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59. When a point moves. uniformly, in a straight line its angular velocity evidently diminishes as it recedes from the point about which the angles are measured, and it may easily be shown that at varies inversely, as the square of the distance from this point. The same proposition is true for any path, when the acceleration is towards the point about which the angles are measured : being werely a different mode of stating the result of $ 48.
60. The intensity of heat and light emanating from a point, or from a uniformly radiating spherical surface, diminishes according to the inverse square of the distance from the centre. Hence the rate at which a planet receives heat and light from the sun varies in simple. proportion to the angular.velocity of the radius-vector. Hence the whole heat and light received by the planet in any time is proportional to the whole angle turned through by its radius-vector in the same time.
61. A further instance of this use of the idea of angular velocity may now. be given, to solve the problem of finding the hodograph. ($ 35) for any case of motion in which the acceleration is directed to a fixed point, and varies inversely as the square of the distance from. that point. The velocity of P, in the hodograph PQ, being the acceleration in the orbit,, varies inversely as the square of the radius-vector; and therefore (8 59) directly as the angular velocity. Hence the arc of PQ, described in any time, is proportional to the corresponding angle-vector in the orbit, i.e.. to the angle through which the tangent to PQ has turned.
Hence (89) the curvature of PQ is constant, or P.Q is a circle.
This demonstration, reversed, proves that if the hodograph be a circle, and the acceleration be towards a fixed point, the acceleration varies inversely as the square of the distance of the moving point 0 from the fixed point.
62. From SS 61, 52, it follows that when a particle. moves with acceleration towards a fixed point, varying. inversely as the square of the distance, its orbit is a conic section, with this point for one focus. And conversely (SS 47, 51, 52), if the orbit be a conic section, the acceleration, if towards either focus, varies inversely as the square of the distance: or, if a point moves in a conic section, describing equal areas in equal times by a radius-vector through a focus, the acceleration is always towards this focus, and varies inversely as the square of the distance. Compare this with the first and second of Kepler's Laws, $ 48.
63. All motion that we are, or can be, acquainted with, is Relative merely. We can calculate from astronomical data for any instant athe direction in which, and the velocity with which, we are moving on account of the earth's diurnal rotation. We may compound this with the (equally calculable) velocity of the earth in its orbit. This resultant again we may compound with the (roughly-known) velocity
of the sun relatively to the so-called fixed stars; but, even if all these elements were accurately known, it could not be said that we had attained any idea of an absolute velocity; for it is only the sun's relative motion among the stars that we can observe; and, in all probability, sun and stars are moving on it may be with inconceivable rapidity) relatively to other bodies in space. We must therefore consider how, from the actual motions of a set of bodies, we may find their relative motions with regard to any one of them; and how, having given the relative motions of all but one with regard to the latter, and the actual motion of the latter, we may find the actual motions of all. The question is very easily answered. Consider for a moment a number of passengers walking on the deck of a steamer. Their relative motions with regard to the deck are what we immediately observe, but if we compound with these the velocity of the steamer itself we get evidently their actual motion relatively to the earth. Again, in order to get the relative motion of all with regard to the deck, we eliminate the motion of the stéamer altogether; that is, we alter the velocity of each relatively to the earth by compounding with it the actual velocity of the vessel taken in a reversed direction.
Hence to find the relative motions of any set of bodies with regard to one of their number, imagine, impressed upon each in composition: with its own motion, a motion equal and opposite to the motion of that one, which will thus be reduced to rest, while the motions of the others will remain the same with regard to it as before.
Thus, to take a very simple example, two trains are running in opposite directions, say north and south, one with a velocity of fifty, the other of thirty, miles an hour. The relative velocity of the second with regard to the first is to be found by imagining impressed on both a southward velocity of fifty miles an hour; the effect of this being to bring the first to rest, and to give the second a southward velocity of eighty miles an hour, which is the required relative motion.
Or, given one train moving north at the rate of thirty miles an hour, and another moving west at the rate of forty miles an hour. The motion of the second relatively to the first is at the rate of fifty miles an hour, in a south-westerly direction inclined to the due west direction at an angle of tan-' . It is needless to multiply such examples, as they must occur to every one.
64. Exactly the same remarks apply to relative as compared with absolute acceleration, as indeed we may see at once, since accelerations are in all cases resolved and compounded by the same law as : velocities.
65. The following proposition in relative motion is of considerable importance:
Any two moving points describe similar paths relatively to each other and relatively to any point which divides in a constant ratio the line joining them.
Let A and B be any simultaneous positions of the points. Take
GA G or Gin AB such that the ratio or
has a constant
GB G'B value. Then, as the form of the relative path depends only upon the length and G'
B direction of the line joining the two points at any instant, it is obvious that these will be the same for A with regard to B, as for B with regard to A, saving only the inversion of the direction of the joining line. Hence B's.path about A is A's about B turned through two right angles. And with regard to G and G' it is evident that the directions remain the same, while the lengths are altered in a given ratio ; but this is the definition of similar curves.
66. An excellent example of the transformation of relative into absolute motion is afforded by the family of Cycloids. We shall in a future section consider their mechanical description, by the rolling of a circle on a fixed straight line or circle. In the meantime, we take a different form of statement, which however leads to precisely. the same result.
The actual path of a point which revolves uniformly in a circle about another point-the latter moving uniformly in a straight line or circle in the same plane-belongs to the family of Cycloids.
67. As an additional illustration of this part of our subject, we may define as follows:
If one point A executes any motion whatever with reference to a second point B; if B executes any other motion with reference to a third point C; and so on--the first is said to execute, with reference to the last, a movement which is the resultant of these several movements.
The relative position, velocity, and acceleration are in such a case the geometfical resultants of the various components combined according to preceding rules.
68. The following practical methods of effecting such a combination in the simple case of the movements of two points are useful in scientific illustrations and in certain mechanical arrange. ments. Let two moving points be joined by a uniform elastic string; the middle point of this string will evidently execute a movement which is half the resultant of the motions of the two points. But for drawing, or engraving, or for other mechanical applications, the following method is preferable ;
CF and ED are rods of equal length moving freely round a pivot .at P; which passes through the middle point
B of each-CA, AD, EB, and BF are rods of half the length of the two former, and so pivoted to them as to form a pair of equal thombi CD, EF, whose angles can be altered: at will. Whatever motions, whether in a plane, or in space of three dimensions, be given to A and B, P will evidently be subjected to half
D thetr resultant
69. Amongst the most important classes of motions which we have to consider in Natural Philosophy, there is one, namely, Har. monic Motion, which is of such immense use, not only in ordinary kinetics, but in the theories of sound, light, heat, etc., that we make no apology for entering here into some little detail regarding it. 70. Def. When a point moves uniformly in a circle, the per. А
pendicular QP drawn from its position at any instant to a fixed diameter AA of the circle, intersects the diameter in' a point P, whose position changes by a simple harmonic motion.
Thus, if a planet or satellite, or one of the constituents of a double star, be supposed to move uniformly in a circular orbit about its primary, and be viewed from a very distant position in the plane of its orbit
, it will appear A
to move backwards and forwards in a straight line with a simple harmonic motion. This is nearly the case with such bodies as the satellites of Jupiter when seen from the earth.
Physically, the interest of such motions consists in the fact of their being approximately those of the simplest vibrations of sounding bodies such as a tuning-fork or pianoforte-wire; whence their name; and of the various media in which waves of sound, light, heat, etc., are propagated.
71. The Amplitude of a simple harmonic motion is the range on one side or the other of the middle point of the course, i. e. Õ A or QA' in the figure
An arc of the circle referred to, or any convenient angular reckoning of it, measured from any fixed point to the uniformly moving point Q, is the Argument of the harmonic motion.
[The distance of a point, performing a simple harmonic motion, from the middle of its course or range, is a simple harmonic function of the time; that is to say
a cos (nt – ë), where a, n, e are constants, and t represents time. The argument of this function is what we have defined as the argument of the motion. In the formula above, the argument is nt - e.]
The Epoch in a simple harmonic motion is the interval of time which elapses from the era of reckoning till the moving point first comes to its greatest elongation in the direction reckoned as positive, from its mean position or the middle of its range. [In the formula above, put in the form
e cos (1-3) is the epoch.) Epoch in angular measure is the angle described on the circle of reference in the period of time defined as the epoch. [In the formula, e is the epoch in angular measure.]
The Period of a simple harmonic motion is the time which elapses
from any instant until the moving point again moves in the same direction through the same position, and is evidently the time of revolution in the auxiliary circle. [In the formula the period is
The Phase of a simple harmonic motion at any instant is an expression used to designate the part of its whole period which it has reached. It is borrowed from the popular expression phases of the moon. Thus for Simple Harmonic Motion we may call the first or zero-phase that of passing through the middle position in the positive direction. Then follow the successive phases quarter-period, half-period, three-quarters-period, and complete period or return to zero-phase. Sometimes it is convenient to reckon phase by a number or numerical expression, which may be either a reckoning of angle or a reckoning of time, or a fraction or multiple of the period. Thus the positive maximum phase may sometimes be called the 90° phase or the phase, or the three-hour phase, if the period be 12 hours, or the quarter-period phase. Or, again, the phase of half way down from positive maximum may be described as the 120° phase or the
phase, or the į period phase. This particular way of specifying 3 phase is simply a statement of the argument as defined above and measured from the point corresponding to positive motion through the middle position.
72. Those common kinds of mechanism, for producing rectilineal from circular motion, or vice versa, in which a crank moving in a circle works in a straight slot belonging to a body which can only move in a straight line, fulfil strictly the definition of a simple harmonic motion in the part of which the motion is rectilineal, if the motion of the rotating part is uniform.
The motion of the treadle in a spinning-whed approximates to the same condition when the wheel moves uniformly; the approximation being the closer, the smaller is the angular motion of the treadle and of the connecting string. It is also approximated to more or less closely in the motion of the piston of a steam-engine connected, by any of the several methods in use, with the crank, provided always the ro
Å tatory motion of the crank be uniform.
73. The velocity of a point executing a Q Р simple harmonic motion is a simple harmonic function of the time, a quarter of a period
o earlier in phase than the displacement, and having its maximum value equal to the velocity in the circular motion by which the given function is defined.
A For, in the fig., if V be the velocity in the circle, it may be represented by in' a direction perpendicular to its own, and