A ring and a solid sphere rotating about an axis passing through their centres have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is 2 x . The value of x is _____.

Question

Aicharya · Accepted Answer

The moment of inertia of the ring about an axis perpendicular to the plane of ring is $I_{r} = m k^{2}$ .
The moment of inertia of the sphere is
$I_{s} = \frac{2}{5} m {R^{2}}_{s}$
It is given that
$m k^{2} = m {R^{2}}_{r} = \frac{2}{5} m {R_{s}}^{2} . . . (i)$
From equation (i)
$R_{r} = \sqrt{\frac{2}{5}} R_{s}$
From the question, it can be written that
$\frac{R_{r}}{R_{s}} = \sqrt{\frac{2}{x}}$
$\Rightarrow \frac{R_{r}}{R_{s}} = \sqrt{\frac{2}{5}} \Rightarrow x = 5$

A ring and a solid sphere rotating about an axis passing through their centres have same radii of gyration.…

Solution