Page:Elementary Text-book of Physics (Anthony, 1897).djvu/43

outward along the normal to the path. This reaction is sometimes called a centrifugal force. There are certain cases in which it may be treated as if it were a real force, determining the motion of a body.

28. Work and Energy.— If the point of application of a force $F$ move through a distance $s$, making the angle $\alpha$ with the direction of the force, the product $FS\cos \alpha$ is defined as the work done by the force during the motion. If the force or the angle between, the direction of the force and the displacement vary during the displacement, the work done may be found by dividing the path of the point into portions so small that $FS\cos \alpha$ may be considered constant for each one of them. By forming the product $FS\cos \alpha$ for each portion of the path, and adding all such products, the work done in the path is obtained.

In the defined sense of the term, no work is done upon a body by a force unless it is accompanied by a change of position, and the amount of work is independent of the time taken to perform it. Both of these statements need to be made, because of our natural tendency to confound work with conscious effort, and to estimate it by the effect on ourselves.

If work be done upon a particle which is perfectly free to move, its velocity will increase. In this case the force $F$ is measured by $ma$, where $m$ is the mass of the particle and $a$ its acceleration. We may suppose that the particle has the velocity $v_{0}$ when it enters upon the distance $s$, and that the distance $s$ coincides with the direction of the force. Using equation (10), we then have

$Fs=mas={\tfrac {1}{2}}mv^{2}-{\tfrac {1}{2}}mv_{0}^{2}.$

(20)

The product ${\tfrac {1}{2}}mv^{2}$ is called the kinetic energy of the particle. The equation shows that the work done upon the particle by a constant force is equal to the kinetic energy which it gains during the motion. If the direction of the motion or the magnitude of the force vary, we may divide the path into small portions, for each of which the force may be considered constant. Forming the equation just proved for each of these portions and adding the