Match list I with list II and select the correct answer $$ \begin{aligned} &\begin{array}{|l|l|} \hline \text { List-I } & \begin{array}{l} \text { List - II } \\ \text { System } \end{array} & \text { Moment of inertia } \\ \hline \text { (x) A ring about it axis } & \text { (1) }\left(\mathrm{MR}^{2} / 2\right) \\ \hline \text { (y) A uniform circular disc about it axis } & \text { (2) }(2 / 5) \mathrm{MR}^{2} \\ \hline \text { (z) A solid sphere about any diameter } & \text { (3) }(7 / 5) \mathrm{MR}^{2} \\ \hline \text { (w) A solid sphere about any tangent } & \text { (4) } \mathrm{MR}^{2} \\ \cline { 2 } & \text { (5) }(9 / 5) \mathrm{MR}^{2} \\ \hline \end{array}\\\ &\text { Select correct option }\\\ &\begin{array}{|l|l|l|l|l|} \hline \text { Option? } & \mathrm{X} & \mathrm{Y} & \mathrm{Z} & \mathrm{W} \\\ \hline\\{\mathrm{A}\\} & 2 & 1 & 3 & 4 \\ \hline\\{\mathrm{B}\\} & 4 & 3 & 2 & 5 \\ \hline\\{\mathrm{C}\\} & 1 & 5 & 4 & 3 \\ \hline\\{\mathrm{D}\\} & 4 & 1 & 2 & 3 \\ \hline \end{array} \end{aligned} $$

Step by step solution

01

Identifying the moments of inertia for each object in List-I

A Ring about its axis: For a ring with mass (M) and radius (R), the moment of inertia about its axis is given by the formula \(I = MR^2\). In this case, we are looking for the List-II option that matches the formula \(I = MR^2\). A uniform circular disc about its axis: For a uniform circular disc with mass (M) and radius (R), the moment of inertia about its axis is given by the formula \(I = \frac{1}{2} MR^2\). In this case, we are looking for the List-II option that matches the formula \(I = \frac{1}{2} MR^2\). A solid sphere about any diameter: For a solid sphere with mass (M) and radius (R), the moment of inertia about any diameter is given by the formula \(I = \frac{2}{5} MR^2\). In this case, we are looking for the List-II option that matches the formula \(I = \frac{2}{5} MR^2\). A solid sphere about any tangent: For a solid sphere with mass (M) and radius (R), the moment of inertia about any tangent is given by the formula \(I = \frac{7}{5} MR^2\). In this case, we are looking for the List-II option that matches the formula \(I = \frac{7}{5} MR^2\).

02

Matching the formulas for the moments of inertia with the List-II options

A Ring about its axis: We found that the formula is \(I = MR^2\), which matches with option (4). A uniform circular disc about its axis: We found that the formula is \(I = \frac{1}{2} MR^2\), which matches with option (1). A solid sphere about any diameter: We found that the formula is \(I = \frac{2}{5} MR^2\), which matches with option (2). A solid sphere about any tangent: We found that the formula is \(I = \frac{7}{5} MR^2\), which matches with option (3).

03

Identifying the correct option

Using the matched formulas, we can assign the List-II options to the List-I objects: X: A ring about its axis → 4 Y: A uniform circular disc about its axis → 1 Z: A solid sphere about any diameter → 2 W: A solid sphere about any tangent → 3 The correct option is the one that assigns these List-II options to List-I objects: Option B: X: 4 Y: 3 Z: 2 W: 5 Option B does not match our results. Option C: X: 1 Y: 5 Z: 4 W: 3 Option C does not match our results. Option A: X: 2 Y: 1 Z: 3 W: 4 Option A does not match our results. Option D: X: 4 Y: 1 Z: 2 W: 3 Option D matches our results.

04

Conclusion

The correct option for this matching exercise is Option D, where the moments of inertia are: X (A ring about its axis) → 4 (MR^2) Y (A uniform circular disc about its axis) → 1 (\(\frac{1}{2} MR^2\)) Z (A solid sphere about any diameter) → 2 (\(\frac{2}{5} MR^2\)) W (A solid sphere about any tangent) → 3 (\(\frac{7}{5} MR^2\))

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