156 WORK 



If any particle is in equilibrium, the resultant force acting 

 on it vanishes, so that the work done in any small displacement 

 of the particle vanishes to a higher order than the displacement. 

 If a rigid body, or system of rigid bodies or particles, is in equilib- 

 rium, and any small displacement is given to it, the work done 

 on each particle is nil, so that the aggregate work done is nil. 



123. The forces acting on the particles of the system may, as 

 in 50, be divided into two classes : 



(a) forces applied to the bodies from outside ; 



(b) pairs of actions and reactions acting between the particles 

 of the bodies, or between two bodies in contact. 



In calculating the work done in a small displacement, we must 

 take account of the work done against all the forces of both classes, 

 but shall find that a great number of the terms arising from the 

 forces of the second class cut one another out. 



124. Let us first consider the pair of forces which constitute 

 the action and reaction between two particles P, Q of a rigid body. 

 Let the amount of each force be R, its direction being QP or P( 



according as it acts on P or Q. Let 

 the effect of a small displacement 

 to move P, Q to P r , Q' respectively, and 

 let P r p, Q f q be perpendiculars drawn 

 from P f , Q' to PQ. The work done 



against the force R acting on P is R X Pp, while that done against 

 the force R acting on Q is R X Qq. Thus the total work performed 



= X(Pp-Qq) 



= R(PQ projection of P'Q' on PQ). 

 Since the body is rigid, the length P'Q' is equal to the length 

 PQ, and since the displacement is, by hypothesis, small, the angl< 

 between P'Q' and PQ is small. Thus the projection of P'Q 1 on P( 

 = P'Q', except for small quantities of order higher than the first 



so that the work performed vanishes, 



