MECHANICS 677 vbsolute nd elative. ewton s om- lents on
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[uences if second aw. When a body, originally at rest, begins to move, we con clude that force is acting on it. And when a moving body is seen to change either the speed or the direction of its motion, we conclude that this is due to force. 1 6. But there is much more than this, even in the First Law. What is "rest" 1 ? The answer must be that the term is relative. Absolute rest and absolute motion are terms to which we find it impossible to assign a meaning. Maxwell has well said (in his Matter and Motion} : " All our knowledge, both of time and place, is essentially relative. When a man has acquired the habit of putting words together, without troubling himself to form the thoughts which ought to correspond to them, it is easy for him to frame an antithesis between this relative knowledge and a so-called absolute knowledge, and to point out our ignorance of the absolute position of a point as an instance of the limitation of our faculties. Any one, however, who will try to imagine the state of a mind conscious of knowing the absolute position of a point will ever after be content with our relative knowledge." As will be seen later, the First Law gives us also a physical definition of " time," and physical modes of measuring it. 7. Newton s own comment on this law is as follows : Projectilia perse verant in motibus suis, nisi quatenus a resistentia aeris retardantur, et vi gravitatis impelluntur deorsum. Trochus, cujus partes cohserendo perpetuo retrahunt sese a motibus rectilineis, non cessat rotari, nisi quatenus ab aere retardatur. Majora autem planetarum et cometarum corpora motus suos et progressives et circulares, in spatiis minus resistentibus factos, conservant diutius. It is particularly worthy of notice that we have here the undisturbed rotation of a body about an axis intro duced as another of those "states" in which it will con tinue, in virtue of the First Law, until force acts to compel it to change that state. Also it is to be noticed that Newton adduces a hoop, whose axis is fixed in direction both in the body and in space, as an example of this new form of state maintained in virtue of inertia. Later, it will be seen that the same thing is true of a body free in space and rotating about the principal axis of greatest or of least moment of inertia through its centre of mass. 2 8. Law II. What Newton designates by the word motus is, as he has clearly pointed out, the same as is expressed by quantitas motus, that for which we now usually employ the term " momentum." Its numerical value depends not only on the rate of motion, but also on the amount of matter, or " mass," of the moving body, and is directly proportional to either of these when the other is unaltered. But it is regarded by Newton as having direction as well as magnitude. It is, in fact, what in the language of quaternions is called a " vector." The change of such a quantity may be either in numerical magnitude, or in direction, alone, or simultaneously in both. We now see what this Second Law enables us to do. For (a) Given the mass of a body, the force acting on it, and the time during which it acts, we can calculate the change of motion. This is the direct problem of dynamics of a particle. (b) Given the mass, and the change of velocity, we can 1 The words we have italicized will be seen to have very important bearings on certain old errors which even now crop up, and which have introduced one of the most inappropriate and apparently ineradicable of terms (" centrifugal force") into the usual vocabulary of our subject. 2 It is also, in a partial sense, true of a free body of which two principal axes through the centre of mass have equal moments of inertia. In that case, as we will show later, even when couples act upon the body, provided they be in planes passing through the third axis, the rate of rotation about that axis remains unaltered, though its direction in space changes. This is approximately the case of the earth. The attractions of the sun and moon on the protuberant parts about the equator produce " precession " and "nutation," but do not influence the length of the day. calculate the magnitude and direction of the force acting. This is the inverse problem. (c) We can compare, and so measure, forces by the changes of motion they produce in one and the same body. (d) We can compare the masses of different bodies by finding what changes of velocity one and the same force produces in them. (e) We can find the one force which is equivalent, in its action, to any given set of forces. For, however many changes of motion may be produced by the separate forces, they must obviously be capable of being compounded into a single change, and we can calculate what force would produce that. 3 9. Hitherto, we have spoken of the motion of a body, Necessity thus implying (except, of course, in the case of Newton s fo ^ a hoop or that of the earth) that all its parts are moving in l exactly the same way. From this point of view every body, however large, may be treated as if it were a single particle. But when the parts of a body have different velocities, as when a rigid body is rotating, or as when a non-rigid body is suffering a change of form, the-question becomes much more complex. We cannot at this stage enter into a full explanation, but will take a couple of very simple cases to *show the nature of the new difficulties, and thence the necessity for an additional law. Suppose a bullet to be thrown in any direction. If we know with what force the earth attracts it, the calculation of the path it will pursue depends on the Second Law, which gives all the necessary preliminary information. But let two bullets be tied together by a string : we know by trial that each moves, in general, in a manner very different from that in which it would move if free. The path of each is now, usually, a tortuous curve, while its free path would be plane. It is no longer subject to gravity alone but also to what is called the " tension " of the string. If we knew the amount of this tension on either of the bullets and its direction, we could calculate, by the help of the Second Law alone, all the circumstances of the motion of that bullet. But how are we to find this tension 1 Is it even the same for each bullet 1 This, if answered in the affirmative, would simplify matters considerably, but we should still require to know the amount and direction of the tension. It is clear that, without a further axiom, we cannot advance to a solution of the question. 10. Law III. Furnished with this, in addition to our Conse- previous information, we can attack the question with more quences hope. We see by this law that, whatever force be exerted ? * UI by the string on one of the bullets, an equal and opposite force, which must therefore be in the direction of the Rigid string, is exerted on the other. Still, the magnitude of con " these equal forces remains to be found. But the string in no way interferes with the motion of either bullet unless it is tight, i.e., unless the distance between the bullets is equal to the length of the string. Hence, whenever the un known force comes into play, at the same time there comes in a geometrical relation of relative position between the two bullets. This supplies the additional equation necessary for the determination of the new unknown quantity. 11. As an additional illustration, suppose the string to Variable be made of india-rubber. The Third Law tells us that the con -. tensions it exerts on the bullets are still equal and opposite. st But we no longer have the geometrical condition we had before. We have, however, what is quite sufficient, a 3 It is to be observed here that Newton s silence is as expressive as his speech. When he says "change of motion" we understand that it does not matter what the original motion was ; and, when he men tions only one force, he implies that the effect of any one force is the same whether others are also at work or not. In fact, with Newton there can be no balancing of forces, though there may be balancing of
the effects of forces, a very different thing.