MECHANICS. 



ir 



n o 

 to 



parallel ogram Cnom, the sides 

 and m C being perpendicular 

 PCB. 



By (9.) it appears, that if C o, (Jig. 

 13), be taken to represent the whole at- 

 traction on the point C, it is equivalent 

 Jig. 13. 



to two separate attractions represented 

 in intensity and direction by the lines 

 C n and C m, and its effect is the same as 

 the united effects of these two would 

 be. Now it is obvious, that a force 

 acting- from C, in the direction C n, 

 would have no effect in producing mo- 

 tion, but would be resisted by the fixed 

 point C, against which it would press, 

 while the other force C m, perpendicular 

 to C P, would tend to turn the body 

 round C P, so as to bring the point C 

 to the line P D, directly below the point 

 of suspension P, in which position, after 

 some oscillations, it would rest. 



(42.) From this investigation, it fol- 

 lows, that if the parallel actions of the 

 force of gravity on the particles of a 

 body be capable of being represented by 

 an equivalent force, acting at a single 

 point, that point may be determined by 

 the properties which we have just ex- 

 plained. Let a body which is bounded 

 by two parallel planes, be suspended 

 from any point taken at pleasure in it. 

 It will be found that there is but one 

 position in which it will hang steadily 

 at rest, and without swinging. To the 

 point of suspension let a plumb line be 

 attached, and let the line in which it 

 touches the plane surface of the sus- 

 pended body be marked. Let the body 

 be now suspended from some other 

 point in its plane surface, and let another 

 line be drawn upon it in the direction 

 of the plumb line. This process being 

 applied to any number of different 

 .points in the surface of the body, and a 



number of such lines being drawn upon 

 it in the direction of the plumb line, it 

 will be found that all these lines will 

 intersect each other in the same point. 

 It follows, therefore, that this point has 

 the property mentioned in (41.), of set- 

 tling itself vertically under the point of 

 suspension when the body is in equili- 

 brium. 



Next let the point thus determined 

 be made the point of suspension, and it 

 will be found that the body will rest in 

 any position in which it may be placed, 

 and that it will not, under any circum- 

 stances, vibrate or swing. 



Again, let the body be suspended by 

 any point different from that which we 

 have here determined, and let it be so 

 placed that this point shall be placed 

 vertically over the point of suspension. 

 It will be found that the body will re- 

 main in equilibrium so long as its posi- 

 tion is not changed ; but upon the least 

 impulse which moves the point in 

 question from its position, it will turn 

 round the point of suspension, and settle, 

 after some vibrations, into the position 

 directly under the point of suspension. 



The point, the existence and proper- 

 ties of which are thus established, is 

 then the centre of gravity. 



In the preceding experiment, we have 

 selected a body bounded by parallel 

 planes, for the purpose of simplifying 

 the experimental process. Strictly 

 speaking, the centre of gravity is not at 

 the intersection of the lines determined 

 by the plumb line on the plane surface, 

 but, if a line be drawn perpendicular to 

 the plane surface through the body, it 

 will be at the middle point of this line. 



If we could conveniently pierce the 

 dimensions of a body by straight lines, 

 the centre of gravity of any body, what- 

 ever be its figure, could be found ex- 

 perimentally by the same process. If 

 it be successively suspended by several 

 points, and pierced by straight lines, in 

 each case passing in a vertical direction 

 through the point of suspension, it 

 would be found that, however numerous 

 these lines might be, they would all in- 

 tersect in one point, which would be 

 the centre of gravity of the body. 



(43.) By these properties of the centre 

 of gravity, mechanical problems respect- 

 ing the effects of the weights of bodies are 

 susceptible of considerable simplifica- 

 tion; for, in stead of taking into considera- 

 tion the separate effects of the attraction 

 of gravitation on the several particles 

 of which a body is composed, it will be 

 c 



