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Match List-I with List-II
List - I List - II
A. Moment of inertia of solid sphere of radius R about any tangent. I. $\frac{5}{3} {MR}^{2}$
B. Moment of inertia of hollow sphere of radius R about any tangent. II. $\frac{7}{5} {MR}^{2}$
C. Moment of inertia of circular ring of radius R about its diameter. III. $\frac{1}{4} {MR}^{2}$
D. Moment of inertia of a circular disc of radius R about any diameter IV. $\frac{1}{2} {MR}^{2}$
Choose the correct answer from the options given below:

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

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

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

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

answer is A.

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Detailed Solution

Solid sphere

$\begin{array}{l} I_{0} = I_{com} + {MR}^{2} \\ I_{0} = \frac{2}{5} {MR}^{2} + {MR}^{2} \\ I_{0} = \frac{7}{5} {MR}^{2} \end{array}$
Hollow sphere

$\begin{array}{l} I_{0} = I_{com} + {MR}^{2} \\ = \frac{2}{3} {MR}^{2} + {MR}^{2} = \frac{5}{3} {MR}^{2} \end{array}$

$I_{1} + I_{2} + I_{3}$ (Perpendicular axis theorem)
By symmetry

MOI about 1” and 2” Axis are same i.e.
$\begin{array}{l} I_{1} = I_{2} \\ ∴ 2 I_{1} = I_{3} = {MR}^{2} (I_{com} = {MR}^{2}) \\ I_{1} = \frac{{MR}^{2}}{2} \end{array}$
Similarly in disc
$\begin{array}{l} 2 I_{1} = \frac{{MR}^{2}}{2} \{I_{com} = \frac{{MR}^{2}}{2}\} \\ I_{1} = \frac{{MR}^{2}}{4} \end{array}$

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