Experimental Mechanics / A Course of Lectures Delivered at the Royal College of Science for Ireland

by Robert S. Ball

THE CONICAL PENDULUM.

648. I have here a tripod ([Fig. 92]) which supports a heavy ball of cast iron by a string 6' long. If I withdraw the ball from its position of rest, and merely release it, the ball vibrates to and fro, the string continues in the same plane, and the motion is that already discussed in the circular pendulum. If at the same instant that I release the ball, I impart to it a slight push in a direction not passing through the position of rest, the ball describes a curved path, returning to the point from which it started. This motion is that of the conical pendulum, because the string supporting the ball describes a cone.

649. In order to examine the nature of the motion, we can make the ball depict its own path. At the opposite point of the ball to that from which it is suspended, a hole is drilled, and in this I have fitted a camel’s hair paint-brush filled with ink. I bring a sheet of paper on a drawing-board under the vibrating ball; and you see the brush traces an ellipse upon the paper, which I quickly withdraw.

650. By starting the ball in different ways, I can make it describe very different ellipses: here is one that is extremely long and narrow, and here another almost circular. When the magnitude of the initial velocity is properly adjusted, and its direction is perpendicular to the radius, I can make the string describe a right cone, and the ball a horizontal circle, but it requires some care and several trials in order to succeed in this. The ellipse may also become very narrow, so that we pass by insensible gradations to the circular pendulum, in which the brush traces a straight line.

Fig. 92.

651. When the ball is moving in a circle, its velocity is uniform; when moving in an ellipse, its velocity is greatest at the extremities of the least axes of this ellipse, and least at the extremities of the greatest axes; but, when the ball is vibrating to and fro, as in the ordinary circular pendulum, the velocity is greatest at the middle of each vibration, and vanishes of course each time the pendulum attains the extremity of its swing. It is worthy of notice that under all circumstances the brush traces an ellipse upon the paper; for the circle and the straight line are only extreme cases, the one being a very round ellipse and the other a very thin one. If, however, the arc of vibration be large the movement is by no means so simple.

652. How are we to account for the elliptic movement? To do so fully would require more calculation than can be admitted here, but we may give a general account of the phenomenon.

Let us suppose that the ellipse a c b d, [Fig. 93], is the path described by a particle when suspended by a string from a point vertically above o, the centre of the ellipse. To produce this motion I withdraw the particle from its position of rest at o to a. If merely released, the particle would swing over to b, and back again to a; but I do not simply release it, I impart a velocity impelling it in the direction a t. Through o draw c d parallel to a t. If I had taken the particle at o, and, without withdrawing it from its position of rest, had started it off in the direction o d, the particle would continue for ever to vibrate backwards and forwards from c to d. Hence, when I release the particle at a, and give it a velocity in the direction a t, the particle commences to move under the action of two distinct vibrations, one parallel to a b, the other parallel to c d, and we have to find the effect of these two vibrations impressed simultaneously upon the same particle. They are performed in the same time, since all vibrations are isochronous. We must conceive one motion starting from a a towards o at the same moment that the other commences to start from o towards d. After the lapse of a short time, the body has moved through a y in its oscillation towards o, and in the same time through o z in its oscillation towards d; it is therefore found at x. Now, when the particle has moved through a distance equal and parallel to a o, it must be found at the point d, because the motion from o to d takes the same time as from a to o. Similarly the body must pass through b, because the time occupied by going from a to b, would have been sufficient for the journey from o to d, and back again. The particle is found at p, because, after the vibration returning from b has arrived at q, the movement from d to o has travelled on to r. In this way the particle may be traced completely round its path by the composition of the two motions. It can be proved that for small motions the path is an ellipse, by reasoning founded upon the fact that the vibrations are isochronous.

Fig. 93.

653. Close examination reveals a very interesting circumstance connected with this experiment. It may be observed that the ellipse described by the body is not quite fixed in position, but that it gradually moves round in its plane. Thus, in [Fig. 92], the ellipse which is being traced out by the brush will gradually change its position to the dotted line shown on the board. The axis of the ellipse revolves in the same direction as that in which the ball is moving. This phenomenon is more marked with an ellipse whose dimensions are considerable in proportion to the length of the string. In fact, if the ellipse be very small, the change of position is imperceptible. The cause of this change is to be found in the fact already mentioned ([Art. 598]), that though the vibrations of a pendulum are very nearly isochronous, yet they are not absolutely so; the vibrations through a long arc taking a minute portion of time longer than those through a short arc.

This difference only becomes appreciable when the larger arc is of considerable magnitude with reference to the length of the pendulum.

Fig. 94.

654. How this causes displacement of the ellipse may be explained by [Fig. 94]. The particle is describing the figure a d c b in the direction shown by the arrows. This motion may be conceived to be compounded of vibrations a c and b d, if we imagine the particle to have been started from a with the right velocity perpendicular to o a. At the point a, the entire motion is for the instant perpendicular to o a; in fact, the motion is then exclusively due to the vibration b d, and there is no movement parallel to o a. We may define the extremity of the major axis of the ellipse to be the position of the particle, when the motion parallel to that axis vanishes. Of course this applies equally to the other extremity of the axis c, and similarly at the points b or d there is no motion of the particle parallel to b d.

655. Let us follow the particle, starting from a until it returns there again. The movement is compounded of two vibrations, one from a to c and back again, the other along b d; from o to d, then from d to b, then from b to o, taking exactly double the time of one vibration from d to b. If the time of vibration along a c were exactly equal to that along b d, these two vibrations would bring the particle back to a precisely under the original circumstances. But they do not take place in the same time; the motion along a c takes a shade longer, so that, when the motion parallel to a c has ceased, the motion along d b has gone past o to a point q, very near o. Let a p = o q, and when the motion parallel to a c has vanished, the particle will be found at p; hence p must be the extremity of the major axis of the ellipse. In the next revolution, the extremity of the axis will advance a little more, and thus the ellipse moves round gradually.