Page:Encyclopædia Britannica, Ninth Edition, v. 12.djvu/52

42 HODOGRAPH From this definition we have the following important fundamental property which belongs to all hodographs, viz., that at any point the tangent to the hodograph is parallel to the direction, and the velocity in the hodograph equal to the magnitude of the resultant acceleration at the corre sponding point of the orbit. This will be evident if we consider that, since radii vectores of the hodograph represent velocities in the orbit, the elementary arc between two con secutive radii vectores of the hodograph represents the velocity which must be compounded with the velocity of the moving point at the beginning of any short interval of time to get the velocity at the end of that interval, that is ti say, represents the change of velocity for that interval. Hence the elementary arc divided by the element of time is the rate of change of valocity of the moving-point, or in other words, the velocity in the hodograph is the accelera tion in the orbi&amp;gt;. Analytically thus (Thomson and Tait, Xat. Phil.}: Let x, y, z be the coordinates of P in the orbit, f, y, those of the corre sponding point T in the hodograph, then dx _dy f_dz therefore dl- (1). dt* ~dt* df 2 Also, if s be the arc of the hodograph, d* /{ ( ^ 2 f dr &amp;gt;&quot; fd 2 di = = V dt) + dt) + dt) di? . . (2). Equation (1) shows that the tangent to the hodograph is parallel to the line of resultant acceleration, and (2) that the velocity in the hodograph is equal to the acceleration. Every orbit must clearly have a hodograph, and, con versely, every hodograph a corresponding orbit ; and, theoretically speaking, it is possible to deduce the one from the other, having given the other circumstances of the motion. We give a few examples : 1. For uniform motion in a straight line the hodograph is easily seen to be a point. 2. For uniform or variable acceleration in a straight line the hodograph is the line described by a point moving with uniform or variable velocity. 3. For uniform circular mo tion the hodograph is also a circle. In this case it may be useful to show how the con ception of the hodograph leads easily to the ordinary expres sion for the so-called centri fugal force. Let P (fig. 2) describe the circumference PPjP,, with uni form velocity v, and from the centre draw OT, OT 1, OT 2 , &c., equal to each other and parallel to the tangents at P, PI. P 2 respectively, then Fi S- 2 - TT,T 2 is the hodograph circle. Also let a equal the acceleration of P, which also equals the velocity of T ; then, since T describes the hodograph uniformly in the same time that P describes the orbit, we have v_ OP _ r_ _ t-s !=s /&quot;m &. a Ol v r It is evident that the tangent at T is parallel to PO the direction of acceleration at P. 4. For simple harmonic motion the hodograph is also simple harmonic motion, and similarly for elliptic harmonic motion. For the former we have .9 = a cos (nt + e) ; .-. v = s= -na sin which indicates simple harmonic motion with changed amplitude and phase. 5. For parabolic motion the hodograph is a straight line. Let OF (fig. 3) be the velocity of projection. Resolve it verti cally and horizontally; the horizontal component OH is constant, so that the hodograph must be the vertical line FHT. The velocity at any point P of the parabola is evidently represented by the lino OT drawn parallel to the tangent at P. Analytically thus : If x, y be the coordinates of the moving point and f and 77 those of the hodograph, we have =, y = -/(/being the vertical acceleration) ; 1 = 0, i - -/; which indicates a vertical straight line velocity /. described with uniform Fi S- 3 - Fig. 4. 6. For central forces the hodograph will vary with the law of acceleration towards the centre. Hamilton showed that, for the Newtonian law of the inverse square of the distance, the hodooraph is always a circle, and for that reason he designated that law the laio of the circular hodograph. Assuming the centre of force as origin for the hodograph, we see that, from definition, the tangent and radius vector at any point P of the orbit are respectively parallel to the radius vector and tan gent at the corresponding point T of the hodograph (fig. 4). These four lines thus enclose a parallelogram PT, whose shape is constantly changing, although its area is constant, being equal to the constant rectangle YT contained by the velocity and the perpendicular on the tangent. As usual, let TY = vp = h. Also the angle between two consecutive tangents to the hodograph is equal to the angle between the corresponding radii vectores of the orbit, and hence, if 6 be the angle between OP and some initial line, and s the arc of the hodograph, the ordinary formula for curvature gives 1 __ M _ 9 p (Is $ but h = r&quot;e, and = velocity in hodograph = acceleration in orbit = P (sav) : T 7 &amp;gt; ^ a most useful expression for the radius of curvature of the hodo graph to any central orbit. If the acceleration vary inversely as the square of the distance, P = -% &amp;gt; where M is the mass at the centre acting upon unit mass in the orbit. Substituting in (3), we get M p = - = constant. n Hence, for this law of force, and evidently for it only, the hodo graph is a circle. Assuming that the hodograph is a circle, we can show that the orbit for this law of force must be a conic section. Let CTB (fig. 5) be the hodograph circle, the origin, and H the centre ; and let the OTT rati HT = * DraW Or&amp;gt; P arallel to tlie angcnt at T and, there fore, perpendicular to TG . Take OP such that the parallelogram contained by OP and OT = a constant = h. Draw OA perpendi cular to CB ; and let POA-fl; then, OP.TG-&; OP. = A; but TH = p = ^, and HG = HOcos = M + MA h h ) - M 1 + e cos