First slide

Radius of Gyration

Question

The ratio of radii of gyration of a circular ring and a circular disc, of the same mass and radius, about an axis passing through their centres and perpendicular to their planes are

Moderate

A

$1 : \sqrt{2}$
B

$3 : 2$
C

$2 : 1$
D

$\sqrt{2} : 1$

Solution

Let M and R be mass and radius of the ring and the disc respectively. Then,
Moment of inertia of ring about an axis passing through its centre and perpendicular to its plane is
$I_{ring} = M R^{2}$
Moment of inertia of disc about the same axis is
$I_{disc} = \frac{M R^{2}}{2}$
$As I = M k^{2} where k is the radius of gyration$
$∴ I_{ring} = M k_{ring}^{2} = M R^{2} or k_{ring} = R and I_{disc} = M k_{disc}^{2} = \frac{M R^{2}}{2} or k_{disc} = \frac{R}{\sqrt{2}} ∴ \frac{k_{ring}}{k_{disc}} = \frac{R}{R / \sqrt{2}} = \frac{\sqrt{2}}{1} k_{ring} : k_{disc} = \sqrt{2} : 1$

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