Consider four objects, each having the same mass and the same radius:

1. a solid sphere

2. a hollow sphere

3. a flat disk in the x, y plane

4. a hoop in the x, y plane

The order of increasing rotational inertia about an axis through the center of mass and parallel to the z axis is:

1. Solid Sphere: 0.4 kg m²

2. Flat Disk in x, y plane: 0.5 kg m²

3. A Hollow Sphere: 0.67 kg m²

4. Hoop in x, y plane: 1 kg m²

Explanation:

The formulae for the rotation inertia of given shapes about z-axis are given as follows:

1. Solid Sphere: (2/5) (MR²)

2. A Hollow Sphere: (2/3) (MR²)

3. Flat Disk in x, y plane = (1/2) (MR²)

4. Hoop in the x, y plane = MR²

where,

M = mass

R = Radius

Since, it is stated that all shapes have same mass and radius. Therefore, we assume a mass of 1 kg and radius of 1 m for each shape. Hence values become:

1. Solid Sphere: (2/5) (1 kg) (1 m) ²

2. A Hollow Sphere: (2/3) (1 kg) (1 m) ²

3. Flat Disk in x, y plane = (1/2) (1 kg) (1 m) ²

4. Hoop in the x, y plane = (1 kg) (1 m) ²

Solving the values, we get:

1. Solid Sphere: 0.4 kg m²

2. A Hollow Sphere: 0.67 kg m²

3. Flat Disk in x, y plane = 0.5 kg m²

4. Hoop in the x, y plane = 1 kg m²

Therefore, it is clear from above calculations, that the order of increasing of rotational inertia of these shapes will be as follows:

1. Solid Sphere: 0.4 kg m²

2. Flat Disk in x, y plane: 0.5 kg m²

3. A Hollow Sphere: 0.67 kg m²

4. Hoop in x, y plane: 1 kg m²