70 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



We shall see later, that whereas positional forces are usually 

 conservative, and motional forces not, there are certain conservative 

 motional forces. 



Kinetic energy being defined as 2 mv 2 is of the dimensions 



-7?r~ ' the same as those of work. Potential Energy is defined as 



work. The C. Gr. S. unit of energy is, therefore, the erg. 



We have in this chapter been concerned with the line integral 

 of the force exerted on a moving point resolved in the direction of 

 the motion of the point of application. This has been called the 

 work of the force, and is physically a quantity of fundamental 

 importance. We have occasionally to consider the time -integral of 

 a force, that is, if F be a force always in the same direction, the 

 quantity 



which has received the name of the impulse of the force during the 

 time from to ^. The effect of a force may be measured either 

 by the work or by the impulse, but it is to be observed that the 

 information obtained when one or the other of these two quantities 

 is given is of a quite different nature. Supposing the force is 

 constant in magnitude and direction, the work done is equal to the 

 force times the distance moved, and a knowledge of the work tells 

 us how far the point of application will be moved by the given force, 

 while the impulse is equal to the force times the interval of time, 

 and tells us how long the point will move under the application of 

 the given force. If the force is variable, considering the significa- 

 tion of a definite integral as a mean 1 ), we may say that the work 

 is the mean with respect to distance of the force multiplied by the 

 length of the path, while the impulse is the mean with respect to 

 the time multiplied by the duration of the motion. Thus the work 

 answers the question "how far", while the impulse answers the 

 question a how long". The work is a scalar quantity, its element 

 being the geometric product jf the force and the displacement. For 

 the element of impulse, however, we have, using equation 7), 3, 



Fdt = Xdt cos (Fx) + Ydt cos (Fy) -f Zdt cos (F0) 



thus the element is the component in the direction of the force of 

 the vector whose components are 



______ d! x =Xdt, dI y 



1) See footnote, 34, p. 98. 



