Rankers Physics
Topic: Rotational Motion
Subtopic: Parallel and Perpendicular Axis Theorem

Three rings each of mass m and radius r are so placed that they touch each other. The radius of gyration of the system about the axis as shown in the figure isImage related to
\[ \sqrt{\frac{5}{3}}r \]
\[ \sqrt{\frac{5}{6}}r \]
\[ \sqrt{\frac{7}{2}}r \]
\[ \sqrt{\frac{7}{6}}r \]

Solution:

Moment of Inertia of ring about it's diameter is mR²/2.  ( Using Perpendicular axis theorem)

Moment of Inertia about tangential axis in plane of ring will be mR²/2 + mR²= 3/2 mR²

Total moment of inertia about the axis shown in figure

=  (3/2 mR² )× 2 + 1/2 mR²= 7/2 mR²

Radius of Gyration is k then 3m×k²= 7/2 mR²

so,

\[ k= \sqrt{\frac{7}{6}}r \]

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