SECTION IV.

GRAVITY.

33. LET P,P,... P1 be any number of parallel forces acting

2

at points 4,4,....4, which are rigidly connected together. Join A,,, and in this line take G, such that

AG, A,G,:: P: P1............(1),

1

2

1

2

then, by Art. (23), the resultant of P, and P2 is equal to P1 + P1, and passes through G, in a direction parallel to P1

1

then the resultant of P1, P, and P, equals

2

3

...

1

.......(2),

2

P1 + P2 + P ̧ is parallel to them and passes through G2: it is manifest from the form (2) that the position of G, is independent of the directions of the parallel forces.

2

By proceeding in a similar manner it might be shewn that the resultant of all the forces P1, P2....P is equal to the sum of them, is parallel to them, and always passes through a point whose position with reference to „............. is independent of the directions of the forces.

...

Now the weight of a body is the force which is brought into action by the attraction of the earth upon each of its

E

particles: but this attraction produces a system of vertical, and consequently parallel forces, whose points of application are the several particles of the body; therefore the weight of a body is equivalent to the resultant of this system; i.e. it is a force equal to the sum of these forces, is parallel to them, and passes through a point in the body whose position in the body is invariable, whatever be the direction of the separate forces with reference to any fixed line in the body; or in other words, whatever be the position of the body. This point is called the Centre of Gravity of the body.

34. It is evident from this definition that there is but one such point for any given body.

35. If the centre of gravity of two portions which constitute the whole of a body be known, the position of the centre of gravity of the body itself can be easily found.

Thus, suppose the body to be made up of any two parts, as

ABC, ABD: let G, be the centre of gravity of the first, W, its weight; and G2 the centre of gravity, and W, the weight, of the second.

2

Then the action of gravity upon ABC is equivalent to the application of a vertical force W, to the point G, similarly, the action of gravity upon ABD is equivalent to a force W acting vertically at G2.

2

Join GG, and in this line take G such that

GG: GG:: W2: W1;

1

2

2

1

2

then, by the principles of parallel forces, W, at G, and W, at G may be replaced by a force W, W, at G; i.e. the action of gravity upon the whole body is equivalent to that of a force equal to W1+ W2 at G; therefore G is by definition the centre of gravity of the whole body.

1

2

It is evident that if any two of the points G1, G2 be given, the other may be obtained.

36. And generally, if a body can be divided into any number of parts, whose centres of gravity are known, the action of gravity upon the whole body is equivalent to the application of the weights of the several parts at their respective centres of gravity; and hence the centre of gravity of the whole can be easily found by the aid of the propositions that have been given with regard to systems of parallel forces.

If the centres of gravity of the different parts, into which we have supposed a solid divided, all lie in one straight line, it will follow from Art. (23) that the centre of gravity of the whole body will also lie in that line: this is a fact of considerable practical importance, as will be observed in some of the examples appended to this section.

37. If a body or system of heavy particles be in equilibrium under the action of gravity and the resistances of surfaces or fixed points only, it is clear that the resultant of these resistances must be vertical and must pass through the centre of gravity of the body or system: if these two conditions be satisfied, the magnitude of this resultant need not be cared for; it is the nature of forces of resistance to be exerted to a sufficient intensity for producing equilibrium, and not to a greater. This consideration affords an easy means of solution to many of the simpler cases of equilibrium.

Thus, if a body be standing upon a smooth horizontal plane having one or more points in the same straight line, in contact

with the plane, the resistances at these points are vertical, and all act in the same direction; hence their resultant will be vertical, and will pass through some point of this line which lies between the two extreme points of contact, but which is à priori indeterminate, because the resistances at the several points are so if however the direction of the weight of the body, i.e. the vertical line through the centre of gravity of the body, meet this line, just so much force will be called forth at each point of contact as will make the resultant of their reactions coincide with this vertical line, and equilibrium will subsist.

But if the direction of the weight of the body pass outside this line, then it cannot be counteracted by the resultant of the resistances which, as was above mentioned, must meet the line, and therefore the body will fall.

Fig. 1.

Fig. 2.

Figs. (1) and (2) represent cases where the direction of the weight of the body falls within and without respectively the

line in which the points of contact lie; equilibrium will subsist in the first but not in the second case.

If the equilibrium be produced by the body's being suspended from one point only, or by its touching a surface in one

A

point only, the preceding reasoning shews that this point of suspension or contact, as A, must be in the same vertical line with the centre of gravity of the body G.

This suggests a method for practically finding the centre of gravity of a body in many cases. Let the body be suspended successively by two separate points in its surface, and when in each case the body has found its position of equilibrium, let the vertical in it which passes through the point of suspension be observed; the point in which the two vertical lines intersect is the centre of gravity required.

38. DEF. A body is said to rest in Stable Equilibrium when upon being disturbed in a very slight degree from its position of equilibrium, it will, upon the disturbing cause being withdrawn, return to its first position.