12 INTRODUCTION TO DYNAMICS. [25. 



25. Three Particles. We find first the centroid P 1 of m z at 

 P 2 and m s at P B (Fig. 2) by Art. 24 ; then, by the same rule, 

 the centroid G of m%-}-m B at P' and m 1 at Pj. We might have 

 begun with P 3 and P v finding P" ; or with P : and /> 2 , finding 



/"". lies at the intersection 

 of the three lines PJ>\ P^P", 

 P z P' n , and can therefore be 

 constructed graphically. 



P' 26. Four Particles. Find the 



Fig ' 2> centroid P' of w x at ^ and m 2 



at Pgj also the centroid P" of w 3 at P 3 and ^/ 4 at P\ then 

 the centroid ^ of m 1 -\-m 2 at P r and m z -\-m at /*". 



The four particles can be arranged in groups of two in three 

 different ways. There are therefore three lines, like P'P", on 

 each of which G lies. Any two of these are sufficient to con- 

 struct G geometrically. 



27. The centroid of a homogeneous rectilinear segment (thin 

 rod or wire of constant cross-section) is evidently at its middle 

 point. 



28. If the density of a rectilinear segment be proportional to the 

 nth power of the distance from one end, say p = kx n , we have 



r 



- Jo 



n+i 



where / is the length of the segment. 



(a) For n = o, this gives x=^l which determines the centroid 

 of a homogeneous straight segment (see Art. 27). 



(b) For n= I, we have x\ /. This determines the distance, 

 from the vertex, of the centroid of a homogeneous triangular 

 area. For such an area can be resolved (Fig. 3) by parallels 

 to the base into elements each of which may be regarded as 



