CHAPTER V.

OF THE CENTRE OF GRAVITY.

69. THE attraction of the earth on any body would, if unopposed, draw it towards the surface of the earth.

The direction in which a particle would fall freely at any place is called the vertical line at that place. It coincides with the direction of a plumb-line, or the normal to the surface of standing water.

A plane perpendicular to this vertical line is said to be horizontal.

If we regard the earth as a sphere (which is very nearly the case), the vertical lines would all converge to the centre, and therefore the directions of the forces which the earth exerts on the different particles composing a body are not parallel, strictly speaking. But since the dimension of any body we shall have to consider is very small compared with the radius of the earth, we may consider these directions to be appreciably parallel, and the resultant attraction on the body or system equal to the sum of the attractions on the constituent particles; i.e. the weight of the whole equal to the sum of the weights of the several parts.

The object of the present chapter is to shew that for every body or system of particles there exists a point through which the resultant attraction of the earth may be supposed to act; i.e. a point at which we may suppose the weight of the body to be collected,-a point whose position depends only on the relative arrangement of the particles composing

the body or system, and on the relative constitution of these particles. If this point then were in rigid connexion with all the parts of the system, all positions of the body or system would be positions of equilibrium, if this point were supported.

Such a point in a body or system is called the centre of gravity of the body or system, and we give the following definition. The point at which the weight of a body or system may always be supposed to act, whatever be the. position of the body or system with respect to a horizontal plane, is called the centre of gravity of the body.

70. We shall first shew that such a point exists in any system of particles.

PROP. Every system of heavy particles has one and only one centre of gravity.

First let us consider two heavy particles A, B, whose weights are P, Q, and suppose them

B

connected by a rigid rod without

E

weight. Now, since P and Q act through A and B in parallel directions and towards the same parts,

=

they are equivalent to a single resultant, the magnitude of which P+Q, and which acts through a point E in the line AB, such that P: Q=BE: AE; and since the position of E in the line AB does not at all involve the direction of action of gravity, if this point E were supported, this system of two particles would balance about E in any position. E then is the centre of gravity of A, B, and the statical effect of P and Q will be the same as if they were collected into one particle and placed at E.

Again, if there are three particles A, B, C whose weights are P, Q, R, we can take E the centre of gravity of P, Q as before, and suppose P+Q placed at E instead of A and B, and we then have two particles at E and C whose weights are P+Q and R; these then, as before, have a centre of gravity at a point Fin the line EC, such that

and we may suppose P, Q, R all collected at F so far as their statical effect is concerned. And so on whatever be the number of particles, so that every system of heavy particles has a centre of gravity.

Also a system of particles can have but one centre of gravity. For, if possible, let a system have two such points G and G', and let the system be turned about if necessary till the line joining G, G' is horizontal. Then we have the weight of the system acting in a vertical line through G, and also in another vertical line through G'; which is impossible, since it cannot act in two different lines at the same time.

We should arrive at the same point G in whatever order we may take the points A, B, C...

COR. 1. Since every continuous body is an aggregation of a great number of particles, every body has a centre of gravity through which the resultant weight of the particles acts: and we may suppose the weight of the whole body collected at its centre of gravity.

And we may proceed to find the centre of gravity of a system of bodies by supposing them to be a series of heavy particles, the weights of which are equal to the weights of the bodies, and which are in the position of the centres of gravity of the several bodies.

COR. 2. The determination of the successive points E, F, &c. in the previous proposition does not require the actual weights P, Q, R, but only their ratios. Hence if the weights of the several parts of a system be all diminished or all increased in any the same proportion, the position of the centre of gravity will not be altered.

COR. 3. Since the weights P, Q, R...are equivalent to a series of parallel forces acting at the points A, B, C..., and the position of the centre of gravity does not depend on the direction in which these forces act, but only on their relative magnitude and their points of application; it would therefore remain in the same position if the direction of these forces were turned about their points of application in any manner, still remaining parallel. Hence the point under consideration is sometimes called the centre of parallel forces.

71. Having given the centre of gravity of a body and also of a part of the body, to find the centre of gravity of the remaining part.

Let w1, w, be the weights of the two parts of the body; G1, G2 their respective centres of

gravity: then G the centre of

gravity of the whole body must be

a point in the straight line which joins G, G2, such that

w1. GG2=w1. GG1. .

2

G

G2

it to G, making GG,=. GG1, and thus the position of G,

the point required is determined.

72. Before proceeding to give a general method of finding the centre of gravity of any system of particles, we will give a few examples of finding the centre of gravity,-premising that when we speak of a line, or plane, or surface as having a centre of gravity, we suppose it to be made up of equal particles of matter uniformly diffused over it: unless some other supposition is stated.

I. To find the centre of gravity of a right line.

Considering it as a line of equal particles uniformly arranged, it is clear that the middle point of the line is its centre of gravity. For we may divide the line into a series of pairs of equal elements, the particles composing any pair being equidistant from the middle point. Hence the centre of gravity of each pair is at the middle point, and therefore the centre of gravity of the whole is there also.

II. To find the centre of gravity of a parallelogram.

Let ABCD be a parallelogram regarded as a uniform lamina of matter, and draw the line EF parallel to AB or CD and bisect

ing AD and BC,-and also the line

A

H

E

B

C

HK parallel to AD and bisecting AB and CD. The point G in which HK, EF intersect is the centre of gravity required. For by drawing lines parallel to BC and at equal distances from each other, we may divide the parallelogram AC into a number of equal small parallelograms whose lengths are all equal BC and breadths as small as we please; and we may take the breadths so small that each may be regarded as a line of particles, the centre of gravity of which is at its middle point,