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Resultant and Component Forces.

17

the boy in drawing the carriage would be represented both in direction and intensity by the line AB.

23. Resultant and component forces.-When a body is acted upon by only a single force, it is clear that, if it is not hindered by any obstacle, it will move in the direction of this force; but if it is simultaneously acted upon by several forces in different directions, its direction will not, speaking generally, coincide with that of any of these forces. If two men, for example, on the banks of a river, tow a boat by means of ropes, as shown in fig. 9, the boat follows

neither the direction AB, nor the direction AC, in which these men are respectively pulling, but takes an intermediate direction, AE; that is, it moves as if it were acted upon by a single force in the direction AE.

B

Paral

The single force, which we conceive as having the direction AE, producing the same effect as the forces of traction of these two men, is called the resultant of these two forces; and conversely these, in reference to their resultant, are spoken of as the components. 24. Value of the resultant of two concurring forces, lelogram of forces.- When two forces having different directions are applied to the same point of a body, as represented in fig. 9, there is a very simple ratio between their intensities and that of their resultant, which is of great importance from the number of its application.

A

D

Fig. 10.

C

It will first of all be necessary to define the word parallelogram,

C

of which we shall make use. The parallelogram is a geometrical figure, formed of four right lines, each pair of which is parallel (fig. 10), that is, the two lines AB and CD are parallel, and also the lines AD and BC. These lines form the sides of the parallelogram, and the points A, B, C, D, the angles. The diagonal is the line, like AC, joining two opposite angles A and C.

In treatises on mechanics proofs are given of the following important theorem, which is known as the principle of the parallelogram of forces :

When two forces applied at the same point A (fig. 11) are represented in direction, and in intensity by the sides AB and AD of the

B

Fig. 11.

parallelogram ABCD, their resultant is represented both as to its intensity and direction by the diagonal AC of this parallelogram.

That is, that the point A being simultaneously acted upon by two forces, whose directions and intensities are respectively represented by AB and AD; this point moves in the direction AC exactly as if it were acted upon by a single force, the direction and intensity of which are represented by the line AC.

Frequent applications are met with of the principle of the parallelogram of forces. Thus, in the flight of a bird, when the wings strike against the air, a resistance is offered which is equal to an impulsive force from back to front in the directions AH and AK (fig. 12); hence representing, by AB and AD, the intensities and directions of these impulsive forces, if the parallelogram be completed,

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Parallelogram of Forces.

19

we shall find that the resultant, or the single force which makes the bird advance, is represented in direction and magnitude by the

diagonal AC. The same reasoning applies to the swimming both of men and fishes.

25. Another effect of the parallelogram of forces.-We have seen that, in accordance with the principle of the parallelogram of forces, two forces applied at the same point of a body may be re

ducing together the same effect as the first. This force is then said to be decomposed into two others.

It is but seldom indeed that the action of a force is entirely utilised; it may almost always be decomposed into two others, only one of which produces a useful effect. Thus when the wind blows against the sails of a vessel, not quite directly, but a little on one side, as shown in fig. 13, the effect of the wind in the direction va may be decomposed into two others, one in the direction ca, and the other in a lateral direction ba. The first moves the vessel, the second only guides it.

26. Case in which the forces are parallel. Value of the resultant. In the case of the boat drawn by a rope (fig. 9), the forces were concurrent, that is, their directions if produced would meet in one point; but it may happen that the forces applied to the same body are parallel, and then two cases present themselves; that is, they either act in the same direction as in the case of two horses drawing a carriage; or they may act in opposite directions, when a steamer for instance ascends a river, the current acts in opposition to the force which urges the steamer. It can be proved that, in the first case, the resultant of the forces is equal to their sum; and that in the second it is equal to their difference.

27. Equilibrium of forces.-When several forces act upon a body at the same time, they do not always put it in motion; it may happen that while some of these forces tend to produce motion in a certain direction, the others tend to produce an equal and contrary motion in the opposite direction. It is clear that in this case, since the forces just neutralise each other, no effect can be produced. Whenever several forces applied to the same body thus mutually destroy each other, we have what is called equilibrium.

The simplest case of equilibrium is that of two equal and opposite forces applied at the same point of a body. For instance, if two men pull at a cord with the same intensity, one in one direc

Fig. 14.

tion, and the other in the opposite one, equilibrium will be produced (fig. 14). In like manner if, in a well, two buckets of the same size,

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Centrifugal Force.

21

each full of water, are suspended at the end of a rope which passes round a pulley, the weight of one holds the other in equilibrium. The bodies which we consider ordinarily to be in a state of rest, are really in a state of equilibrium. For instance, when a body rests on a table, there is equilibrium between the force of gravity which tends to make the body fall, and the resistance which the table offers to the fall. If the weight of the body exceeds this resistance, equilibrium is destroyed, the table is broken, and the body falls.

28. Centrifugal force. We shall conclude these notions about forces by mentioning a force to which curvilinear motion is due, namely centrifugal force. This may be explained as follows. Whenever a body has been put in motion in a particular direction, in virtue of its inertia, it tends always to move in this direction. Hence whenever a line is seen to move in a circle, this can only be