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process is precisely similar to that which we have already explained for the measurement of velocity and acceleration.
We may also speak of the angular velocity of a moving plane with respect to a fixed one, as the rate of increase of the angle contained by them; but unless their line of intersection remain fixed, or at all events parallel to itself, a somewhat more laboured statement is required to give a complete specification of the motion.
55. The unit angular velocity is that of a point which describes, or would describe, unit angle about a fixed point in unit of time. The usual unit angle is (as explained in treatises on plane trigonometry) that which subtends at the centre of a circle an arc whose length is equal
180° to the radius; being an angle of = 57°29578...=57°17'44":8 nearly.
56. The angular velocity of a point in a plane is evidently to be found by dividing the velocity perpendicular to the radius-vector by the length of the radius-vector.
57. When the angular velocity is variable its rate of increase or diminution is called the Angular Acceleration, and is measured with reference to the same unit angle.
58. When one point describes uniformly a circle about another, the time of describing a complete circumference being T, we have the angle 27 described uniformly in T; and, therefore, the angular
27 velocity is
271 a planet's motion, it is useful to introduce the quantity which is then called the mean angular velocity.
T' 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 it 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 merely 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 ($ 59) directly as the angular velocity. Hence the arc of Pl, 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 (§ 9) the curvature of PQ is constant, or PQ 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 from the fixed point.
62. From $$ 61, 52, it follows that when a particle moves with acceleration towards a fixed point, varying inversely as the square of the istance, its orbit is a conic section, with his point for one focus. And conversely ($$ 47, 51, 62), 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.
63. All motion that we are, or can be, acquainted with, is Relative merely. We can calculate from astronomical data for any instant the 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 steamer 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 relatively to it moving south at the rate of twentyfive miles an hour, the actual motion of the second is thirty miles. north, and twenty-five south, per hour, i. e. five miles north. It is needless to multiply such examples, as they must occur to every
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'A G or G' in AB such that the ratio
has a constant GB G'B
value. Then, as the form of the relative GO À
path depends only upon the length and 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 enunciation, 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 geometrical 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 arrangements. 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
E of each-CA, AD, EB, and BF are rods of
B half the length of the two former, and so pivotted to them as to form a pair of equal rhombi CD, EF, whose angles can be altered C
P 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 their resultant.
69. Amongst the most important classes of motions which we have to consider in Natural Philosophy, there is one, namely, Harmonic 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 perpendicular QP drawn from its position at any
A instant to a fixed diameter AA of the circle, intersects the diameter in a point P, whose position changes by a simple harmonic motion.
Q 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
A appear 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. OA or OA' in the figure.
An arc of the circle referred to, measured from any fixed point to the uniformly moving point l, 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. The argument of this function is what we have defined as the argument of the motion.]
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. Epoch in angular measure is the angle described on the circle of reference in the period of time defined as the epoch.
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.
The Phase of a simple harmonic motion at any instant is the fraction of the whole period which has elapsed since the moving point last passed through its middle position in the positive direction.
72. Those common kinds of mechanism, for producing rectilineal from circular motion, or vice versá, 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-wheel 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 А
connected, by any of the several methods in use, with the crank, provided always the rotatory motion of the crank be uniform.
73. The velocity of a point executing a
simple harmonic motion is a simple harmonic 0
function of the time, a quarter of a period earlier in phase than the displacement, and having its maximum value equal to the
velocity in the circular motion by which the A'
given function is defined. For, in the fig., if V be the velocity in the circle, it may be represented by. OQ in a direction perpendicular to its own, and