## To determine the rotational inertia of the array about the y-axis, you need to consider the distribution of mass and the distances from the axis of rotation for each particle.

In this case, since the particles are arranged in a cross shape, there are two particles along the x-axis and two particles along the y-axis. However, only the two particles along the x-axis contribute to the rotational inertia about the y-axis, as they are farther away from this axis.

To calculate the rotational inertia of each particle, you can use the formula I = mr^2, where m is the mass and r is the perpendicular distance from the axis of rotation. Since all particles are identical and have a mass of 2.0 kg, the rotational inertia of one particle is (2.0 kg)(1.0 m)^2 = 2 kgm^2.

Since there are two particles contributing to the rotational inertia, you can multiply the inertia of one particle by 2. So the total rotational inertia of the array about the y-axis is 2 kgm^2 * 2 = 4 kgm^2.

For the second part of your question, considering the four objects and their rotational inertia about an axis through the center of mass and parallel to the z-axis, you are correct in your reasoning.

The order of increasing rotational inertia is as follows:

1. Solid sphere: A solid sphere has its mass distributed uniformly throughout, and the majority of the mass is concentrated farther away from the axis. Therefore, it has the highest rotational inertia among the given objects.

2. Hollow sphere: A hollow sphere has a greater portion of its mass distributed closer to the axis compared to a solid sphere. Hence, it has a lower rotational inertia than the solid sphere.

3. Flat disk in the xy plane: A flat disk has its mass evenly distributed throughout its surface, resulting in a smaller rotational inertia compared to a solid or hollow sphere.

4. Hoop in the xy plane: A hoop consists of a ring of mass, with all mass points equidistant from the axis of rotation. Hence, it has the lowest rotational inertia among the given objects.

Your reasoning for the hollow sphere and flat disk is correct. The hollow sphere has points of mass closer to the axis near the top, whereas the flat disk has mass distributed relatively closer to the axis compared to the hollow sphere. However, the hoop has equally spaced points of mass from the axis, leading to the lowest rotational inertia among the given objects.