Ncert Exercises & Solutions

With reference to the figure of a cube of edge $a$ and mass $m,$ the following statements are given. ( $O$ is the centre of the cube).

Choose the correct statements:

(a)	The moment of inertia of the cube about the $z$ -axis is $I_z=I_x+I_y$
(b)	The moment of inertia of the cube about the $z'$ -axis is ${I_z}'=I_z+\frac{{ma}^2}{2}$
(c)	The moment of inertia of the cube about $z''$ -axis is = $=I_z+\frac{{ma}^2}{2}$
(d)	$I_x=I_y$

Choose the correct option from the given options:

1.	(a, c)	2.	(a, d)
3.	(b, d)	4.	(b, c)

Hint: Apply the parallel axis theorem

${I}={I}_{{cm}}+{mr}^2$

Explanation: Let us analyse each statement one by one.

In case (a);
The theorem of perpendicular axes applies only to laminar (plane) objects, and in the given case, it is applied along a 3D cube. Thus, this is false.
In case (b);
$z'$ is a parallel axis at a distance of $\frac{a}{\sqrt 2}$ from $z.$ By the parallel axis theorem,
$\Rightarrow I_{z'}=I_z+m\left(\frac{a}{\sqrt{2}}\right)^2=I_z+\frac{m a^2}{2}$
Thus, the given statement is true.
In case (c);
$z"$ is not parallel to $z.$ Thus, the theorem of the parallel axis cannot be applied. Thus, this is false.
In case (d);
As $x$ and $y$ -axes are symmetrical, the inertia about them will be equal, $I_x=I_y.$ Thus, this statement is true.
Therefore, statements (b) and (d) are true for the given conditions.
Hence, option (3) is the correct answer.

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