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.
[Illustration: 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.
[Illustration: 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.