Assertion (A): If the moment of inertia of a non-uniform thin circular ring is same about two different axes parallel to each other and lying in the plane of ring, then both the axis can be at same distance from geometrical centre of the ring.nReason (R): From parallel axis theorem \(I = I_{cm} + md^2\), (where terms have usual meaning). Moment of inertia of a body about two axes parallel to each other and at a same distance from centre of mass of the body is same.
Both (A) & (R) are true and the (R) is the correct explanation of the (A)
Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
(A) is true but (R) is false
Both (A) and (R) are false
Solution:
Assertion (A) is true: If two axes have the same moment of inertia, they must be equidistant from the center of mass. It is possible for them to also be equidistant from the geometrical center (e.g., if the CM coincides with the geometrical center). Reason (R) is true: The parallel axis theorem states \(I = I_{cm} + md^2\), so if two parallel axes are at the same distance \(d\) from the CM, their moments of inertia will be equal. Both statements are true, but R does not directly explain A.
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