The radius of gyration of a body about an axis at $6 \mathrm{~cm}$ from its centre of mass is $10 \mathrm{~cm}$. Find its radius of gyration about a parallel axis through its centre of mass.
Q 96.6
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Let $O$ be a point at $6 \mathrm{~cm}$ from the centre of mass of the body.
Let $\mathrm{I}=\mathrm{Ml}$ about an axis through $\mathrm{O}$,
$\mathrm{k}=$ radius of gyration about the axis through $\mathrm{O}$,
$\mathrm{I}_{\mathrm{CM}}=\mathrm{Ml}$ about a parallel axis through the centre of mass of the body,
$\mathrm{k}_{\mathrm{CM}}=$ radius of gyration about a parallel axis through the centre of mass,
$M=$ mass of the body,
$\mathrm{h}=$ distance between the two axes.
Data : $\mathrm{h}=6 \mathrm{~cm}, \mathrm{k}=10 \mathrm{~cm}$
By the theorem of parallel axis,
$
\mathrm{I}=\mathrm{I}_{\mathrm{CM}}+\mathrm{Mh}^2
$
Also, $\mathrm{I}=\mathrm{Mk}^2$ and $\mathrm{I}_{\mathrm{CM}}=M k_{\mathrm{CM}}^2$
$
\begin{aligned}
& \therefore M k^2=M k_{\mathrm{CM}}^2+M h^2 \\
& \therefore k^2=k_{\mathrm{CM}}^2+h^2 \\
& \therefore 100=k_{\mathrm{CM}}^2+36 \quad \therefore k_{\mathrm{CM}}=8 \mathrm{~cm}
\end{aligned}
$
The radius of gyration about a parallel axis through its centre of mass is $8 \mathrm{~cm}$.
Let $\mathrm{I}=\mathrm{Ml}$ about an axis through $\mathrm{O}$,
$\mathrm{k}=$ radius of gyration about the axis through $\mathrm{O}$,
$\mathrm{I}_{\mathrm{CM}}=\mathrm{Ml}$ about a parallel axis through the centre of mass of the body,
$\mathrm{k}_{\mathrm{CM}}=$ radius of gyration about a parallel axis through the centre of mass,
$M=$ mass of the body,
$\mathrm{h}=$ distance between the two axes.
Data : $\mathrm{h}=6 \mathrm{~cm}, \mathrm{k}=10 \mathrm{~cm}$
By the theorem of parallel axis,
$
\mathrm{I}=\mathrm{I}_{\mathrm{CM}}+\mathrm{Mh}^2
$
Also, $\mathrm{I}=\mathrm{Mk}^2$ and $\mathrm{I}_{\mathrm{CM}}=M k_{\mathrm{CM}}^2$
$
\begin{aligned}
& \therefore M k^2=M k_{\mathrm{CM}}^2+M h^2 \\
& \therefore k^2=k_{\mathrm{CM}}^2+h^2 \\
& \therefore 100=k_{\mathrm{CM}}^2+36 \quad \therefore k_{\mathrm{CM}}=8 \mathrm{~cm}
\end{aligned}
$
The radius of gyration about a parallel axis through its centre of mass is $8 \mathrm{~cm}$.
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