Question
State the expression for the Ml of a thin spherical shell (i.e., a thin-walled hollow sphere) about its diameter. Hence obtain the expression for its $\mathrm{Ml}$ about a tangent.
✓
Answer
Consider a uniform, thin-walled hollow sphere radius $\mathrm{R}$ and mass $\mathrm{M}$. An axis along its diameter is an axis of spherical symmetry through its centre of mass. The Ml of the thin spherical shell about its diameter is
$
\mathrm{I}_{\mathrm{CM}}=\frac{2}{3} \mathrm{MR}^2
$
Let $\mathrm{I}$ be its $\mathrm{MI}$ about a tangent parallel to the diameter. Here, $\mathrm{h}=\mathrm{R}=$ distance between the two axes. Then, according to the theorem of parallel axis,
$
\begin{aligned}
& k_{\mathrm{CM}}=\sqrt{\frac{I_{\mathrm{CM}}}{M}}=\sqrt{\frac{2}{3}} R \simeq 0.8165 R \text { and } \\
& \left.k=\sqrt{\frac{I}{M}}=\sqrt{\frac{5}{3}} R \simeq 1.291 R\right]
\end{aligned}
$
$
\mathrm{I}_{\mathrm{CM}}=\frac{2}{3} \mathrm{MR}^2
$
Let $\mathrm{I}$ be its $\mathrm{MI}$ about a tangent parallel to the diameter. Here, $\mathrm{h}=\mathrm{R}=$ distance between the two axes. Then, according to the theorem of parallel axis,
$
\begin{aligned}
& k_{\mathrm{CM}}=\sqrt{\frac{I_{\mathrm{CM}}}{M}}=\sqrt{\frac{2}{3}} R \simeq 0.8165 R \text { and } \\
& \left.k=\sqrt{\frac{I}{M}}=\sqrt{\frac{5}{3}} R \simeq 1.291 R\right]
\end{aligned}
$
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