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.
THE COMPOSITION OF VIBRATIONS.
656. We have learned to regard the elliptic motion in the conical pendulum as compounded of two vibrations. The importance of the composition of vibrations justifies us in examining this subject experimentally in another way. The apparatus which we shall employ is represented in [Fig. 95].
a is a ball of cast iron weighing 25 lbs., suspended from the tripod by a cord: this ball itself forms the support of another pendulum, b. The second pendulum is very light, being merely a globe of glass filled with sand. Through a hole at the bottom of the glass the sand runs out upon a drawing-board placed underneath to receive it.
