Question

The radius of gyration of a body about an axis at a distance of $12 \mathrm{cm}$ from its centre of mass is $13 \mathrm{cm} .$ Then the radius of gyration of the body about a parallel axis through its C.O.M is (in $\mathrm{cm}$ ):
(A) 25
(B) 625
(C) 2.5
(D) 5

Answer 1

Hint
We know the radius of gyration or gyradius of a body about an axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated. The distance from an axis at which the mass of a body may be assumed to be concentrated and at which the moment of inertia will be equal to the moment of inertia of the actual mass about the axis, equal to the square root of the quotient of the moment of inertia and the mass. Based on this concept you have to solve this question.

Complete step by step answer
 From the given question, we can determine that,
$\Rightarrow \mathrm{K}_{1}=13 \mathrm{cm}$
$\Rightarrow \mathrm{I}_{\text {com }}+\mathrm{m}(12)^{2}=\mathrm{m}(13)^{2}$
$\Rightarrow \mathrm{I}_{\mathrm{com}}=25 \mathrm{m}$
Now $\mathrm{K}_{2}$ is radius of gyration about $\mathrm{COM}$. $\mathrm{mK}_{2}^{2}=25 \mathrm{m}$
$\Rightarrow \mathrm{K}_{2}=5 \mathrm{cm}$
Therefore, the correct answer is Option (D).

Note
We know the radius of gyration of a body depends on the axis of rotation and also on distribution of mass of the body on this axis, hence it is not constant. The knowledge of mass and radius of gyration of the body about a given axis of rotation gives the value of its moment of inertia about the same axis, even if we do not know the actual shape of the body. Since, the radius of gyration of a body is defined about its axis of rotation it will change if we change the axis of rotation of the object. It is a scalar quantity. It is termed as a square root of ratio of moment of inertia to the cross-sectional area of material. It is the measure of slenderness of the area of the cross section of the column.