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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

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'A

Gor G'in AB such that the ratio

B

or

has a constant

GB G'B

value. Then, as the form of the relative G' A G 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.

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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 :

E

CF and ED are rods of equal length moving freely round a pivot at P, which passes through the middle point of each-CA, AD, EB, and BF are rods of 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 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 Ꭺ

their resultant.

D

F

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 Q moves uniformly in a circle, the per

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.

A

D

Α'

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 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 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. 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 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-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 connected, by any of the several methods in use, with the crank, provided always the rotatory motion of the crank be uniform.

A

Α'

73. The velocity of a point executing a simple harmonic motion is a simple harmonic 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 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

therefore by OP and PQ in directions perpendicular to those lines. That is, the velocity of P in the simple harmonic motion is

PQ

00

V or

Ꮴ оо

PQ; which, when P passes through O, becomes V.

74. The acceleration of a point executing a simple harmonic motion is at any time simply proportional to the displacement from the middle point, but in opposite direction, or always towards the middle point. Its maximum value is that with which a velocity equal to that of the circular motion would be acquired in the time in which an arc equal to the radius is described.

For in the fig., the acceleration of Q (by § 36) is

V2

QQ along QO. Supposing, for a moment, QO to represent the magnitude of this acceleration, we may resolve it into QP, PO. The acceleration of P is therefore represented on the same scale by PO. Its magnitude V2 PO V2 PO, which is proportional to PO, and

is therefore

=

00 00 ᎤᏅ v2

has at A its maximum yalue,

an acceleration under which the

1

velocity

Qo'
would be acquired in the time.

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have in simple harmonic motion

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where T is the time of describing the circle, or the period of the harmonic motion.

S

R

A

P

75. Any two simple harmonic motions in one line, and of one period, give, when compounded, a single simple harmonic motion; of the same period; of amplitude equal to the diagonal of a parallelogram described on lengths equal to their amplitudes measured on lines meeting at an angle equal to their difference of epochs; and of epoch differing from their epochs by angles equal to those which this diagonal makes with the two sides of the parallelogram. P and P' be two points executing simple harmonic motions of one period, and in one line B'BCAA'. Let Q and

Let

be the uniformly moving points in the relative circles. On CQ and CO

P

C

B

B'

describe a parallelogram SQCQ; and through S draw SR perpendicular to B'A' produced. We have obviously P'R = CP (being projections of the equal and parallel lines Q'S, CQ, on CR). Hence CR CP+ CP'; and therefore the point R executes the

=

resultant of the motions P and P'. But CS, the diagonal of the parallelogram, is constant (since the angular velocities of CQ and CO are equal, and therefore the angle QCQ is constant), and revolves with the same angular velocity as CQ or CQ; and therefore the resultant motion is simple harmonic, of amplitude CS, and of epoch exceeding that of the motion of P, and falling short of that of the motion of P', by the angles QCS and SCQ' respectively.

76. [The construction described in the preceding section exhibits the resultant of two simple harmonic motions, whether of the same period or not. Only, if they are not of the same period, the diagonal of the parallelogram will not be constant, but will diminish from a maximum value, the sum of the component amplitudes, which it has at the instant when the phases of the component motions agree; to a minimum, the difference of those amplitudes, which is its value when the phases differ by half a period. Its direction, which always must be nearer to the greater than to the less of the two radii constituting the sides of the parallelogram, will oscillate on each side of the greater radius to a maximum deviation amounting on either side to the angle whose sine is the less radius divided by the greater, and reached when the less radius deviates by this together with a quarter circumference, from the greater. The full period of this oscillation is the time in which either radius gains a full turn on the other. The resultant motion is therefore not simple harmonic, but is, as it were, simple harmonic with periodically increasing and diminishing amplitude, and with periodical acceleration and retardation of phase. This view is most appropriate for the case in which the periods of the two component motions are nearly equal, but the amplitude of one of them much greater than that of the other.

To find the time and the amount of the maximum acceleration or retardation of phase, let CA be equal to the greater half-amplitude.

From A as centre, with AB the less halfamplitude as radius, describe a circle. CB touching this circle represents the most deviated resultant. Hence CBA is a right angle; A and

sin BCA =

AB
CA

77. A most interesting application of this case of the composition of harmonic motions is to the lunar and solar tides; which, except in tidal rivers, or long channels or bays, follow each very nearly the simple harmonic law, and produce, as the actual result, a variation of level equal to the sum of variations that would be produced by the two causes separately.

The amount of the lunar tide in the equilibrium theory is about 2.1 times that of the solar. Hence the spring tides of this theory are 3·1, and the neap tides only 1.1, each reckoned in terms of the solar tide; and at spring and neap tides the hour of high water is that of the lunar tide alone. The greatest deviation of the actual

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