8 INTRODUCTION TO DYNAMICS. [17. 



The centroid (Art. 14) is a point such that, for any plane 

 passing through it, the moment of the system is equal to zero. 

 Such a point exists for every mass, volume, area, or arc. The 

 centroid coincides, of course, with the centre, when such a 

 centre exists and the distribution of mass is uniform. 



17. 'As soon as p is given either as a constant or as a function 

 of the co-ordinates, the problem of determining the centroid of 

 a continuous mass is merely a problem in integration. To 

 simplify the integrations, it is of importance to select the 

 element in a convenient way conformably to the nature of the 

 particular problem. 



Considerations of symmetry and other geometrical properties 

 will frequently make it possible to determine the centroid with- 

 out resorting to integration. Thus, in a homogeneous mass, 

 any plane of symmetry, or any axis of symmetry, must contain 

 the centroid, since for such a plane or line the sum of the 

 moments is evidently zero (see Art. 47). 



It is to be observed that the whole discussion is entirely 

 independent of the physical nature of the masses m which 

 appear here simply as numerical coefficients, or "weights," 

 attached to the points (comp. Art. 5). Some of the masses 

 might even be negative. 



It will be shown later that the centre of gravity, as well as 

 the centre of inertia, of a body coincides with its centroid. 



18. The centroid can be defined without any reference to a 

 co-ordinate system as follows. 



As in Art. 12, let there be given a system of n points 

 />!, P 2 , . .. P n (Fig. i) whose masses are m v m 2 , ... m n . Taking 

 an arbitrary origin O and putting OP l = r l , OP 2 = r 2 , ... OP n = r n) 

 we may represent the moments m\r^ m 2 r 2 , ...m n r n of the 

 given masses with respect to O (Art. 11) by lengths (vectors) 

 laid off on OP V OP^ . . . OP n . The moment of the system can 

 then be defined as the geometric sum of these vectors. It is- 

 therefore found by geometrically adding these vectors ; i.e. we 



