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Question
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.
Solution
Moment of inertia of the disc about the centre and perpendicular to the plane of the disc = \[\frac{1}{2}mr^2\]
Radius of gyration of the disc about a point = Radius of the disc
\[\text{Therefore, }m k^2 = \frac{1}{2}m r^2 + m d^2\]
(k = Radius of gyration about acceleration point; d = Distance of that point from the centre)
\[\Rightarrow K^2 = \frac{r^2}{2} + d^2 \]
\[ \Rightarrow r^2 = \frac{r^2}{2} + d^2 \]
\[ \Rightarrow \frac{r^2}{2} = d^2 \]
\[ \Rightarrow d = \frac{r}{\sqrt{2}}\]
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