Integrating, and determining the constants so that, when =0, t = and i> = V, we obtain tan - l ~r- = tan 1 k k Let T be the time at which the speed becomes zero, and h the corresponding value of x, then i. V P T = A tan -i * ma ft* l </ A Z<7 After this the particle begins to return ; the resistance therefore acts against gravity, and the equation of motion is dt Integrating, and determining the constants so that, when a = /i and < = T, we obtain ff (ft -as)- log *L. A; 2 AT - tr (It must be remembered that u is now negative.) Let U be the speed with which the particle returns to the point of projection ; then, putting z = in the latter equation, we obtain or, substituting for h its value, whence JL_A TJ a V 2 rminal It is to be observed that k is the " terminal velocity," as it is ocity. called, i.e., the speed to which that of a falling body continually tends, whether its original speed have been greater or less than this limit. It is to be observed also that (strictly) we should write g (1 - a) for g, where a is the specific gravity of air, to take account of the apparent loss of weight of a raindrop on account of immersion in air. When k is very large, i.e., the absolute amount of resistance very small, as in the case of air, the general integrals in the second case above become, by expanding the logarithms, 2*7, 2y 2 V s k k r 3& ! i) 2 i; 4 mple mdu- of which the terms independent of Jc are 0--00-T), v 2 = 2#(fr - x). These, if we remember that t - T is the time of fall, and h - x the space fallen through, are at once recognized as the ordinary formulae of 28. The modification due to the resistance is shown approxi mately by the second terms on the right-hand side of the develop ments above. The necessity for this double investigation, one part for the ascent, the other for the descent, is due to the non-conservative, or "dissipative," character of the force of resistance. 134. As an illustration of constraint by a smooth curve, let us take the case of a simple pendulum. Let O (fig. 40) be the point of suspension, P the position of the bob at any time t. Then, if PG represent the weight of the bob, and be resolved into PH, HG respectively along, and per pendicular to, the tangent at P, we see that PH produces the acceleration of the motion, while the tension of the cord balances HG and also furnishes the ac celeration perpendicular to the direction of motion which is required to produce the curvature of the path. PH is 40. 705 (cssteris paribus) proportional to the sine of PGH, that is, of POA. Hence the acceleration is proportional to the sine of the angular displacement. When that angle is small it may be used in place of its sine. Hence, for small vibra tions, the acceleration is proportional to the displacement, and the motion is "simple harmonic." The time of vibra tion, being ( 51) 2?r / f a si acement/ acceleration, is here 27r A / , approximately. The rigorous solution of the pendulum problem requires the use of elliptic functions. 135. Some very curious properties of pendulum motion are easily proved by geometrical processes. The whole theory of the motion in a vertical plane of a particle attached by a weightless rod to a fixed point, whether it oscillate as a pen dulum or perform continuous rota tions, may be deduced from the two following propositions, which are easily established by geometrical processes in which corresponding in definitely small motions are compared. (1) To comparg different cases of continuous rotation. Let DA (fig. 41) be taken equal to the tangent from D to the circle BPC , whose centre C is vertically under D. Let PAQ be any line through A, cutting Fig. 41. in Q the semicircle on AC. Also make DE = DA. Then, if P move under gravity with speed due to the level of D, Q moves with speed due to the level of E, the acceleration due to gravity being reduced in the ratio AC 2 : 2BC 2 . (2) To compare continuous rotation with oscillation. Let two circles touch one another at their lowest points O (fig. 42) ; compare the arcual motions of points P and p, which are always in the same horizontal line. Draw the ~o~ horizontal tangent AB, Then, if P move, Fi S- 42 - with speed due to g and level or, continuously in its circle, p oscillates with speed due to level AB and acceleration aO 2 9 :^ 136. Two particles are projected from the same point, Motion in the same direction, and with the same speed, but at in ver - different instants, in a smooth circular tube of small bore * ca * whose plane is vertical, to show p/ p that the line joining them con stantly touches another circle. Let the tube be called the circle A, and the horizontal line, to the level of which the speed is due, L. Let M, M (fig. 43) be simultaneous positions of the particles. Suppose that MM passes into its next position by turning about O, these two lines will intercept two indefinitely small arcs at M and M which (by a property of the circle) are in the ratio MO : OM . Let another circle B be described touching MM in 0, and such that L is the radical axis of A and B. Let MP, M P be drawn perpen dicular to L. Let M M cut L in C. Then, by the pro perty of the radical axis, CO 2 = CM. CM ; from which we have, by geometry, CM:CM = OM 2 :OM 2 or OM 2 :OM 2 = PM:P M . But (speed at M) 2 : (speed at M ) 2 = PM = I XV.
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