Page:Elementary Text-book of Physics (Anthony, 1897).djvu/44

equations thus obtained, we obtain for this general case the same result as that already obtained for the special case. The forces introduced by constraints need not be considered, since they are always perpendicular to the path, and so do no work.

When several forces act at a point, the work done by them during any small displacement of the point is equal to the work done by their resultant; for the sum of the projections of all the forces on the line of direction of the resultant is equal to the resultant, and the sum of the projections of each of these projections upon the direction of motion or the projection of the resultant upon the direction of motion is equal to the sum of the projections of each force upon the direction of motion. If, then, several forces do work on a particle, the kinetic energy gained by the particle will be equal to $Rs\cos {\alpha }$, where $R$ is the resultant of the forces, and $\alpha$ the angle between its direction and the direction of the displacement $s$. Let us suppose that the forces are so related that $R=0$. Then the work done by one of the forces must be equal and opposite to that done by the others, the particle will move with a constant velocity, and no kinetic energy will be gained. If any of the forces against which work is done are such that they depend only upon the position of the particle in the field, the work that is done against these forces is equal to that which is done by them if the particle traverse the path in the opposite direction. Such forces are called conservative forces. Other forces, which are not functions of the position of the particle only, but depend on its motion or some other property, are called non-conservative forces. When a particle acted on by conservative forces is so displaced that work is done against those forces, it is said to have acquired potential energy. The measure of the potential energy acquired is the work done against the conservative forces.

Energy is frequently defined as the capacity for doing work. The propriety of this definition is obvious in the case of potential energy; for the particle, acted on by conservative forces, and left free, will move under the action of these forces, and they will