Question 46 of 280

Match List-I with List-II List-I List-II (A) Moment of inertia of solid sphere
of radius $R$ about any tangent. (I) $\dfrac{5}{3}{MR}^{2}$ (B) Moment of inertia of hollow sphere of radius
$\left(R\right)$ about any tangent. (II) $\dfrac{7}{5}{MR}^{2}$ (C) Moment of inertia of circular ring of radius
$\left(R\right)$ about its diameter. (III) $\dfrac{1}{4}{MR}^{2}$ (D) Moment of inertia of circular disc of radius
$\left(R\right)$ about any diameter. (IV) $\dfrac{1}{2}{MR}^{2}$

A$A-II,B-I,C-IV,D-III$B$A-I,B-II,C-IV,D-III$C$A-II,B-I,C-III,D-IV$D$A-I,B-II,C-III,D-IV$

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Answer: (A) $A-II,B-I,C-IV,D-III$

For solid sphere

${I}_{0}={I}_{com}+M{R}^{2}$ (Using parallel axis theorem)
${I}_{0}=\dfrac{2}{5}M{R}^{2}+M{R}^{2}$
${I}_{0}=\dfrac{7}{5}M{R}^{2}$$\Rightarrow A\rightarrow II$
For hollow sphere

$I={I}_{com}+M{R}^{2}\Rightarrow I=\dfrac{5}{3}M{R}^{2}\Rightarrow B\rightarrow I$
For Circular ring

${I}_{1}+{I}_{2}={I}_{3}$ ( Using perpendicular axis theorem)
By symmetry MOI
About $1$ and $2$ Axis are same i.e.
${I}_{1}={I}_{2}$
$∴2{I}_{1}={I}_{3}=M{R}^{2}\left(∵{I}_{com}=M{R}^{2}\right)$
${I}_{1}=\dfrac{M{R}^{2}}{2}\Rightarrow C\rightarrow IV$
Similarly in disc
$2{I}_{1}=\dfrac{M{R}^{2}}{2}\left\{∵{I}_{com}=\dfrac{M{R}^{2}}{2}\right\}$
${I}_{1}=\dfrac{M{R}^{2}}{4}\Rightarrow D\rightarrow III$