Final answer:
The rotational inertia of four identical particles, each of mass m, placed at the vertices of a square with side length a and connected by massless rods, about an axis that passes through midpoints of opposite sides in the plane of the square, is I = (1/2)ma².
Step-by-step explanation:
The question asks us to determine the rotational inertia, also known as moment of inertia, of four identical particles of mass m each, located at the vertices of a square with side length a, all connected by massless rods forming the sides of the square. The axis of rotation passes through the midpoints of opposite sides and lies in the plane of the square.
When calculating the moment of inertia for this system, we must remember that each particle's contribution to the total moment of inertia is given by the formula I = mr² where r is the distance from the particle to the axis of rotation. Because we are dealing with a square, the particles on the axis of rotation contribute zero to the moment of inertia as their distance r is zero. However, the two particles equidistant from the axis contribute with the square of their distances from the axis.
The distance of these two particles from the axis is a/2, giving us a total rotational inertia for the system: I = 2m(a/2)², which simplifies to I = (1/2)ma². This is the rotational inertia of our system about the given axis.