To calculate the moment of inertia of the system with respect to one side, we need to consider the individual moments of inertia of each sphere and their distances from the axis of rotation.
The moment of inertia of a solid sphere with mass "m" and radius "r" about an axis passing through its center is given by:
I = (2/5) * m * r^2
Now, let's consider one of the spheres and calculate its moment of inertia about the side of the square. The distance from the center of the sphere to the side of the square is "a/2". Since the moment of inertia of each sphere is the same, we can denote it as "I_s".
The moment of inertia of this sphere about the side of the square is given by the parallel axis theorem:
I_s' = I_s + m * d^2
where "d" is the distance between the axis of rotation (side of the square) and the axis passing through the center of the sphere. In this case, "d" is equal to "a/2".
Substituting the values into the equation:
I_s' = (2/5) * m * r^2 + m * (a/2)^2
Now, we have the moment of inertia of one sphere about the side of the square. Since there are four identical spheres in the system, we multiply this value by four to get the total moment of inertia of the system:
I_total = 4 * I_s'
I_total = 4 * [(2/5) * m * r^2 + m * (a/2)^2]
Simplifying further:
I_total = (8/5) * m * r^2 + 2 * m * a^2
So, the moment of inertia of the whole system with respect to any one of the sides is given by (8/5) times the moment of inertia of an individual sphere plus 2 times the moment of inertia of the square formed by the spheres.