CENTRE OP GRAVITY.] 



MECHANICAL PHILOSOPHY STATICS. 



711 



will be parallel to each other ; since if in still water, at 

 a distance of 200 yards from each other, we were to sink 

 two plummets a mile and a-half in length, they would 

 not deviate further than 3 inches from perfect parallelism. 

 It is true that a large mass of matter, such as a moun- 

 tain, will slightly dt fleet, as according to the theory of 

 gravitation it ought, the plumb-line from the true ver- 

 tical ; but in ordinary cases this source of error may be 

 neglected. Again, the attraction of the earth will vary 

 according to the distance from its centre of the particle 

 or body attracted ; but this variation may also be ne- 

 glected for bodies which are small when compared with 

 the magnitude of the earth. 



Hence, in most cases, even such, for instance, as the 

 largest line-of- battle ship, without introducing any 

 sensible error, we may regard the attraction of the earth, 

 on any mass of matter, as acting on every material 

 particle composing that mass, in directions parallel to the 

 plumb-line suspended near it. 



The centre of the parallel forces, produced by the 

 weights of the material particles of which any heavy body 

 is composed, is called the centre of gravity of that body. 



Hence, Prop. XVII. enables us to find the position of 

 the centre of gravity of any number of heavy particles 

 whose weights and positions are known. 



From the properties of the centre of parallel forces we 

 have already demonstrated, it follows that the centre of 

 gravity of a heavy body is that point within or without the 

 body at'which the whole of its weight may be conceived to 

 act ; and the body will proilnce the same mechanical effect, 

 as if we were to suppose the whole of its weight concen- 

 trated in that point. This enables us to extend Prop. 

 XVII., to find the common centre of gravity of several 

 bodies, whose weights and the position of whose respec- 

 tive centres of gravity are known. 



Again, because the parallel forces exerted by the 

 weights of the material particles of a body will have the 

 same direction in whatever position it is placed, its centre 

 of gravity will not change its position with respect to 

 the body for any change in the position of the body. 

 Hence, if the centre of gravity be fixed, the body will 

 balance about it in every position, because the resultant 

 of the weights of every one of its elementary particles 

 will pass through the fixed point (the centre of gravity) 

 in every position in which the body can be placed. 



PROPOSITION XIX. 



If a heavy body be in equilibrium when suspended from a 

 point, or when resting on a point in contact with another 

 body, its centre of gravity will be in the vertical line 

 passing through the point of suspension or contact. 

 Let (Fig. 00*,) 1, 2, and 3 represent a section of the 

 heavy body, taken through the plane passing through its 

 centre of gravity G, and its point of suspension or con- 

 tact A. 



In (1) we suppose the body supported by a pin A C, 



Fig. 60*. 



passing through a hole at A, about which it can move 

 freely ; in (2) it is suspended from a point at A, by a 



string fixed at C ; and in (3) it is supported by another 

 body with which it is in contact at A. 



In all three cases let the vertical be represented by 

 the plumb-line AB ; the centre of gravity must lie in that 

 line. If it do not, let it have some other position, as G. 

 From our definition of the centre of gravity, the 

 weight of the body will produce the same effect upon the 

 body as if, being destitute of weight, a force equal to its 

 weight, acting vertically downwards, were applied to its 

 centre of gravity. 



Let G W = W represent this force, the weight of the 

 body, in magnitude and direction, and we then have the 

 case of a rigid body without weight in equilibrium when, 

 acted on by the force W acting at G, and the reaction or 

 tension at the point A. 



Through A draw A D perpendicular to G \V or G W 

 produced. Then by the principle of the transmission of 

 force, W may be transferred from G to D, and W A D 

 will be the moment of a force tending to twist the body 

 round the point A, which is counteracted by no other 

 force. Consequently, equilibrium can only exist if W 

 or A D be equal to nothing. But by the terms of the 

 proposition, W cannot be equal to nothing. Hence, there 

 can only be equilibrium where A D is nothing, in which 

 case G must lie in the line A I!. 



In the same manner it can be shown that if a heavy 

 body balance on a given straight line as, for instance, 

 the sharp edge of another body which is a straight lino 

 its centre of gravity will lie in a straight line. 



If any homogeneous heavy body be of a form which 

 is symmetrical with respect to a certain point or line, 

 the centre of gravity will be in that point or line ; for 

 the very idea of symmetry requires, that if any point be 

 taken in the body, there must be another point in that 

 body equidistant from the point or line about which it is 

 symmetrical. Hence, since the centre of gravity is the 

 centre of the parallel forces produced by the weights of 

 the material particles composing the body ; if it be homo* 

 geneous, that is, composed of particles of tho same weight, 

 Fig. 61. 



and distributed uniformly through- 

 out its substance, any one particle 

 in the body will be balanced about 

 the point or line around which its 

 form is symmetrical, by another 

 particle equal to it in weight, and 

 equidistant from that point or 

 line. 



The centre of gravity, therefore, 

 of a sphere, will be the centre of 

 the sphere ; that of a cube or ob- 

 lique parallel opiped (Fig. 61 1), 

 the point where two diagonals intersect each other. The 

 centra of gravity of a right (2) or oblique (J) cylinder 

 will be in the middle of its axis ; and that of a ring (4) 

 will be the centre of the ring. 



Tho centre of gravity of a hollow sphere or cylinder 

 will be the same as if it were solid, provided it be of the 

 same thickness throughout. From this it follows that the 

 centre of gravity of a body need not necessarily be a 

 point within it. 



EXPERIMENTAL DETERMINATION OF THE 

 CENTRE OF GRAVITY. The preceding proposition 

 affords a means, in many instances, of determining the 

 >osition of the centre of gravity of a body experimen- 

 ally. 



