60 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



direction of the normal to the surface. But owing to the constraint 

 the motion can be only tangential, consequently the particle cannot 

 move, and the applied forces together with the reaction produce 

 equilibrium. 



The Principle of Virtual Work is as follows. If any system of 

 as many bodies or particles as we please, each acted upon by any 

 forces whatsoever, is in equilibrium, and a small arbitrary virtual 

 displacement is given to each point of the system, the work done 

 by all the forces will vanish (at least to the first order of small 

 quantities). For instance a particle placed on a smooth surface under 

 the action of gravity experiences a force mg vertically downward. 

 If we displace it a distance ds the work done by the force will be 

 mgdz, if the z coordinate is taken positively downward. We may 

 write this 



and if this vanishes whatever the value of ds for all directions of 

 displacement on the surface v -=- must be zero, that is the tangent 



plane to the surface is horizontal. But the particle is in equilibrium 

 at such a point. 



Conversely, if the surface is not horizontal, dW will not vanish 

 for all possible displacements, neither will the particle be in equili- 

 brium. (It is to be noticed that in the neighbourhood of a point 

 where the tangent plane is horizontal dz is proportional to ds 2 , so 

 that the work, although vanishing to the first order, does not vanish 

 to the second, z is in this case a maximum or minimum.) 



Simple illustrations of the principle of virtual work are furnished 

 by the so-called mechanical powers. Consider in particular the 

 pulley. The mechanical advantage or multiplying power as regards 

 force, that is the ratio of the force sustained by the movable block 

 to the tension on the cord, is equal to n, the number of cords 

 coming from the movable block, for the fundamental assumption is 

 that the tension of the cord is everywhere the same. If the end of 

 the cord is displaced a small distance in its own direction, the block 

 is displaced l/n ih of that distance, consequently the work of the 

 two equilibrating forces is equal in absolute magnitude, but one 

 being positive and the other negative, their sum is zero. 



By means of this principle Lagrange gave a simple general 

 proof of the principle of virtual work. He supposed each force 

 applied to a point of the system to be replaced by a pull of a block 

 of pulleys, the number of pulleys in each block being so chosen 

 that the proper force could be produced by the tension of a single 

 cord passing over all the pulleys and fastened to a weight at one 



