Question:
$\text{Match List-I with List-II:}$ $\begin{array}{|c|c|} \hline \text{List-I} & \text{List-II} \\ \hline \text{(A) Moment of inertia of a solid sphere of radius } R \text{ about any tangent} & \text{(I) } \frac{5}{3} MR^2 \\ \text{(B) Moment of inertia of a hollow sphere of radius } R \text{ about any tangent} & \text{(II) } \frac{7}{5} MR^2 \\ \text{(C) Moment of inertia of a circular ring of radius } R \text{ about its diameter} & \text{(III) } \frac{1}{4} MR^2 \\ \text{(D) Moment of inertia of a circular disk of radius } R \text{ about any diameter} & \text{(IV) } \frac{1}{2} MR^2 \\ \hline \end{array}$ $\text{Codes:}$ $\text{1. A-II, B-I, C-IV, D-III}$ $\text{2. A-I, B-II, C-IV, D-III}$ $\text{3. A-II, B-I, C-III, D-IV}$ $\text{4. A-I, B-II, C-III, D-IV}$
Solution:
$\text{Hint: Use the concept of the parallel axis theorem \& perpendicular axis theorem.}$ $\text{Step 1: Find the moment of inertia of a solid sphere about any tangent.}$ $I_0 = I_{\text{com}} + MR^2$ $\Rightarrow I_0 = \frac{2}{5} MR^2 + MR^2$ $\Rightarrow I_0 = \frac{7}{5} MR^2$ $\text{Hence, A} \rightarrow \text{II}$ $\text{Step 2: Find the moment of inertia of a hollow sphere about any tangent.}$ $I_0 = I_{\text{com}} + MR^2$ $\Rightarrow I_0 = \frac{2}{3} MR^2 + MR^2$ $\Rightarrow I_0 = \frac{5}{3} MR^2$ $\text{Hence, B} \rightarrow \text{I}$ $\text{Step 3: Find the moment of inertia of a circular ring about its diameter.}$ $I_1 + I_2 = I_3 \quad \cdots (1)$ $\text{By symmetry MOI about 1 and 2 axis are the same i.e.:}$ $I_1 = I_2 \quad \cdots (2)$ $\text{By using equations (1\&2):}$ $2I_1 = MR^2 \quad \left[ \because I_3 = I_{\text{com}} = MR^2 \right]$ $\Rightarrow I_1 = \frac{MR^2}{2}$ $\text{Hence, C} \rightarrow \text{IV}$ $\text{Step 4: Find the moment of inertia of a circular disk about any diameter.}$ $2I_1 = \frac{MR^2}{2} \quad \left[ \because I_{\text{com}} = \frac{MR^2}{2} \right]$ $\Rightarrow I_1 = \frac{MR^2}{4}$ $\text{Hence, D} \rightarrow \text{III}$ $\text{Therefore, the correct match is}$ $\text{A - II, B - I, C - IV, D - III.}$ $\text{Hence, option (1) is the correct answer.}$