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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.