We can see here that:
(a) Rotational inertia about an axis passing through the midpoints of opposite sides in the plane of the square: 8.00 kgm².
(b) Rotational inertia about an axis passing through the midpoint of one side perpendicular to the plane of the square: 12.00 kgm².
(c) Rotational inertia about an axis lying in the plane of the square and passing through two diagonally opposite particles: 8.00 kgm².
To find the rotational inertia of a rigid body, we need to consider the mass distribution and the axis of rotation. Let's calculate the rotational inertia for each case:
(a) Axis passing through the midpoints of opposite sides in the plane of the square:
In this case, the axis of rotation passes through the center of mass of the system. The rotational inertia of a point particle is given by the formula:
I = mr²,
where I = rotational inertia,
m = mass, and
r = perpendicular distance from the axis of rotation to the particle.
Since the four particles are identical and equidistant from the axis of rotation, we can calculate the rotational inertia of one particle and multiply it by four.
The distance from the center of mass to each particle is half the length of a side of the square, which is 2 m. Therefore, r = 2 m.
The rotational inertia of one particle is I = m×r² = 0.50 kg × (2 m)² = 2.00 kgm².
Since there are four particles, the total rotational inertia is 4 × 2.00 kgm² = 8.00 kgm².
(b) Axis passing through the midpoint of one side perpendicular to the plane of the square:
The rotational inertia about the axis through the center of mass is already calculated in part (a) as 8.00 kgm².
The distance between the two axes is the length of one side of the square, which is 2 m. Therefore, d = 2 m.
The rotational inertia about the axis passing through the midpoint of one side perpendicular to the plane of the square is I = 8.00 kgm² + (0.50 kg * (2 m)²) = 12.00 kgm².
(c) Axis lying in the plane of the square and passing through two diagonally opposite particles:
In this case, we can consider the axis passing through the center of mass and perpendicular to the plane of the square.
The rotational inertia about this axis is the same as the rotational inertia about the axis passing through the midpoints of opposite sides (part a), which is:
8.00 kgm².