Motivation for Concept

Real-world systems often involve rotating objects or objects which must be prevented from rotating. In some cases, the rotation is constrained to occur about an obvious axis (e.g. a door will rotate about the line defined by its hinges, a fixed pulley will rotate about its axle, etc.). In some cases, however, there is no obvious fixed axis of rotation. For example, a rolling ball or wheel translates and rotates, and a collision involving a rotating object will often result in a change in the "physical" axis of rotation. In these cases, it is important to understand the rules for choosing an axis that will allow (and ideally, simplify) analysis of the situation.

Choice of Axis in Common Situations

• Pure Rotation about a Fixed Axle: The clearest situation is the case of a system that is executing pure rotation about an axle which is fixed in an inertial frame. A more common possibility is that a significant component of the system is constrained to rotate about a fixed axle. In either case, it is usually advantageous to choose the axis of rotation for the problem to coincide with the fixed axle.

• Rotation and Translation of a Single Rigid Body: When a single rigid body is chosen as the system of interest and the center of mass of the rigid body is accelerating, the safest choice for the axis of rotation is to ensure that it passes through the body's center of mass.

• Zero Net Torque: For cases in which the net external torque is zero, which includes the special case of statics, the choice of axis is completely arbitrary. There are two rules of thumb for selecting an appropriate axis in this case:
  1. If the system is completely static, place the axis in a position that makes the moment arm zero for one or more forces, so that they produce zero torque.
  2. If a collision is taking place which conserves angular momentum, place the axis in a position that yields simple expressions for the angular momentum before and after the collision.