153-157] VIRTUAL WORK. 139 



the similar resolved parts of the velocity just before the impulse, 

 X, Y, Z the sums of the resolved parts parallel to the axes of the 

 external impulses applied to m, X , F , Z f the sums of the similar 

 resolved parts of the internal impulses, T and T the kinetic 

 energies of the system just after and just before the impulses. 



We have such equations as 



Also T - T = JSm (tf 2 + 2/ 2 4- s 2 ) - J2m (f 2 + rf + ) 

 = -JSm (x % ) (x + j) + two similar terms 

 = J2 CJ&quot; -f Z ) (*+?)+ two similar terms. 

 Thus, /i6 change of kinetic energy produced by impulses is the 

 sum of the products of all the impulses and the arithmetic means of 

 the velocities, in their directions, of the particles to which they are 

 applied just before and just after the impulsive actions. 



It is very important to notice that the internal impulses may 

 not be omitted from the equation here obtained, just as the internal 

 forces may not be omitted from the equation of energy of Article 

 151. 



156. Virtual Work. We have defined the work done by a force when 

 the particle on which it acts undergoes any finite or infinitesimal displacement. 



Suppose the particle is in motion in any manner, then our definition 

 applies not only to the actual displacement of the particle but also to any 

 other possible displacement. 



The work done by a force in an infinitesimal displacement which is not 

 necessarily the actual displacement is called the virtual work of the force for that 

 displacement, and the displacement itself is called a virtual displacement. 



157. Principle of Virtual Work. The sum of the virtual works of all the 

 forces on a system in equilibrium vanishes in every infinitesimal displacement. 



The sum in question vanishes, not by the quantities that express the 

 displacements becoming indefinitely small, but by the factors that multiply 

 these quantities in the sum being indefinitely small. 



We have already seen that the work of a system of forces acting on a 

 particle is equal to the work of their resultant, and this applies to infinitesimal 

 displacements. Now when the particle is in equilibrium under the forces the 

 resultant vanishes identically. Hence the virtual work of the forces vanishes. 



Again, for a system of particles in equilibrium under the action of a system 

 of forces each particle is in equilibrium, and therefore the principle enunciated 

 holds good for each particle. It follows by addition that the principle holds 

 for the system. 



