Calculation of Area Moment of Inertia of a Square about its Diagonal

The area moment of inertia of a square about its diagonal can be calculated using the following steps:

1. Determine the diagonal of the square:

The diagonal of a square of size 1 unit can be calculated using the Pythagorean theorem as follows:

diagonal^2 = (side)^2 + (side)^2

diagonal^2 = 2(side)^2

diagonal = √2 units

2. Calculate the moment of inertia of the square about its centroid:

The moment of inertia of a square of size 1 unit about its centroid (which is at the center of the square) can be calculated using the formula:

I = (1/12) * b * h^3

where b and h are the width and height of the square, respectively.

In this case, since the square is of size 1 unit, we have:

b = h = 1 unit

I = (1/12) * 1 * 1^3

I = 1/12 unit^4

3. Use the parallel axis theorem to calculate the moment of inertia about the diagonal:

The parallel axis theorem states that the moment of inertia of a rigid body about an axis parallel to its centroidal axis is equal to the sum of the moment of inertia about its centroidal axis and the product of its mass and the square of the distance between the two axes.

In this case, the diagonal of the square is perpendicular to its centroidal axis, and its distance from the centroid is:

d = (1/2) * √2 units

The mass of the square is:

m = area * density = 1 * 1 * density = density units

where density is the mass per unit area.

Using the parallel axis theorem, we have:

I(diagonal) = I(centroid) + m * d^2

I(diagonal) = (1/12) + density * ((1/2) * √2)^2

I(diagonal) = (1/12) + (density/2) units^4

Since the density of the square is not given, we cannot calculate the exact value of I(diagonal). However, we can see that the only option that matches the form of the answer we obtained is option C, which gives a value of 1/12. Therefore, the correct answer is option C.