Work done by a variable force: general definition

The work W done by a constant force on a system is defined by

W = = Fd cos θ = F_x d_x + F_y d_y(1)

The vector is the displacement of the point at which the force is acting. For details, please see Page 2.

If the force varies while the displacement occurs, either in magnitude or direction, Eq.(1) does not apply. Eq.(1) requires a definite force vector, but there is no one force vector that one could use during the entire time interval while the displacement is occurring if the force is varying.

However, although a force may be varying during a finite time interval, during an infinitesimal time interval it cannot be anything but constant. There is not enough time for the force to change during such an interval. Therefore, Eq.(1) does apply to an infinitesimal time interval during which the point at which the force is acting undergoes an infinitesimal displacement d. The corresponding infinitesimal work dW done by the force on the system is defined by

dW = d = F|d| cos θ = F_xdx + F_ydy.(2)

In three dimensions, a third term F_zdz needs to be added on the right-hand side of Eq.(2). In one dimension, if the motion is along an x-axis, the term F_xdx is sufficient.

The net work W done on the system during a finite time interval is the sum of all the infinitesimal amounts of work dW done during the infinitesimal intervals that make up the finite interval. The path taken by the point at which the force is acting will in general be curved. The infinitesimal amounts of work dW need to be calculated for the infinitesimal displacements that make up this curved path.

Technically, the sum W of infinitely many infinitesimal amounts dW is what is called an "integral". Calculus provides rules for working out such integrals.