THE CYCLOID.

616. If the amplitude of the vibration of a circular pendulum bear a large proportion to the radius, the time of oscillation is slightly greater than if the amplitude be very small. The isochronism of the pendulum is only true for small arcs.

617. But there is a curve in which a weight may be made to move where the time of vibration is precisely the same, whatever be the amplitude. This curve is called a cycloid. It is the path described by a nail in the circumference of a wheel, as the wheel rolls along the ground. Thus, if a circle be rolled underneath the line a b ([Fig. 85]), a point on its circumference describes the cycloid a d c p b. The lower part of this curve does not differ very much from a circle whose centre is a certain point o above the curve.

618. Suppose we had a piece of wire carefully shaped to the cycloidal curve a d c p b, and that a ring could slide along it without friction, it would be found that, whether the ring be allowed to drop from c, p or b, it would fall to d precisely in the same time, and would then run up the wire to a distance from d on the other side equal to that from which it had originally started. In the oscillations on the cycloid, the amplitude is absolutely without effect upon the time.

619. As a frictionless wire is impossible, we cannot adopt this method, but we can nevertheless construct a cycloidal pendulum in another way, by utilizing a property of the curve, o a ([Fig. 85]) as a half cycloid; in fact, o a is just the same curve as b d, but placed in a different position, so also is o b. If a string of length o d be suspended from the point o, and have a weight attached to it, the weight will describe the cycloid, provided that the string wrap itself along the arcs o a and o b; thus when the weight has moved from d to p, the string is wrapped along the curve through the space o t, the part t p only being free. This arrangement will always force the point p to move in the cycloidal arc.

Fig. 85.

620. We are now in a condition to ascertain experimentally, whether the time of oscillation in the cycloid is independent of the amplitude. We use for this purpose the apparatus shown in [Fig. 86]. d c e is the arc of the cycloid; two strings are attached at o, and equal weights a, b are suspended from them; c is the lowest point of the curve. The time a will take to fall through the arc a c is of course half the time of its oscillation. If, therefore, I can show that a and b both take the same time to fall down to c, I shall have proved that the vibrations are isochronous.

Fig. 86.

621. Holding, as shown in the figure, a in one hand and b in the other, I release them simultaneously, and you see the result,—they both meet at c: even if I bring a up to e, and bring b down close to c, the result is the same. The motion of a is so rapid that it arrives at c just at the same instant as b. When I bring the two balls on the same side of c, and release them simultaneously, a overtakes b just at the moment when it is passing c. Hence, under all circumstances, the times of descent are equal.

622. It will be noticed that the string attached to the ball b, in the position shown in the figure, is almost as free as if it were merely suspended from o, for it is only when the ball is some distance from the lowest point that the side arcs produce any appreciable effect in curving the string. The ball swings from b to c nearly in a circle of which the centre is at o. Hence, in the circular pendulum, the vibrations when small are isochronous, for in that case the cycloid and the circle become indistinguishable.

LECTURE XIX.
THE COMPOUND PENDULUM AND THE
COMPOSITION OF VIBRATIONS.

The Compound Pendulum.—The Centre of Oscillation.—The Centre of Percussion.—The Conical Pendulum.—The Composition of Vibrations.