Parallel and Perpendicular Axis Theorem

Question 1:

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 is

1. \[ \sqrt{\frac{5}{3}}r \]

2. \[ \sqrt{\frac{5}{6}}r \]

3. \[ \sqrt{\frac{7}{2}}r \]

4. \[ \sqrt{\frac{7}{6}}r \]

View Answer

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,

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