MECHANICS 701 Hence the element of work is This must be integrated between the limits and W of -fc = Z to l- x= > aild tllc rcsult is which, when E is very great compared with W, gives the previous result. Further Comments on the First Two Laivs of Motion, Meal 114. We are now prepared to consider, more closely t sure- than we could at starting, the bearing of the various clauses
of of each of Newton s Laws. Thus, from the first law we
il!< may draw the following immediate consequences. The times during which any particular body, not com pelled by force to alter the speed of its motion, passes through equal distances are equal. And, again, every other body in the universe, not compelled by force to alter the speed of its motion, moves over equal distances in successive intervals, during which t-he particular chosen body moves over equal distances. The earth, in its rota tion about its axis, presents us with a case of motion in which the condition of not being compelled by force to alter its speed is more nearly fulfilled than in any other which we can easily or accurately observe. Hence the numerical measurement of time practically rests on defin ing " equal intervals of time " as times during ivhich the earth turns through equal angles. 115. It has been objected to this statement that we be<nn by defining uniform motion by the description of equal spaces in equal times, and then employ this defini tion as a mode of measuring equal times. The objection, however, is not valid ; for, if we agree to measure equal intervals by the undisturbed motion of any one physical mass, we find that in the successive intervals so determined all other absolutely free physical masses describe successive equal spaces. isure H6. Again, from the second law we see that, if we brce. multiply the change of velocity, geometrically determined, by the mass of the body, we have the change of motion ( 99) referred to in the law as the measure of the force which produces it. In the statement of the second law there is nothing said about the actual motion of the body before it was acted on by the force ; the same force will produce precisely the sains change of motion in a body whether the body be at rest or in motion with any velocity whatever. Again, nothing is said as to the body being under the action of one force only ; so that we may logically put part of the second law in the following (apparently) amplified form : When any forces whatever act on a body, then, whether the body be originally at rest or moving with any velocity and in any direction, each force produces in the body the exact change of motion ivhich it would have produced if it had acted singly on the body originally at rest. m- 117. Since now forces are measured by the changes of sition mo tion they produce, and their directions assigned by the forces. Directions in which these changes are produced, and since the changes of motion of one and the same body are in the directions of and proportional to the changes of velocity, a single force, measured by the resultant change of velocity, and in its direction, will be the equivalent of any number of simultaneously acting forces. Hence The, resultant of any number of forces (applied at one point} is to be found by the same geometrical process as the resultant of any number of simultaneous velocities. of ma
From this follows at once ( 30) the construction of the parallelogram of forces " for finding the resultant of
- wo forces acting at the same point, and the " polygon of
forces " for the resultant of any number of forces acting at a point. And, so far as a single particle is concerned, we lave at once the whole subject of Statics. 118. The second law gives us the means of measuring Measure iorce, and also of measuring the mass of a body. For, if we consider the actions of various forces upon f l (
- he same body for equal times, we evidently have changes
of velocity produced which are proportional to the forces. The changes of velocity, then, give us in this case the means of comparing the magnitudes of different forces. Thus the speeds acquired in one second by the same mass (falling freely) at different parts of the earth s surface give us the relative amounts of the earth s attraction at these places. Again, if equal forces be exerted on different bodies, the changes of velocity produced in equal times must be inversely as the masses of the various bodies. This is approximately the case, for instance, with trains of various lengths drawn by the same locomotive. Again, if we find a case in which different bodies, eacli Gravity, acted on by a force, Acquire in thesame time the same changes of velocity, the forces must be proportional to the masses of the bodies. This, when the resistance of the air is removed, is the case of falling bodies; and from it we conclude that the weight of a body in any given locality, or the force with ivhich the earth attracts it, is proportional to its mass. This is no mere truism, but an important part of the grand Law of Gravitation. Gravity is not, like magnetism for instance, a force depending on the quality as well as on the quantity of matter in a particle. 119. It appears, lastly, from this law that every Transla- theorem of kinematics connected with acceleration has its j; 1 | 1 J r011 counterpart in kinetics. Thus, for instance ( 36), we^^ see that the force under which a particle describes any illto curve may be resolved into two components, one in the kinetics, tangent to the curve, the other towards the centre of cur vature, their magnitudes being the rate of change of momentum in the direction of motion, and the product of the momentum into the angular velocity about the centre of curvature, respectively. In the case of uniform motion, the first of these vanishes, or the whole force is perpen dicular to the direction of motion. When there is no force perpendicular to the direction of motion, there is no curvature, or the path is a straight line. Hence, if we resolve the forces acting on a particle of mass in, whose coordinates are x, y, z, into three rectangular components X, Y, Z, we have the equations originally given by Maclaurin, viz., d< * x -v ^_v = Z m dt?~ m dt?~ m dP In several of the examples which follow, these equations will be somewhat simplified by assuming unity as the mass of the moving particle. When this cannot be done, it is sometimes convenient to assume X, Y, Z as the component forces on unit mass, and the previous equations become m ^=mX &c from which m may of course be omitted. [Some confusion is often introduced by the division of forces into "accelerating" and "moving" forces : and it is even stated occa sionally that the former are of one, and the latter of four linear dimensions. The fact is, however, that an equation such as may be interpreted either as dynamical or as merely kinematics!. If kinematical, the meanings of the terms are obvious ; if dynamical, the unit of mass must be understood as a factor on the left-hand side, and in that case X is the ^-component, per unit of mass, ot the whole force exerted on the moving body.]
If there be no acceleration, we have of course equilibrium among