128 THEORY OF WORK AND ENERGY. [CHAP. VIII. 



140. Work done by the mutual actions between two particles. Let 



A, B be the initial positions of the particles, F the force between them. To 



AM B N 



Fig. 42. 



fix ideas suppose F to act on the particle at A from A towards B. Let the 

 length AB = l. 



Let very small displacements be made so that A comes to A and B 

 comes to B \ let the line A B make with AB an angle 6, which must be very 

 small when the displacements are very small. Also let A B = l , then I -I 

 also is very small. 



Let M, N be the projections of A , B on AB ; to fix ideas suppose M is in 

 AB, and N in AB produced. 



The work done by the two forces F is 



F.AM-F.BN 

 =F(AM-BN] 

 =F(AB-MN) 

 = F (I- 1 cos 6) 



Now 1 - cos 6 is an infinitesimal of the second order when 6 is of the first 

 order, and the work done is therefore F(l- 1 } to the first order of infinitesimals. 



141. Case where the distance is invariable. When the particles move 

 in such a way that the distance between them is unaltered, the sum of the 

 works done by the forces exerted by the first on the second and by the second 

 on the first is an infinitesimal of the second order when the displacements are 

 of the first order. Hence in any finite displacement no work is done by such 

 forces. 



Examples of this are (1) the tension of an inextensible string does no 

 work ; (2) the forces between the particles of a rigid body do no work. 



142. Work of gravitating system. Let m, m be the masses of two 

 particles of the system, r the distance between them, and let this distance 

 become r+8r, the work done is, by Art. 140, 



Hence the work done by the forces between these particles as their 

 distance changes from r to r l is 



,/l 1\ 

 = ymm --- . 

 7 Vi V 



