not generally into those of equilibrium. The speed or velocity of a cannon-ball must be considered at every varying moment of its flight; but the strains and pressures among and on the beams of the roof of a railway station are the same at all moments. Time does not affect the latter unless by wear and tear. With statics, therefore, we commence, and, of course, with the simplest class of questions, those which relate to a force or forces acting on a single point. But here I must turn back to the notion of force, and endeavour to fix it with greater accuracy in your minds. I must show you how it is said to be applied and measured to the body it moves or strains; and this will best be done under the three following heads:—
1. The Direction of a Force.
2. The Point of Application of a Force.
3. The Magnitude of a Force.
1. The Direction of a Force.—In Mechanics, forces are assumed to act in right lines. The assumption is made for the best of reasons namely, that of experience. All the simpler cases of motion confirm it, and all the more complicated can be accounted for by it. A ball falls to the ground in a right line— that which points to the centre of the earth, whence the force of attraction which moves it acts. The billiard-ball moves in a right line; and the calculations of the skilful player, which are based on the supposition that it so moves, are never found to be wrong. A ship, with her sails square set and wind aft, moves in a right line; and to make it leave that line the steersman must put the helm to port or starboard, and by turning the face of the rudder against the water, cause another force to be applied to the ship across the line of its course, and at her stern, turning her round. It is true that the stone thrown obliquely into the air moves in a curved path; but in this case we know that there are two forces not one only acting on it, namely, the original impulse, which makes it move in a right line, and the earth's attraction, which pulls it from that line into a curved course. Moreover, all the calculations on which are based the predictions of astronomers as to the places in which the sun, moon, and planets will be on a certain day, hour, and minute, are based on this assumption, that forces act in right lines; and the predictions invariably prove true. Our first mechanical axiom may, therefore, on the ground of experience be assumed to be true—namely, that the direction in which a force acts is that of a right line. Indeed, it is not easy to conceive how it could act otherwise.
2. The Point of Application of a Force.—The direction of a force being disposed of, we must fix our ideas as to its point of application. The rule is, that any point on the line of its direction may be considered such; but this you must understand with a limitation, or exception, which should not be forgotten. The point of application can only be on so much of the line of direction as lies within the body. For instance, suppose a person to push with an iron rod, which he holds in his hand, at the point a (as in the diagram), against a block of iron which lies on a table. Then, clearly a is the point of application of the force with which he pushes. Let now a hole be drilled through the block in the direction of the push from a to e, into which the rod may fit closely but freely; and also other holes, downwards, b b, c c, d d, to meet the passage, a e, into which thumbscrews, b, c, d, are fitted. Let the rod now be passed through the block so as to emerge at the other side, and clamp it down firmly by the thumb-screw, b. If it be now pushed against the block with the same force as before, it is clear that the force will be arrested by the thumb-screw, b, at b and that b will become its point of application to the body. So, in like manner, may it be applied to c and d, by tightening in succession each screw, while the others are left loose. In all these cases the force is the same, and the direction the same ; but the points of application are different. But will the effects in the several cases be different? No; for the portion of the rod within the block, and extending from a to any of the points of application, performs the same part in transmitting the force from a to the point within, as the iron which was removed did when the force was first applied directly at a. The removed iron has its place filled by an equivalent of that metal in rod, and the body is virtually in its original condition. The force of the hand may still be considered applied at a, thence to be transmitted to b, or c, or d, as we please, by the portion of rod within. The second case becomes identical with the first, and the effects, therefore, must be identical in every respect; and, nothing being changed, intensity, direction, nor effect of the force, it is clearly indifferent which point we make the point of application.
Another instance is the raising of a weight by a rope. Weight and rope together make one body; and whether the lifting power be applied by engine, by horse, or by man, whether it acts over a pulley or not, every point of the strained rope may be considered a point of application. Or let the case be that of three strings attached to a ring, and pulled in different directions by three persons. It makes no difference, in this compound body of ring and strings, whether the hold taken of the latter be long or short—all their points are points of application of their respective forces. We thus see that, in all cases, we may assume that the point of application of a force is any point on so much of its line of direction as lies within the body. To suppose it applied to a point outside would be absurd; for, as we have shown, though a force may act or push through a point of empty space, it can make no impression on that point, either in the way of strain or motion, and therefore cannot come under the consideration of Mechanics.
3. The Magnitude of a Force.—To find a suitable measure of the intensity or magnitude of a force, we must also look to experience. It would be very convenient to measure forces by comparing them with weights; but this is not always practicable, and, even if it were, it would not answer all the purposes of Mechanics. I may as well, therefore, explain to you the perfect method, as that is as simple as any other. Experience teaches that a double force produces a double velocity, a treble force a treble velocity, and so on, in any body to which it is applied. But then a difficulty occurs: the same force will produce different velocities in bodies of different sizes. If it make a ball of one pound weight move at a certain rate, it will give double that speed to a half-pound ball, and half to one of two pounds. As a general rule, the greater the mass of the body, the less the speed produced. Everybody is familiar with this fact. We see, then, that if we desire to measure forces by the velocities they produce, we must try them on bodies of some fixed weight or mass. Tried on this particular mass, experience teaches that that which produces the greater velocity is the greater force. Now, the mass of matter which mechanicians choose for this purpose is that of any substance which is equal in weight to a cubic inch of distilled water. That much matter is designated the Unit of Mass, and for a reason I shall hereafter more fully explain. Imagine, then, a round ball, say of ivory, whose weight is that of a cubic inch of pure water, and suppose that several forces are in succession applied to it; the velocities they produce will be accurate measures of their intensities, or of their magnitudes.
But, then, how are the velocities to be ascertained? Clearly by the spaces the ball would move over in any given time, say the unit of time—a second—on the force being applied to it. Suppose, then, the unit ivory ball, put on a perfectly smooth floor, and then suddenly struck by a blow equal to the force you want to measure. By some means—and there are many which may be devised—manage to ascertain the distance the ball moves over in one second. That space, or length of line, will be the measure of the force; and if any number of such forces be tried in the same way and on the same ball, that which causes it to move over the greater space is the greater force, over a double space a double force, and so on.
The final result, then, is that, in considering a force in Mechanics, we must first suppose drawn within the body a line representing its direction. Then, on that line, let any point be taken for its point of application. Thirdly, on the line of direction so fixed, let as many inches be measured from the point of application as, on any scale you agree to use, represents the space the force would cause the unit ivory ball to move over in one second. Then you have a line which also in magnitude represents the force. Or in fewer words—
A Force is represented, both in magnitude and in direction, by a finite right line passing through its point of application.
If in the above explanations I have succeeded in giving you clear notions of the aim of Mechanics, and of the nature and effects of force, you are prepared for the consideration of a force, or forces, applied to a single point, which will be the subject of our next Lesson.
