Constrained Motion

Abstract

In many cases, the object we are studying is not free to move in every direction. For example, a bead on a wire can only move along the wire. The shape and position of the wire determine the path of the bead, but the bead is still free to move in many different ways along that wire. In this case we call the motion “constrained”. The constraints can be strong, by restricting the motion to be along a given path, or the constraints can be weak, such as for a bead caught between two parallel glass plates, or for a car driving on the terrain. Constraints may also arise because objects are connected to each other: The motion of an individual atom in a large rigid molecule is constrained by the motion of the whole molecule, and a small part of a spinning wheel is constrained to follow the motion of the wheel. We have learned that the motion of an object can be determined from the forces acting on the object. This is, of course, also the case for constrained motion. But in many cases it is not practical to include the forces that restrict the motion, either because we do not have good models for them, or because we would rather like to determine these constraining forces from what we know about the motion, from the fact that the object follows a particular path. For example, a bead moving along a wire will be affected by normal forces from the wire on the bead, and it is these forces that cause the bead to follow the path given by the wire. However, we may not want to model these forces in detail, instead we are interested in the consequences of constraints—we want to find the normal force when we know that the bead follows the wire. In this chapter, we will discuss constrained motion. Starting from the simplest case of motion constrained to a line, through circular motion, to motion constrained to be along a general curve.