The moment of inertia is a fundamental concept in mechanical engineering that plays a significant role in designing rotationally symmetric objects such as flywheels, gears, and other mechanical systems. In this context, we will discuss the moment of inertia of a flywheel rotating about its centroidal axis and how it relates to the moment of inertia about an axis parallel to it.

For a flywheel approximated as a solid disc, consider an infinitesimal differential element with an arbitrary distance from the centroidal axis. The moment of inertia for this differential element is calculated by multiplying the area of the differential element with the square of the distance from its centroidal axis. This equation is integrated over the entire disc area to determine the disc's moment of inertia.

Here, the first integral term represents the moment of inertia around the object's centroid, while the third integral term equals the total area. The second integral term is zero, as this axis passes through the centroid. The moment of inertia about an axis parallel to the centroidal axis equals the moment of inertia around the centroid plus the product of the area and distance squared. This is called the parallel-axis theorem for an area.

This theorem is an essential tool in mechanical engineering applications that involve the moment of inertia of an area. For example, when designing a flywheel or other rotating components, engineers must consider the moment of inertia around its centroidal axis and other axes parallel to the centroidal axis. This helps ensure that the flywheel's design is optimized for its intended application.

In addition, summing the moments of inertia along the planar axes gives the polar moment of inertia about an arbitrary point. This parameter is vital in mechanical engineering applications involving torques in different directions. By considering both the moment of inertia about the centroidal axis and the moment of inertia about the axes parallel to it, mechanical engineers can design more efficient and effective systems.