Explanation:

The moment of inertia of an object is a measure of its resistance to rotational motion about a particular axis. In this case, we have four rods forming a square of side 2a, and we want to find the moment of inertia of this system about an axis passing through its center of mass and perpendicular to the plane of the body.

Step 1: Identifying the axis of rotation

In order to calculate the moment of inertia, we need to identify the axis of rotation. In this case, the axis of rotation is passing through the center of mass of the square and is perpendicular to the plane of the square.

Step 2: Dividing the square into four equal parts

To simplify the calculation, we can divide the square into four equal parts by drawing two perpendicular lines passing through its center. This will create four identical rods, each with a length of a and a mass of m.

Step 3: Calculating the moment of inertia of a single rod

The moment of inertia of a single rod about an axis passing through its center and perpendicular to its length can be calculated using the formula:

I = (1/12) * m * L^2

where I is the moment of inertia, m is the mass of the rod, and L is the length of the rod.

Step 4: Applying the parallel axis theorem

Since the rods are not rotating about their centers, but about an axis passing through the center of mass of the square, we need to apply the parallel axis theorem. The parallel axis theorem states that the moment of inertia of a body about an axis parallel to and a distance d from an axis through its center of mass is given by:

I = I_cm + m * d^2

where I is the moment of inertia about the parallel axis, I_cm is the moment of inertia about the axis through the center of mass, m is the mass of the body, and d is the distance between the two axes.

Step 5: Calculating the moment of inertia of the square

Using the parallel axis theorem, we can calculate the moment of inertia of each rod about the axis passing through the center of mass of the square. Since all four rods are identical, their moment of inertia about this axis will be the same.

Finally, we can calculate the moment of inertia of the entire square by summing up the moment of inertia of each rod about the axis passing through the center of mass.

Conclusion:

In conclusion, the moment of inertia of four rods forming a square of side 2a about an axis passing through its center of mass and perpendicular to the plane of the body can be calculated by dividing the square into four equal parts, calculating the moment of inertia of each rod about its own center, and then applying the parallel axis theorem to find the moment of inertia of the entire square.

16masqure /3 means 16*m*a square (ak square )/3