A ring and a solid sphere rotating about an axis passing through their centres have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is $$\sqrt{\dfrac{2}{x}}$$. The value of $$x$$ is ______.

A ring and a solid sphere rotate about an axis passing through their centres with the same radii of gyration. The axis is perpendicular to the plane of the ring.

For a ring about an axis perpendicular to its plane through the center: $$I_{\text{ring}} = MR_1^2$$

Radius of gyration: $$k_1 = R_1$$

For a solid sphere about an axis through its center: $$I_{\text{sphere}} = \dfrac{2}{5}MR_2^2$$

Radius of gyration: $$k_2 = R_2\sqrt{\dfrac{2}{5}}$$

$$k_1 = k_2 \Rightarrow R_1 = R_2\sqrt{\dfrac{2}{5}}$$

$$\dfrac{R_1}{R_2} = \sqrt{\dfrac{2}{5}}$$

$$\dfrac{R_1}{R_2} = \sqrt{\dfrac{2}{x}}$$

$$x = 5$$

The value of $$x$$ is $$\boxed{5}$$.