Understand the configuration

Four particles of mass $1\,\text{kg}$ are placed at the four corners of a square of side $2\,\text{m}$.

We need the moment of inertia about an axis:

• perpendicular to the plane of the square,
• passing through one vertex.

Use the formula for point masses

For discrete particles, $I = \sum m r^2$ where $r$ is the perpendicular distance of each particle from the axis.

Find distances of all four particles from the axis

Let the axis pass through vertex $A$.

The four vertices are:

• $A$: on the axis, so $r=0$
• $B$: adjacent vertex, so $r=2\,\text{m}$
• $D$: adjacent vertex, so $r=2\,\text{m}$
• $C$: opposite vertex, so diagonal distance $r = \sqrt{2^2+2^2} = 2\sqrt{2}\,\text{m}$

Compute individual contributions

Since each mass is $1\,\text{kg}$:

• At $A$: $I_A = 1\cdot 0^2 = 0$

• At $B$: $I_B = 1\cdot 2^2 = 4$

• At $D$: $I_D = 1\cdot 2^2 = 4$

• At $C$: $I_C = 1\cdot (2\sqrt{2})^2 = 8$

Add them

$I = 0+4+4+8 = 16\,\text{kg m}^2$

Final answer

$\boxed{16}$

Comparison with stored answer

Stored correct answer = $16$

This matches the derived answer.