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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.
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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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