When springs are connected in parallel, the effective spring constant is:

$$k_b = K + 2K = 3K$$

The time period is:

$$T_b = 2\pi\sqrt{\frac{m}{3K}}$$

Finding the ratio:

$$\frac{T_b^2}{T_a^2} = \frac{m/3K}{3m/2K} = \frac{m}{3K} \times \frac{2K}{3m} = \frac{2}{9}$$

$$T_b^2 = \frac{2}{9} \times T_a^2 = \frac{2}{9} \times 9 = 2$$

$$T_b = \sqrt{2} \text{ s}$$

Therefore, $$x = \textbf{2}$$.