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Given below are two statements :

Question:

Given below are two statements : one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$. Assertion A : Moment of inertia of a circular disc of mass ' $M$ ' and radius ' $R$ ' about $X, Y$ axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be $I_{x}, I_{y}$ and $I_{z}$ respectively. The respective radii of gyration about all the three axes will be the same.

Reason $\mathbf{R}$ : A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below :

  1. Both $\mathbf{A}$ and $\mathbf{R}$ are correct but $\mathbf{R}$ is NOT the correct explanation of $\mathbf{A}$.

  2. $\mathbf{A}$ is not correct but $\mathbf{R}$ is correct.

  3. $\mathbf{A}$ is correct but $\mathbf{R}$ is not correct.

  4. Both $\mathbf{A}$ and $\mathbf{R}$ are correct and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.


Correct Option: , 2

Solution:

$\mathrm{I}_{\mathrm{z}}=\mathrm{I}_{\mathrm{x}}+\mathrm{I}_{\mathrm{y}}$ (using perpendicular axis theorem)

$\& \mathrm{I}=\mathrm{mk}^{2}(\mathrm{~K}$ : radius of gyration $)$

so $\mathrm{mK}_{\mathrm{z}}^{2}=\mathrm{mK}_{\mathrm{x}}^{2}+\mathrm{mK}_{\mathrm{y}}^{2}$

$\mathrm{K}_{\mathrm{z}}^{2}=\mathrm{K}_{\mathrm{x}}^{2}+\mathrm{K}_{\mathrm{y}}^{2}$

so radius of gyration about axes $\mathrm{x}, \mathrm{y} \& \mathrm{z}$ won't be same hense asseration $A$ is not correct reason $R$ is correct statement (property of a rigid body)

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