Equation of Motion: Rotation About a Fixed Axis

Consider a flywheel, having an uneven mass distribution, rotating steadily around a fixed axis. As this rotation occurs, the center of mass of the flywheel traces a circular path. Understanding the acceleration of this center of mass requires observing both its tangential and normal components.

The tangential component is dependent on the direction of the angular acceleration of the flywheel. The tangential component of the acceleration propels the flywheel along its path. On the other hand, the normal component is always directed along the radius towards point O. Point O lies on the axis of rotation along which the flywheel spins.

A crucial aspect of this scenario is the moment applied to the flywheel's center of mass. This is calculated by multiplying the moment of inertia of the center of mass by its angular acceleration. The equation for this moment can be articulated in terms of the moment about point O, which effectively eliminates any unknown forces acting on the body. Interestingly, the moment resulting from the normal component of acceleration is not taken into account in these calculations. The reason for this exclusion is that the normal component of the acceleration passes through point O and is parallel to the radial vector, resulting in no moment.

To further refine this understanding, one could employ the parallel axis theorem. This allows the moment equation to be expressed in terms of the moment of inertia about point O, providing a more detailed view of the flywheel's motion.