The radius of gyration of a uniform disc about a line perpendicular to the disc equals its radius. Find the distance of the line from the centre.

The moment of inertia about the centre and perpendicular to the plane of the disc of radius and mass  $m$ is  $m{r}^2$

According to the question the radius of gyration of the disc about a point = radius of the disc.

Therefore, $m{k}^2=\frac{1}{2}m{r}^2+m{d}^2$

$\begin{array}{l}\Rightarrow K^2=\frac{r^2}{2}+d^2\\ \Rightarrow r^2=\frac{r^2}{2}+d^2(K=r)\\ \Rightarrow \frac{r^2}{2}=d^2\\ \Rightarrow d=\frac{r}{\sqrt{2}}\end{array}$