Final answer:
A rod or other shape rotated about a vertical axis forms a solid cylinder by rotating 360 degrees, and the concept of rotational inertia is essential for understanding the dynamics of such rotational motion in physics.
Step-by-step explanation:
A rotated about a vertical axis forms a solid cylinder when the plane figure is circular and rotates through 360 degrees around the axis. Objects such as disks can generate a cylinder through this rotational motion. In relation to the question, a rod rotated about an axis through its center perpendicular to its length has a moment of inertia represented as I = MR² where M is the mass of the disk and R is the radius of the disk, which is also the radius of the resulting cylinder.
In mathematics and physics, these characteristics of rotated shapes are often discussed in the context of geometric transformations and rotational inertia. For example, the moment of inertia is a property that measures an object's resistance to changes in its rotation and is crucial in the study of rotational dynamics.
Such principles also help in understanding phenomena such as how rotational speeds change when objects with different moments of inertia interact, as in the case of two cylinders with different rotational inertias coupling together, resulting in a new angular speed and kinetic energy for the system.