1. Given data

• Mass of each solid sphere: $m = 2\,\text{kg}$
• Radius of each sphere: $R = 10\,\text{cm} = 0.1\,\text{m}$
• Distance between centres: $40\,\text{cm} = 0.4\,\text{m}$

The axis is perpendicular to the rod and passes through the midpoint of the rod.

So, the distance of each sphere’s centre from the axis is $d = \frac{0.4}{2} = 0.2\,\text{m}$

2. Moment of inertia of one solid sphere about its own centre

For a solid sphere, $I_{\text{cm}} = \frac{2}{5}mR^2$

Substitute values: $I_{\text{cm}} = \frac{2}{5}(2)(0.1)^2$ $I_{\text{cm}} = \frac{4}{5}\times 0.01 = 0.008\,\text{kg m}^2$

3. Use parallel axis theorem for one sphere

Moment of inertia of one sphere about the given axis: $I_1 = I_{\text{cm}} + md^2$ $I_1 = 0.008 + 2(0.2)^2$ $I_1 = 0.008 + 2(0.04)$ $I_1 = 0.008 + 0.08 = 0.088\,\text{kg m}^2$

4. For two identical spheres

$I_{\text{total}} = 2 \times 0.088 = 0.176\,\text{kg m}^2$

5. Write in the required form

We need $I = \underline{\hspace{1cm}} \times 10^{-3}\,\text{kg m}^2$

Since $0.176 = 176 \times 10^{-3}$

So the required integer is $\boxed{176}$

6. Comparison with stored answer

Stored correct answer = $176$

Our derived answer also = $176$

So, the answer agrees with the stored correct answer.