Find the ratio of the radius of gyration of a solid sphere about its diameter to the radius of gyration of a hollow sphere about its tangent, given that both the spheres have the same radius.
Q 91
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The radius of gyration of a body about a given axis, $\mathrm{k}=\sqrt{I / M}$, where $\mathrm{M}$ and $\mathrm{I}$ are respectively the mass of the body and its moment of inertia (MI) about the axis.
For a solid sphere rotating about its diameter,
$
k_{\mathrm{SS}}=\sqrt{\frac{2}{5}} R
$
For a hollow sphere rotating about its diameter,
$
I_{\mathrm{HS}}=\frac{2}{3} M R^2
$
For a hollow sphere rotating about its tangent,
$
I_{\mathrm{HS}}=\frac{2}{3} M R^2+M R^2=\frac{5}{3} M R^2
$
so that, its radius of gyration for rotation about a tangent is
$
k_{\mathrm{HS}}^{\prime}=\sqrt{\frac{5}{3}} R
$
The required ratio, $\frac{k_{\mathrm{SS}}}{k_{\mathrm{HS}}^{\prime}}=\sqrt{\frac{2}{5}} \times \sqrt{\frac{3}{5}}=\frac{\sqrt{6}}{5}$
For a solid sphere rotating about its diameter,
$
k_{\mathrm{SS}}=\sqrt{\frac{2}{5}} R
$
For a hollow sphere rotating about its diameter,
$
I_{\mathrm{HS}}=\frac{2}{3} M R^2
$
For a hollow sphere rotating about its tangent,
$
I_{\mathrm{HS}}=\frac{2}{3} M R^2+M R^2=\frac{5}{3} M R^2
$
so that, its radius of gyration for rotation about a tangent is
$
k_{\mathrm{HS}}^{\prime}=\sqrt{\frac{5}{3}} R
$
The required ratio, $\frac{k_{\mathrm{SS}}}{k_{\mathrm{HS}}^{\prime}}=\sqrt{\frac{2}{5}} \times \sqrt{\frac{3}{5}}=\frac{\sqrt{6}}{5}$
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