Question

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason 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 $I_x, I_y, & I_z$, respectively. The respective radii of gyration about all the three axes will be the same.
Reason 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 :

• A. Both A and R are correct and R is the correct explanation of A.
• B. Both A and R are correct but R is NOT the correct explanation of A.
• C. A is correct but R is not correct.
• D. A is not correct but R is correct.

Hint:

Remember that radius of gyration is not a constant for a body; it varies based on how the mass is distributed relative to the axis of rotation. Only for a thin spherical shell is $k$ constant about any diameter.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The radius of gyration (\(k\)) of a rigid body about a given axis is the distance from the axis at which the entire mass of the body could be concentrated without changing its moment of inertia about that axis.
The moment of inertia (\(I\)) of a disc depends on the choice of the axis of rotation.
Step 2: Key Formula or Approach:
1. Moment of inertia relation to radius of gyration: \(I = Mk^2 \implies k = \sqrt{\frac{I}{M}}\).
2. Perpendicular axis theorem: \(I_z = I_x + I_y\).
3. For a disc: \(I_x = I_y = \frac{MR^2}{4}\) and \(I_z = \frac{MR^2}{2}\).
Step 3: Detailed Explanation:
For the X and Y axes (diameters in the plane of the disc):
\[ k_x = \sqrt{\frac{I_x}{M}} = \sqrt{\frac{MR^2/4}{M}} = \frac{R}{2} \]
\[ k_y = \sqrt{\frac{I_y}{M}} = \sqrt{\frac{MR^2/4}{M}} = \frac{R}{2} \]
For the Z-axis (perpendicular to the plane and through the center):
\[ k_z = \sqrt{\frac{I_z}{M}} = \sqrt{\frac{MR^2/2}{M}} = \frac{R}{\sqrt{2}} \]
Since \(\frac{R}{2} \neq \frac{R}{\sqrt{2}}\), the radii of gyration are not the same about all three axes. Thus, Assertion A is false.
Reason R defines a rigid body as one having a fixed mass and shape, which is a true statement in classical mechanics.
Step 4: Final Answer:
Assertion A is false because radius of gyration depends on the axis, while Reason R is a true general statement about rigid bodies.