Question

The moment of inertia of a solid sphere about an axis passing through the centre of gravity is $\dfrac{2}{{5M{R^2}}}$, then its radius of gyration about a parallel axis at a distance $2R$ from first axis is
A. $5R$
B. $\sqrt[2]{{\dfrac{{22}}{{5R}}}}$
C. $\dfrac{5}{{2R}}$
D. $\sqrt[2]{{\dfrac{{12}}{{5R}}}}$

Answer 1

The parallel axis theorem, also known as the Huygens–Steiner theorem or simply as Steiner's theorem, can be used to calculate the moment of inertia or the second moment of area of a rigid body about either axis, provided the body's moment of inertia about a parallel axis.

Complete step by step answer:
According to the parallel axis theorem, the moment of inertia of a body about an axis parallel to the body passing through its centre is equal to the sum of the moment of inertia of the body about the axis passing through the centre and the product of the mass of the body times the square of the distance between the two axes. The statement of the parallel axis theorem is as follows:
${I{\text{ }} = {\text{ }}{I_c}\; + {\text{ }}M{h^2}}$
By parallel axis theorem, the moment of inertia at $2R$ is

\Rightarrow I{\text{ }}= 22 \dfrac{M{R^2}}{5} $$ The radius of gyration is $$M{K^2}= {\text{ }}22M{R^2}/5$$ $$\therefore K{\text{ }} = {\text{ }}\surd \left( {22/5} \right) \times R$$ **Hence, the radius of gyration is $${\text{ }}\surd \left( {22/5} \right) \times R$$.** **Note:** The moment of inertia is determined solely by the body's geometry and the direction of the rotational axis; it is unaffected by the forces involved in the movement. The mass distribution of a body or a system of rotating particles with respect to a rotation axis is reflected by the moment of inertia. The mass of the body, its axis of rotation, and its shape and size all influence the body's moment of inertia.