radius r. curvature Infinite tortuosity will be easily understood, by considering Integral a helix, of inclination a, described on a right circular cylinder of of a curve 1 that of The curvature in a circular section being 2 p cos3 a (compare sin a cos a r • the helix is, of course, π the curvature and tortuosity 4 tan a × curvature. Hence, if a = The tortuosity is helix being 2πr tan a ==- 2π √ 1 tortuosity, being - tan a or ρ 1 Let the curvature be denoted by ρ so that cos2a α= or , p = Let p ρ √7 (1–2)3, is, in the limit, 2 √pr, which is infinitely small. Thus the motion of a point in the ρ (1–2) 3, √pr will in the limit be a mean proportional between the curvature of the circular section The acceleration (or force) required to produce such a motion of a point (or material particle) will be afterwards investigated (§ 35 d.). line. 14. A chain, cord, or fine wire, or a fine fibre, filament, or Flexible hair, may suggest what is not to be found among natural or artificial productions, a perfectly flexible and inextensible line. The elementary kinematics of this subject require no investigation. The mathematical condition to be expressed in any case of it is simply that the distance measured along the line from any one point to any other, remains constant, however the line be bent. 15. The use of a cord in mechanism presents us with many practical applications of this theory, which are in general extremely simple; although curious, and not always very easy, geometrical problems occur in connexion with it. We shall say nothing here about the theory of knots, knitting, weaving, Flexible line. Evolute. plaiting, etc., but we intend to return to the subject, under vortex-motion in Hydrokinetics. 16. In the mechanical tracing of curves, a flexible and inextensible cord is often supposed. Thus, in drawing an ellipse, the focal property of the curve shows us that by fixing the ends of such a cord to the foci and keeping it stretched by a pencil, the pencil will trace the curve. By a ruler moveable about one focus, and a string attached to a point in the ruler and to the other focus, the hyperbola may be described by the help of its analogous focal property; and so on. 17. But the consideration of evolutes is of some importance in Natural Philosophy, especially in certain dynamical and optical questions, and we shall therefore devote a section or two to this application of kinematics. Def. If a flexible and inextensible string be fixed at one. point of a plane curve, and stretched along the curve, and be then unwound in the plane of the curve, its extremity will describe an Involute of the curve. The original curve, considered with reference to the other, is called the Evolute. 18. It will be observed that we speak of an involute, and of the evolute, of a curve. In fact, as will be easily seen, a curve can have but one evolute, but it has an infinite number of involutes. For all that we have to do to vary an involute, is to change the point of the curve from which the tracing point starts, or consider the involutes described by different points of the string, and these will, in general, be different curves. The following section shows that there is but one evolute. Q 19. Let AB be any curve, PQ a portion of an involute, pP, q? positions of the free part of the string. It will be seen at once that these must be tangents to the arc AB at p and q. Also (see § 90), the string at any stage, as pP, revolves about p. Hence pP is normal to the curve PQ. And thus the evolute of PQ is a definite curve, P B viz., the envelope of the normals drawn at every point of PQ, q p or, which is the same thing, the locus of the centres of curva- Evoluto. ture of the curve PQ. And we may merely mention, as an obvious result of the mode of tracing, that the arc pq is equal to the difference of qQ and pP, or that the arc pA is equal to pP. 20. The rate of motion of a point, or its rate of change of Velocity. position, is called its Velocity. It is greater or less as the space passed over in a given time is greater or less: and it may be uniform, i.e., the same at every instant; or it may be variable. Uniform velocity is measured by the space passed over in unit of time, and is, in general, expressed in feet per second; if very great, as in the case of light, it is sometimes popularly reckoned in miles per second. It is to be observed, that time. is here used in the abstract sense of a uniformly increasing quantity—what in the differential calculus is called an independent variable. Its physical definition is given in the next chapter. 21. Thus a point, which moves uniformly with velocity v, describes a space of v feet each second, and therefore vt feet in t seconds, t being any number whatever. Putting s for the space described in t seconds, we have s = vt. Thus with unit velocity a point describes unit of space in unit. of time. 22. It is well to observe here, that since, by our formula, we have generally and since nothing has been said as to the magnitudes of s and t, we may take these as small as we choose. Thus we get the same result whether we derive v from the space described in a million seconds, or from that described in a millionth of a second. This idea is very useful, as it makes our results intelligible when a variable velocity has to be measured, and we find ourselves obliged to approximate to its value by considering the space described in an interval so short, that during its lapse the velocity does not sensibly alter in value. Velocity. 23. When the point does not move uniformly, the velocity is variable, or different at different successive instants; but we define the average velocity during any time as the space described in that time, divided by the time, and, the less the interval is, the more nearly does the average velocity coincide with the actual velocity at any instant of the interval. Or again, we define the exact velocity at any instant as the space which the point would have described in one second, if for one second its velocity remained unchanged. That there is at every instant a definite value of the velocity of any moving body, is evident to all, and is matter of everyday conversation. Thus, a railway train, after starting, gradually increases its speed, and every one understands what is meant by saying that at a particular instant it moves at the rate of ten or of fifty miles an hour,—although, in the course of an hour, it may not have moved a mile altogether. Indeed, we may imagine, at any instant during the motion, the steam to be so adjusted as to keep the train running for some time at a perfectly uniform velocity. This would be the velocity which the train had at the instant in question. Without supposing any such definite adjustment of the driving power to be made, we can evidently obtain an approximation to this instantaneous velocity by considering the motion for so short a time, that during it the actual variation of speed may be small enough to be neglected. 24. In fact, if v be the velocity at either beginning or end, or at any instant of the interval, and s the space actually S described in time t, the equation v= is more and more nearly t true, as the velocity is more nearly uniform during the interval t; so that if we take the interval small enough the equation may be made as nearly exact as we choose. Thus the set of values Space described in one second, Ten times the space described in the first tenth of a second, hundredth and so on, give nearer and nearer approximations to the velocity at the beginning of the first second. The whole foundation of the differential calculus is, in fact, contained in this simple Velocity. question, "What is the rate at which the space described increases?" i.e., What is the velocity of the moving point? Newton's notation for the velocity, i. e. the rate at which s increases, or the fluxion of s, is s. This notation is very convenient, as it saves the introduction of a second letter. Let a point which has described a space s in time t proceed to describe an additional space ds in time St, and let v, be the greatest, and v, the least, velocity which it has during the interval St. Then, evidently, 2 2 But as St diminishes, the values of v, and v, become more and 29 > v ds of velocity. 25. The preceding definition of velocity is equally applica- Resolution ble whether the point move in a straight or curved line; but, since in the latter case the direction of motion continually changes, the mere amount of the velocity is not sufficient completely to describe the motion, and we must have in every such case additional data to remove the uncertainty. In such cases as this the method commonly employed, whether we deal with velocities, or as we shall do farther on with accelerations and forces, consists mainly in studying, not the velocity, acceleration, or force, directly, but its components parallel to any three assumed directions at right angles to each other. Thus, for a train moving up an incline in a NE direction, we may have given the whole velocity and the steepness of the incline, or we may express the same ideas thus--the train is moving simultaneously northward, eastward, and upwardand the motion as to amount and direction will be completely known if we know separately the northward, eastward, and upward velocities-these being called the components of the whole velocity in the three mutually perpendicular directions N, E, and up. |