Question

One solid sphere \[A\] and another hollow sphere \[B\] are of same mass and same outer radius. Their moments of inertia about their diameters are respectively \[{I_A}\] and \[{I_B}\] such that:
A) ${{\text{I}}_{\text{A}}} = {{\text{I}}_{\text{B}}}$
B) ${{\text{I}}_{\text{A}}} > {{\text{I}}_{\text{B}}}$
C) ${{\text{I}}_{\text{A}}} < {{\text{I}}_{\text{B}}}$
D) ${{\text{I}}_{\text{A}}}/{{\text{I}}_{\text{B}}} = {{\text{d}}_{\text{A}}}/{{\text{d}}_{\text{B}}}$

Answer 1

Hint: Moment of inertia: It is a quantity that expresses the body's tendency to resist angular acceleration, which is calculated as the sum of the products of the mass of each particle in the body and the square of its distance from the axis of rotation.
Moment of inertia of an object is a tensor quantity.
Tensor: It sets a relationship between sets of algebraic objects related to a vector space.

Formula used: Moment of inertia of the solid sphere \[A\] is given by the equation, ${{\text{I}}_{\text{A}}} = \dfrac{{2{{\text{m}}_{\text{A}}}{{\text{R}}_{\text{A}}}^2}}{5}$,
Here \[{m_A}\]is the mass of the solid sphere, \[{R_A}\] is the radius of the solid sphere.
Moment of inertia of the hollow sphere \[B\] is given by the equation, ${{\text{I}}_{\text{B}}} = \dfrac{{2{{\text{m}}_{\text{B}}}{{\text{R}}_{\text{B}}}^2}}{3}$
Here \[\;{m_B}\] is the mass of the hollow sphere, \[{R_B}\] is the radius of the hollow sphere.

Complete step-by-step answer:
According to the given details in the question the mass of the solid sphere and the mass of the hollow sphere is the same and it is also given that the radius of the solid sphere and the radius of the hollow sphere are the same. By using the above formulas of the moment of inertia of solid sphere and moment of inertia of hollow sphere, we get that,
Moment of inertia of the solid sphere A is given by the equation, ${{\text{I}}_{\text{A}}} = \dfrac{{2{{\text{m}}_{\text{A}}}{{\text{R}}_{\text{A}}}^2}}{5}$,${{\text{I}}_{\text{A}}} = \dfrac{{2{\text{m}}{{\text{R}}^2}}}{5}$
Moment of inertia of the hollow sphere is given by the equation, ${{\text{I}}_{\text{B}}} = \dfrac{{2{{\text{m}}_{\text{B}}}{{\text{R}}_{\text{B}}}^2}}{3}$,${{\text{I}}_{\text{B}}} = \dfrac{{2{\text{m}}{{\text{R}}^{}}}}{3}$
Comparing these two equations, we get that, ${{\text{I}}_{\text{A}}} < {{\text{I}}_{\text{B}}}$

Hence the correct option is C.

Note: Moment of inertia of an object depends upon the mass as well as the mass's distribution around its axis.
A body can have different values of moments of inertia about different axes.
Moment of inertia is an inherent property of matter by which it tries to maintain its state of angular motion unless and until it is compelled by external torques.