718 MECHANICS constants by the new conditions. This we leave to the reader. Complex 177. Next let us take the disc of a Complex Pendu- pendu- i um> the motion of two or more pellets attached, at !m- different points, to the same thread, supported at one end. The general solution of this question presents con siderable difficulties, but if we confine our attention to slight disturbances it is easily treated by very elementary processes. In fact, just as a simple pendulum, slightly disturbed in a vertical plane, has simple harmonic motion which may be regarded as the resolved part of conical pendulum motion, so we may treat of a complex conical pendulum, and resolve its motions parallel to any vertical plane. If there be but two masses attached to the string, it is clear that they must, if the motion is to be a persistent conical one, be always in one vertical plane with the point of suspension. And there are obviously two dispositions of the string which are consistent with kinetic stability. Let A (fig. 54) be the point of suspension, then the masses may rotate steadily in either of the two configurations sketched. To keep either mass ^ A moving in a horizontal circle, all that is required is that the resultant force on it shall be horizontal, directed towards the centre, and producing an acceleration equal to V 2 /R, as in 34. Let the whole system turn with angular ve locity w, and let the lengths of the strings be a and 6, their . directions making angles 0, <f> with the vertical. We will treat only the case E in which these angles are so small that the arcs may be written in place of their sines. Then m requires a horizontal resultant force directed towards the axis, and M requires similarly directed. Also, as the strings are both very nearly vertical, the tension of the lower string may be taken as the weight of M, and that of the upper as the sum of the weights of M and m. Treating it, then, as a statical problem, we have for the mass m madia 2 => (M + m)gd - and for M iV! B These formulae correspond to the first configuration, but a change of sign of < adapts them to the second. These two equations involve three unknown quantities o 2 , 0, (f>. But the ratio, only, of and < is involved, so that two equations are sufficient. [We have confined ourselves to small values of and d>, but have not assigned any limit to their smallness ; so that their ratio has still an infinite range of values.] Eliminate the ratio </ # between the two equations ; and we have, putting ^ for g/a) 2 , ty -ma), _. M + m It is clear that, because the right-hand member is essentially less than ah, there are always two real values of i/r, both positive, but one greater than the greater of a, b, the other less than the lesser. These correspond to two values of (f>/0, one positive, the other negative. 178. The most general motion, then, of the double complex pendulum, when it vibrates in one plane, con sists (for each of the masses) of the resultant of two simple harmonic motions, whose periods are lr having one or other of the two positive values given by the equation above, and being therefore the length of the equivalent simple pendulum. Thus the double com plex pendulum supplies at once the mechanical means of tracing (by ink, sand, electric sparks, &c., 156) a graphical representation of the composition of two simple harmonic motions, of different periods, in one line. Analytically thus. For any displacement in one plane we have, 6 and $ being, as before, the deflexions and T, T" the tensions of the strings, ~ sin0= - at d 2
at )
at = mg- Tcos0 + T cos0, )= -T sin0
dt four equations to determine 0, 0, T, and T . They become much more manageable if we assume that and tp are so small that their squares may be neglected. For then we have sin = 0, cos0-=l, &c., and the equations become maS = - T0 + T 0, = mg - T + T , Thus T = M0 , T = (M + m)g , and we have mad = - (M + m)gO + Mgr< , ad + b(f> = - g<f> . Introducing an arbitrary multiplier A, we have If we choose A so that A-M the equation can be put in the form (1): Now (1) is a quadratic equation in A, and has obviously real roots, a positive root greater than M, and a negative root numerically less than m. Write (1) as the equation of an hyperbola, in the form b X - M and we see that A + m = is an asymptote. The branch on the positive side of this asymptote lies mainly below the axis of A. But ^ is positive for A = M, and also for A = 0. Hence /j. must pass through the value zero while A is greater than M, and for another value of A between zero and -m. But it is obvious that, for each of these values of A, w-t-A is positive. Hence the equation may be written where e and n have two sets of real values given as above ; and thus we have the complete solution, with the four requisite arbitrary con stants, in the form + e 2 </> = P 2 cos (n. 2 t + Q J . This applies to every possible set of values of a, l>, m, M ; for, as we have seen, the two values of A are essentially different, at least so long as neither of the masses becomes zero. Thus, in this parti cular case, we are not met by the difficult} 7 of equal roots. But it is very interesting to contrast this case, when m is much greater than M, and = &, with the case discussed in 162 where the point of suspension of a simple pendulum has a horizontal simple harmonic motion of the period of the pendulum and in the vertical plane in which it vibrates. There the oscillations increase inde finitely ; here they are in all cases essentially finite, in accordance with the assumptions made. There is, in fact, no increase of the energy of the system. A very slight modification of the process gives us the result of small displacements not in one plane. Kinetics of a /System of Free Particles. 179. A system of free particles is subject only to their mutual attractions; fco investigate the motion of the
system.