Question

The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is:
A. $2:1$
B. $\sqrt 5 :\sqrt 6$
C. $2:3$
D. $1:\sqrt 2$

Answer 1

After reading the question, as we know that the radius of gyration of a disc or a ring is directly proportional to the moment of inertia. So, we will apply the formula of moment of inertia of both circular ring and circular disc separately. And then we will find the ratio of moment of inertia of the circular disc and the circular ring.

Formula-used:
Moment of inertia in the terms of radius of gyration:
$I = MK$
where, $I$ is the moment of inertia, $M$ is the mass of a body, and $K$ is the radius of gyration.

Complete step by step answer:
As we know that, the radius of gyration is directly proportional or related with the moment of inertia, so we need to apply the formula of moment of inertia in the terms of radius of gyration:-
$I = MK$
As per the question, we will find the moment of inertia of a disc and circular ring about a tangential axis in their planes respectively.
So, the moment of inertia for disc:-
${I_{disc}} = \dfrac{5}{4}{M_{disc}}{R^2}$
The moment of inertia for circular ring:-
${I_{ring}} = \dfrac{3}{2}{M_{ring}}{R^2}$
where, $R$ is the radius.

But as we mentioned above-
$I = MK$
$\Rightarrow K = \sqrt {\dfrac{I}{M}}$
So, the ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is:
$\dfrac{{{K_{disc}}}}{{{K_{ring}}}} = \sqrt {\dfrac{{{I_{disc}}}}{{{I_{ring}}}} \times \dfrac{{{M_{ring}}}}{{{M_{disc}}}}}$

Now, we will separate the ratio of moment of inertia of both the disc and ring:-
$\Rightarrow \dfrac{{{I_{disc}}}}{{{I_{ring}}}} = \sqrt {\dfrac{{\dfrac{5}{4}{M_{disc}}{R^2}}}{{\dfrac{3}{2}{M_{ring}}{R^2}}} \times \dfrac{{{M_{ring}}}}{{{M_{disc}}}}} \\\ \Rightarrow \dfrac{{{I_{disc}}}}{{{I_{ring}}}} = \sqrt {\dfrac{5}{6}}$
We can write the upper equation in the ratio form as:
$\therefore {I_{disc}}:{I_{ring}} = \sqrt 5 :\sqrt 6$
So, as we mentioned above that the radius of gyration is directly proportional to the moment of inertia that means the ratio of moment of inertia of both disc and ring is equals to the ratio of their radii i.e.. $\sqrt 5 :\sqrt 6$ .

Hence, the correct option is B.

Note: If the principal moments of the two-dimensional gyration tensor are not equal, the column will tend to buckle around the axis with the smaller principal moment. For example, a column with an elliptical cross-section will tend to buckle in the direction of the smaller semi axis.