[Illustration: FIG. 87.]
623. Pendulous motion must now be studied in other forms besides that
of the simple pendulum, which consists of a weight and a cord. Any
body rotating about an axis may be made to oscillate by gravity. A
body thus vibrating is called a _compound_ pendulum. The ideal form,
which consists of an indefinitely small weight attached to a perfectly
flexible and imponderable string, is an abstraction which can only be
approximately imitated in nature. It follows that every pendulum used
in our experiments is strictly speaking compound.
624. The first pendulum of this class which we shall notice is that
used in the common clock (Fig. 87). This consists of a wooden or steel
rod A E, to which a brass or leaden bob B is attached. This pendulum is
suspended by means of a steel spring C A, which being very flexible,
allows the vibration to be performed with considerable freedom. The use
of the screw at the end will be explained in Art. 664. A pendulum like
this vibrates isochronously, when the amplitude is small, but it is not
easy to see precisely what is the length of the simple pendulum which
would oscillate in the same time. In the first place, we are uncertain
as to the virtual position of the point of suspension, for the spring,
though flexible, will not yield at the point C to the same extent as a
string; thus the effective point of suspension must be somewhat lower
than C. The other extremity is still more uncertain, for the weight, so
far from being a single point, is not exclusively in the neighbourhood
of the bob, inasmuch as the rod of the pendulum has a mass that is
appreciable. This form of pendulum cannot therefore be used where it is
necessary to determine the length with accuracy.
625. When the length of a pendulum is to be measured, we must adopt
other means of supporting it than that of suspension by a spring, as
otherwise we cannot have a definite point from which to measure. To
illustrate the mode that is to be adopted, I take here an iron bar 6'
long and 1" square, which weighs 19 lbs. I wish to support this at
one end so that it can vibrate freely, and at the same time have a
definite point of suspension. I have here two small prisms of steel E
(Fig. 88) fastened to a brass frame; the faces of the prisms meet at
about an angle of 60° and form the edges about which the oscillation
takes place: this frame and the edges can be placed on the end of the
bar, and can be fixed there by tightening two nuts. The object of
having the edges on a sliding frame is that they may be applicable
to different parts of the bar with facility. In some instruments
used in experiments requiring extreme delicacy, the edges which are
attached to the pendulum are supported upon plates of agate; they are
to be adjusted on the same horizontal line, and the pendulum really
vibrates about this line, as about an axis. For our purpose it will
be sufficient to support the edges upon small pieces of steel. A B,
Fig. 88, represents one side of the top of the iron bar; E is the edge
projecting from it, with its edge perpendicular to the bar. C D is
a steel plate bearing a knife edge on its upper surface; this piece
of steel is firmly secured to the framework. There is of course a
similar piece on the other side, supporting the other edge. The bar,
thus delicately poised, will, when once started, vibrate backwards and
forwards for an hour, as there is very little friction between the
edges and the pieces which support them.
[Illustration: FIG. 88.]
626. The general appearance of the apparatus, when mounted, is shown
in Fig. 89. A B is the bar: at A the two edges are shown, and also
the pieces of steel which support them. The whole is carried by a
horizontal beam bolted to two uprights; and a glance at the figure
will explain the arrangements made to secure the steadiness of the
apparatus; the second pair of edges shown at B will be referred to
presently (Art. 635).
627. This bar, as you see, vibrates to and fro; and we shall determine
the length of a simple pendulum which would vibrate in the same period
of time. The length might be deduced by finding the time of vibration,
and then calculating from Art. 606. This would be the most accurate
mode of proceeding, but I have preferred to adopt a direct method which
does not require calculation. A simple pendulum, consisting of a fine
cord and a small iron sphere C, is mounted behind the edge, Fig. 89.
The point from which the cord is suspended lies exactly in the line of
the two edges, and there is an adjustment for lengthening or shortening
the cord at pleasure.
[Illustration: FIG. 89.]
628. We first try with 6' of cord, so that the simple pendulum shall
have the same length as the bar. Taking the ball in one hand and
the bar in the other, I draw them aside, and you see, when they are
released, that the bar performs two vibrations and returns to my hand
before the ball. Hence the length of the isochronous simple pendulum
is certainly less than the length of the bar; for a pendulum of that
length is too slow.
629. I now shorten the cord until it is only half the length of the
bar; and, repeating the experiment, we find that the ball returns
before the bar, and therefore the simple pendulum is too short. Hence
we learn that the isochronous pendulum is greater than half the length
of the bar, and less than the whole length.
630. Let us finally try a simple pendulum two-thirds of the length of
the bar. I make the experiment, and find that the ball and the bar
return to my hand precisely at the same instant. Therefore two-thirds
of the length of the bar is the length of the isochronous simple
pendulum.
We may state generally that _the time of vibration of a uniform
bar about one end equals that of a simple pendulum whose length is
two-thirds of the bar_; no doubt the bar we have used is not strictly
uniform, because of the edges; but in the positions they occupy, their
influence on the time of vibrations is imperceptible.
632. For this rule to be verified, it is essentially necessary that
the edges be properly situated on the bar; to illustrate this we may
examine the oscillations of the small rod, shown at D (Fig. 89). This
rod is also of iron 24" × 0"·5 × 0"·5, and it is suspended from a point
near the centre by a pair of edges; if the edges could be placed so
that the centre of gravity of the whole lay in the line of the edges,
it is evident that the bar would rest indifferently however it were
placed, and would not oscillate. If then the edges be very near the
centre of gravity, we can easily understand that the oscillations may
be very slow, and this is actually the case in the bar D. By the aid of
the stop-watch, I find that one hundred vibrations are performed in 248
seconds, and that therefore each vibration occupies 2·48 seconds. The
length of the simple pendulum which has 2·48 seconds for its period of
oscillation, is about 20'. Had the edges been at one end, the length of
the simple pendulum would have been
24" × ⅔ = 16".
A bar 72" long will vibrate in a shorter time when the edge is 15"·2
from one end than when it has any other position. The length of the
corresponding simple pendulum is 41"·6.