Equation of Motion: Rotation About a Fixed Axis

For a flywheel rotating about a fixed axis, the center of mass acceleration has two components. The tangential component depends on the direction of angular acceleration and propels the flywheel along its circular path. The normal component is always directed along the radius toward point O on the rotation axis. Together, these components fully describe the center of mass motion.

The moment applied to the flywheel's center of mass equals the product of its moment of inertia and angular acceleration. By expressing this moment in terms of the moment about point O, unknown forces acting on the body can be eliminated. This approach simplifies the analysis of rotational dynamics for bodies with non-uniform mass distribution.

The normal component of acceleration passes through point O and is parallel to the radial vector, producing no moment about that point. Since this component creates zero moment, it is excluded from moment calculations. This simplification allows the moment equation to focus only on forces and accelerations that generate rotational effects about the fixed axis.

The parallel axis theorem allows the moment equation to be expressed in terms of the moment of inertia about point O rather than about the center of mass. This transformation provides a more detailed view of the flywheel's rotational motion and simplifies calculations when the axis of rotation is offset from the center of mass.

As a flywheel with non-uniform mass rotates about a fixed axis, its center of mass traces a circular path around point O. The motion of the center of mass is described by both tangential and normal acceleration components. Understanding this motion is essential for analyzing the dynamics of rigid bodies undergoing rotational motion about a fixed axis.

The tangential component of the center of mass acceleration depends directly on the direction of angular acceleration. As angular acceleration changes, the tangential acceleration adjusts accordingly, propelling the flywheel along its circular path. This relationship is fundamental to understanding how rotational motion affects the linear acceleration of points on a rotating body.

By expressing the moment equation in terms of the moment about point O on the rotation axis, unknown reaction forces are eliminated from the analysis. The moment from the normal acceleration component is excluded since it passes through point O. Using the parallel axis theorem further refines these equations, enabling clearer analysis of rotational dynamics.