615. The time of a single vibration is found, by dividing 241·6 by 300, to be 0·805 second. This is certainly correct to within a thousandth part of a second. We conclude that a pendulum whose length is 25"·37 = 2·114, vibrates in 0·805 second; and from this we find that gravity at Dublin is 2'·114 × (3·1416 / 0.805)² = 32·196. This result agrees with one which has been determined by measurement made with every precaution.
Another method of measuring gravity by the pendulum will be described in the next lecture ([Art. 637]).
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.