232.] THE PRINCIPLE OF VIRTUAL WORK. 143, 



VII. The Principle of Virtual Work. 



231. Work has been defined in Art. 72 as the product of a 

 force into the displacement of its point of application in the 

 direction of the force. 



Thus the expansive force F of the steam in the cylinder of a 

 steam-engine, in pushing the piston through a distance s, is -said 

 to do work, and this work is measured by the product Fs. Simi- 

 larly the force of gravity, i.e. the attractive force of the earth's 

 mass, does work on a falling body. 



The resistance to be overcome by the engine, in the former 

 case, and the resistance of the air in the latter, are also forces 

 acting on the body during its displacement. But as the sense 

 of the displacement is opposite to that of these forces, their 

 work is negative ; work is done against these forces. Thus the 

 muscular force of a man who raises a weight does work against 

 gravity ; if the weight he holds is so heavy as to pull him down, 

 gravity does work against his force ; if he merely tugs at a 

 weight without being able to lift it, the work is zero, because 

 the displacement is zero. 



232. In general, the point of application of a force /''will be 

 acted upon by a number of different forces, so that the displace- 

 ment s of this point will not necessarily take place in the direc- 

 tion of F. In this general case the work of a force is defined as 

 the product of the force into the projection of the displacement of 

 its point of application on the direction of the force. 



In Fig. 74, for instance, the particle P while acted upon by 

 the force F (and any number of other 

 forces) is displaced from P to P 1 ; hence 

 if PP' = s, and KP'PQ = $, the work 

 of the force 



(I) 



It is obvious that this work might also be defined as the 



