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



The potential energy is 



^ 



r l2 



the summation being extended to all the pairs of particles. 

 It is clear that this can be written 



. f /WZ-9 ^ \ fWri Ml"* \ "1 



-i w iy M + - s + ...)+W ' + 8 +...) + ... i 



L V12 r !3 / V12 r 23 / 



and then the factor multiplying m 1 is the potential of the system at the point 

 occupied by m 15 the factor multiplying m 2 is the potential of the system at 

 the point occupied by m 2 , and so on. 



Thus the potential energy of the system is ^2m V, where m is the mass 

 of any particle, and V the potential of the system at the point occupied by it. 



145. Gravity. Consider the particular case of a heavy body near the 

 Earth's surface. We regard the Earth as a spherical body with its mass so 

 arranged that its attraction on a particle at an external point is the same as 

 that of a particle at its centre of mass equal to the mass of the Earth. Let E 

 be this mass. 



Then the particles of the Earth exert on a particle of mass m near its 



JBUj 



surface a force y ^- where r is the distance of the particle from the Earth's 

 centre. 



This is very nearly equal to the weight of the particle, viz. to mg, where g 

 is the acceleration due to gravity. 



K 



We therefore have y-^=g approximately. 



When the particle is displaced so that r becomes r + dr the work done by 

 the forces is - mg8r. 



Now when the particle remains near the Earth's surface at a place, 8r is 

 the height above a horizontal plane through which it is raised, and the height 

 through which the centre of inertia of a system of such particles is raised is 



2m 



Hence the work done in the displacement of the body is Mgh, where M is 

 the mass of the body and h the height through which its centre of inertia 

 falls. 



This is the same result as that differently expressed in Article 139. 



146. Examples of conservative systems. The gravitating system 

 discussed above is an example of a conservative system. No matter by what 

 paths the particles move from one set of positions to another the work done 

 depends only on the initial and final distances. 



