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Relative motion.

If x, y, z, and c', y, z, be the co-ordinates of two points referred to axes regarded as fixed; and ļ, no § their relative co-ordinates—we have

$ = c - a, n= - , = 3 - 2,
and, differentiating,
dě dx' da

dt dt


which give the relative, in terms of the absolute, velocities; and

do dvd du
dta dt dta


proving our assertion about relative and absolute accelerations.

The corresponding expressions in polar co-ordinates in a plane are somewhat complicated, and by no means convenient. The

student can easily write them down for himself. 47. The following proposition in relative motion is of considerable importance

Any two moving points describe similar paths relatively to
each other, or relatively to any point which divides in a con-
stant ratio the line joining them.
Let A and B be any simultaneous positions of the points.

Take G or G' in AB such that the ratio

has a constant value. Then

GB GB as the form of the relative 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 Gand 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.



48. As a good example of relative motion, let us consider that of the two points involved in our definition of the curve of pursuit, $ 40. Since, to find the relative position and motion of the pursuer with regard to the pursued, we must impress on both a velocity equal and opposite to that of the latter, we see



at once that the problem becomes the same as the following. A Relative boat crossing a stream is impelled by the oars with uniform velocity relatively to the water, and always towards a fixed point in the opposite bank; but it is also carried down stream at a uniform rate; determine the path described and the time of crossing. Here, as in the former problem, there are three cases, figured below. In the first, the boat, moving faster than the current, reaches the desired point; in the second, the velocities of boat and stream being equal, the boat gets across only after an infinite time-describing a parabola—but does not land at the desired point, which is

=3 indeed the focus of the parabola, while the landing point is the vertex. In the third case, its proper velocity being less than that of the water, it never reaches the other bank, and is carried indefinitely down stream. The comparison of the figures in § 40 with those in the present section cannot fail to be instructive. They are drawn to the same scale, and for the same relative velocities. The horizontal lines represent the farther bank of the river, and the vertical lines the path of the boat if there were no current.



We leave the solution of this question as an exercise, merely noting that the equation of the curve is

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in one or other of the three cases, according as e is >,

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When e=1 this becomes

ye = a* - 2ax, the parabola.

The time of crossing is


which is finite only for e<1, because of course a negative value
is inadmissible.

Relative motion.

49. Another 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 mean time, we take a different form of enunciation, which, however, leads to precisely the same result.

Find 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.

Take the former case first: let a be the radius of the relative circular orbit, and w the angular velocity in it, v being the velocity of its centre along the straight line.

The relative co-ordinates of the point in the circle are a cos wt and a sin wt, and the actual co-ordinates of the centre are vt and 0. Hence for the actual path

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Hence $ = sin-"* +, an equation which, by giving different values to v and w, may be niade to represent the cycloid itself, or either form of trochoid. See $ 92.

For the epicycloids, let b be the radius of the circle which B describes about A, w, the angular velocity; a the radius of A's path, w the angular velocity.

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x and

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X = a cos wt +b cos w, t, y = a sin wt + b sin w, t, which, by the elimination of t, give an algebraic equation between


whenever w and are commensurable. Thus, for w, = 2w, suppose wt = 0, and we have

= a cos 0 + b cos 20, y = a sin 0 + b sin 20, or, by an easy reduction,

(acé + ya - b) = a' {(x+b)* + y"}.






Put a b for x, i.e., change the origin to a distance AB to the Relative left of O, the equation becomes

a’ (2c2 + y^)= (2c2 + y26x), or, in polar co-ordinates,

a= (r-- 26 cos 0)*, p = a + 26 cos 0, and when 2b = a, r = a (1 + cos 6), the cardioid.

(1+cos ), the cardioid. (See $ 94.) 50. As an additional illustration of this part of our subject, Resultant 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.

51. 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 an 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 of eachCA, AD, EB, and BF, are rods of half the с

F length of the two former, and so pivoted to them as to form a pair of equal rhombi


D 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 their resultant.



52. Amongst the most important classes of motions which Harmonic we have to consider in Natural Philosophy, there is one, namely, Harmonic Motion, which is of such immense use, not only in


Simple harmonic motion.


Harmonic ordinary kinetics, but in the theories of sound, light, heat, etc.,

that we make no apology for entering here into considerable
detail regarding it.
53. Def. When a point @ moves uniformly in a circle, the

perpendicular QP drawn from its position
at any instant to a fixed diameter AA' of

the circle, intersects the diameter in a point P

P, whose position changes by a simple har0

monic motion.

Thus, if a planet or satellite, or one of

the constituents of a double star, supposed A

to move uniformly in a circular orbit about its primary, 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.

54. 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


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.

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