Imagine a flywheel having a non-uniform mass, rotating around a fixed axis. As the flywheel spins, its center of mass moves in a circular path.
The acceleration of the center of mass is described by its tangential and normal components.
The tangential component of acceleration depends on the direction of the angular acceleration of the flywheel. In contrast, the normal component of acceleration is always along the radius and towards the point O.
The moment exerted on the flywheel's center of mass is determined by the product of its moment of inertia of the center of mass and its angular acceleration.
The moment equation can be written in terms of the moment about point O to eliminate any unknown force acting on the body.
Here, the moment due to the normal component of the acceleration is not considered as it passes through point O.
Using the parallel axis theorem, the moment equation can be expressed in terms of the moment of inertia about point O.
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.