The radius of gyration of a solid sphere of mass $M$ and radius $R$ about its diameter is:

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Professor PC Thomas Classes · Answer

The moment of inertia of a solid sphere about its diameter is $I = \frac{2}{5}MR^2$. By definition, the radius of gyration $k$ is given by $I = Mk^2$. Equating the two yields $Mk^2 = \frac{2}{5}MR^2$, which simplifies to $k = \sqrt{\frac{2}{5}}R$.