Question

The moment of inertia of a hollow cubical box of mass M and side a about an axis
passing through the centres of two opposite faces is equal to?
A $\quad \dfrac{5 M a^{2}}{3}$
$\mathrm{B} \quad \dfrac{5 M a^{2}}{6}$
C $\quad \dfrac{5 M a^{2}}{12}$
$\mathrm{D} \quad \dfrac{5 M a^{2}}{18}$

Answer 1

Hint We know that the moment of inertia is a physical quantity which describes how easily a body can be rotated about a given axis. It is a rotational analogue of mass, which describes an object's resistance to translational motion. Inertia is the property of matter which resists change in its state of motion. Rotational inertia is important in almost all physics problems that involve mass in rotational motion. It is used to calculate angular momentum and allows us to explain (via conservation of angular momentum) how rotational motion changes when the distribution of mass changes. The moment of inertia of a rigid body depends only on the mass of the body, its shape and size.

Complete step by step answer
It is known that hollow Cubical box 6 identical Square plates of mass 'm' joined to form a cube of side 'a'
$\mathrm{M}=6 \mathrm{m}$
Now the moment of Inertia of the square plate about COM whose axis is perpendicular to the plane of the square plate.:
$\dfrac{\mathrm{Ma}^{2}}{6}$
We have,
A cube whose axis passes through a centre of mass of two square plates, facing opposite to each other.
Therefore, $\mathrm{I}_{1}=2 \times \dfrac{\mathrm{Ma}^{2}}{6}=2 \times \dfrac{\mathrm{M}}{6} \times \dfrac{\mathrm{a}}{6^{2}}=\dfrac{\mathrm{Ma}^{2}}{18}$
Moment of Inertia of square plate about Com, having axis parallel to plane of square plate. Using Parallel axis theorem, $\mathrm{I}_{2}=\dfrac{\mathrm{Ma}^{2}}{12}$
So, the remaining four square plates. distance from square plate to centre of cube, d $=\dfrac{\mathrm{a}}{2}$
$\mathrm{I}_{\mathrm{rem} 1}=\dfrac{2 \mathrm{Ma}^{2}}{9}$
Finally, we have $\mathrm{I}_{\mathrm{req}}=\mathrm{I}_{\mathrm{rem} 1}+\mathrm{I}_{1}=\dfrac{5 \mathrm{Ma}^{2}}{18}$

Hence, we can say that the correct option is option D.

Note We can conclude that the moment of inertia of a body about an axis parallel to the body passing through its center is equal to the sum of moment of inertia of body about the axis passing through the center and product of mass of the body times the square of distance between the two axes. The theorem of parallel axis states that the moment of inertia of a body about an axis parallel to an axis passing through the centre of mass is equal to the sum of the moment of inertia of body about an axis passing through centre of mass and product of mass and square of the distance between the two axes. The parallel axis theorem states that the moment of inertia of an object around a particular axis is equal to the moment of inertia around a parallel axis that goes through the center of mass, plus the mass of the object, multiplied by the distance to that parallel axis, squared.