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Find the moment of inertia of a thin uni...

Find the moment of inertia of a thin uniform rod about an axis perpendicular to its length and passing through a point which is at a distance of `1/3` from one end. Also find radius of gyration about that axis.

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(i) Let `P` be the point at a distance `l/3` from one end. It is a distance of `(l/2-l/3)=l/6` from the centre as shown in the figure.

By parallel axes theoram `I=I_(C)=Mr^(2)`
`=(MI^(2))/12+M(l/6)^(2)=(Ml^(2))/9`
ii. The radius of gyration, `K=sqrt(I/M)=sqrt((MI^(2))/(9M))=l/3`
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