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).

(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 cube about z'-axis is $I_{z}' = I_{z} + \frac{{ma}^{2}}{2}$
(c) The moment of inertia of cube about z"-axis is $= I_{z} + \frac{{ma}^{2}}{2}$
(d) $I_{x} = I_{y}$

Choose the correct statements:

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

Hint: Recall parallel and perpendicular axis theorems.

Step 1: Interpret option (a).

(a) Theorem of perpendicular axes is applicable only for laminar (plane) objects. Thus option (a) is false.

Step 2: Apply the parallel axis theorem.

(b) As z' || z and distance between them $= a \frac{\sqrt{2}}{2} = \frac{a}{\sqrt{2}}$
Now, by the theorem of parallel axes

$I_{z^{'}} = I_{z} + m (\frac{a}{\sqrt{2}})^{2} = I_{z} + \frac{m a^{2}}{2}$

Hence. choice (b) is true.

c) z" is not parallel to z hence. the theorem of the parallel axis cannot be applied
Thus, option (c) is false.

d) As x and y-axes are symmetrical.

Hence, $I_{x} = I_{y}$
Thus, option (d) is true.

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