Let two forces P and Q act at the point A, the one in the direction A P, the other in the direction A Q, as indicated by the arrows. Let A B represent the magnitude of the force the P, and a c the magnitude of the force Q, and complete the parallelogram A CDB by drawing в D parallel to A c, and CD parallel to a B. Join AD, then the magnitude and direction of the resultant R will be represented by a D, diagonal of the parallelogram. Thus the point A, acted on by two forces in the directions AP and a Q, is exactly in the same case as if it were acted on by a single force in the direction AR, and whose magnitude may be represented by ᎪᎠ . When the components are equal the resultant will bisect the angle which they make with each other, when unequal it will be nearer the greater force. If the point on which the forces act is fixed, the pressure which it sustains is the same as the resultant of the forces; if the point be not sustained, it will move in the direction of the diagonal. In the same manner, whatever be the number of forces applied at a point, their resultant can always be found; for, first, find the resultant of any two of them by the preceding rule; and then take this resultant and another; and so on till the resultant of all the forces is found. 22. Resultant of parallel forces.-The forces in the preceding articles are oblique forces, that is, they have their directions inclined to each other at some angle. But when the forces have their directions parallel, they may generally be replaced by a single force, which is their resultant. Thus if a P and в Q be parallel forces, there will be a single force passing through some point c betwixt A and B, parallel to the components, which will produce precisely the same effect. In these figures the forces act on the same side of A B, or in the same direction, but if one acted on the other side as represented by the dotted lines, there would be a single resultant, except in the case in which the two forces P and Q are exactly equal. Two equal and parallel forces acting on opposite sides of a line constitute what is termed a couple, which does not produce the same effect on a body as a single force; it twists a body; and any body subject only to two such forces can never be at rest. The action of gravity presents us with an example of parallel forces on the grand scale: we shall see that bodies may be considered as having every one of their particles acted on by forces whose directions are parallel, and consequently the whole effect is the same as might be produced by a single force, or by their resultant. 23. The Lever.-In the preceding articles we have seen the general conditions of equilibrium for forces acting at a point; but in practice the forces can rarely be considered as acting in that manner; they are generally applied to different points on a rigid body, and tend to turn it about some axis or centre of motion. In these cases other considerations must be introduced, and as the laws are the same for all these cases, we shall state and illustrate them in their application to the simple case of the equilibrium of a lever. The form and applications of the lever are very varied and numerous; but the simplest case is that of a straight rower pulls, is the power exerted; in all these cases, which are levers of the second kind, the power exerted is less than the resistance overcome. Hence, in levers of the first and second kind, the power acts at a mechanical advantage. In But in the third kind, in which the power acts between the weight and the fulcrum, the force exerted is greater than the resistance overcome. Thus in the common fire tongs, the hand being near the fulcrum exerts a greater force than is exerted at the surface of the substance raised. the shears used for cutting off the wool from sheep, the hand exerts a greater force than the resistance of the substance cut. This may also be well illustrated by a person pushing at a gate near the hinge. The force which he must exert to open the gate is the greater the nearer the hand is placed to the hinge. The limbs of animals are in general levers of this class; the object being to obtain a motion through a larger space than that through which the power acts. Thus the muscle which moves the arm and hand acts near the elbow, so that the hand moves through a large space while the parts near the elbow may be considered as stationary. The considerations of the relative space through which the power and the weight act are of the greatest importance in practical mechanics, and the proposition which is thus arrived at for all the mechanical powers, as well as for the lever, may be expressed in the following terms, what is gained in force is lost in velocity.' Thus in the first kind of lever, in the common poker for instance, the hand moves through a much greater space, in the same time, than the coals raised move through. In the oar, which is a lever of the second class, the hand moves through a greater space than the boat. But in the third kind of lever, as in the common fire tongs, the hand moves through a much less space than the end of the tongs, and just in the same proportion as the power exerted is greater than the resistance overcome. 24. Machines. It would be foreign to our present subject to consider the various combinations which are commonly expressed by the term mechanical powers; but as this term has been much misunderstood from incorrect views of physical principles, we shall endeavour to illustrate these principles by their applications in machines. The first question then is, what is meant by a machine; and to this we reply that a machine is any instrument designed for the modification of force; as, for the transfer of force exerted at one point to another point, or for causing a force which acts in one direction and in one manner at one point, to act in a different direction and in a different manner at another point. Thus a knife is a machine, since the power or force exerted by the hand at the handle is transferred to the substance to be cut. A pair of scissors is a machine. In all machines there are two conventional terms, the power and the resistance; by the power we mean the force whose action and direction is to be changed and modified by the machinery, and by the resistance we mean the obstacle which is to be overcome, or the work which is to be done. Thus, in cutting any thing with a pair of scissors, as we have already seen, the force which the fingers exert is the power, and the force exerted on the substance cut is the resistance; in a steam engine the elastic force of the steam is the power, the friction to be overcome, and the work to be done, together constitute the resistance. We must therefore consider a machine merely as intended to transmit power and to execute work; and the advantage derived from machinery consists in the modifications and adaptations of which a power is susceptible from this very transmission. The simple machines enable us to concentrate and divide any quantity of force which we may possess; a small force may be husbanded so as to produce all at once a prodigious effect; a large force may be subdivided so as to produce ten thousand small effects. These principles are illustrated by what is going on in every factory in the kingdom. The force of wind, water, or steam, by a continuous uniform action, sets in motion a large fly-wheel; this wheel may be employed every few minutes to lift a massive forge hammer, or it may keep thousands of smaller E wheels in operation. The various motions which are going on in a single factory, all derived from one original motion, and all arising from the modifications which force and motion may receive by transmission, are objects most worthy of attentive examination. In all the mechanical agents by which motion is transmitted, as in the lever, the pulley, the wheel and axle, and many others, it may be mathematically demonstrated, that no real power can be gained by their use, however combined. It is certain that whatever force is applied at one point can only be exerted at some other, diminished by friction and other incidental causes; but the gain arises from other sources, as the economy of time and labour, and convenience. There is another principle, that whatever is gained in the rapidity of the execution is compensated by the necessity of exerting additional force: all the combinations of mechanical art can only augment the force communicated to a machine at the expense of the time employed in producing the effect. We will illustrate this by the familiar instance of raising water by a common pump: the pump handle may be either short or long; or, which is the same thing, the person who pumps may at one time lay hold of the middle of the pump handle, and at another time at the end; and he may work the pump in either case; but when his hand is at the middle he must exert a much greater force than when it is at the end, but then his hand moves through a much less space. The advantage thus derived from a long pump handle is, that a weak man moving his arm through a long space at the end of the handle can raise as much water, that is, do as much work, as a strong man moving his arm through a less But unless the weak man moves his arm quicker space. than the strong man, he will take more time in going through his appointed space: then we see that the effect of the lever in this case may be viewed in two ways; first, either as enabling a weak man who moves his arm rapidly to do as much work as a strong man who moves his arm slowly, and thus placing them on an equality, so to speak, as mechanical |