608. The pendulum affords the proper means of measuring the force
of gravity at any place on the earth. We have seen that the time of
vibration can be expressed in terms of the length and the force of
gravity; so conversely, when the length and the time of vibration are
known, the force of gravity can be determined and the expression for it
is—
Length × (3·1416 / Time)².
609. It is impossible to observe the time of a single vibration with
the necessary degree of accuracy; but supposing we consider a large
number of vibrations, say 100, and find the time taken to perform them,
we shall then learn the time of one oscillation by dividing the entire
period by 100. The amplitudes of the oscillations may diminish, but
they are still performed in the same time; and hence, if we are sure
that we have not made a mistake of more than one second in the whole
time, there cannot be an error of more than 0·01 second, in the time
of one oscillation. By taking a still larger number the time may be
determined with the utmost precision, so that this part of the inquiry
presents little difficulty.
610. But the length of the pendulum has also to be ascertained, and
this is a rather baffling problem. The ideal pendulum whose length
is required, is supposed to be composed of a very fine, perfectly
flexible cord, at the end of which a particle without appreciable
size is attached; but this is very different from the pendulum which
we must employ. We are not sure of the exact position of the point of
suspension, and, even if we had a perfect sphere for the weight, the
distance between its centre and the point of suspension is not the
same thing as the length of the simple pendulum that would vibrate
in the same time. Owing to these circumstances, the measurement of
the pendulum is embarrassed by considerable difficulties, which have
however been overcome by ingenious contrivances to be described in the
next chapter.
611. But we shall perform, in a very simple way, an experiment for
determining the force of gravity. I have here a silken thread which is
fastened by being clamped between two pieces of wood. A cast iron ball
2"·54 in diameter is suspended by this piece of silk. The distance from
the point of suspension of the silk to the ball is 24"·07, as well as
it can be measured.
The length of the ideal pendulum which would vibrate isochronously with
this pendulum is 25"·37, being about 0"·03 greater than the distance
from the point of suspension to the centre of the sphere.
612. The length having been ascertained, the next element to be
determined is the time of vibration. For this purpose I use a
stop-watch, which can be started or stopped instantaneously by touching
a little stud: this watch will indicate time accurately to one-fifth
of a second. It is necessary that the pendulum should swing in a small
arc, as otherwise the oscillations are not strictly isochronous. Quite
sufficient amplitude is obtained by allowing the ball to rotate to and
fro through a few tenths of an inch.
613. In order to observe the movement easily, I have mounted a
little telescope, through which I can view the top of the ball. In
the eye-piece of the telescope a vertical wire is fastened, and I
count each vibration just as the silken thread passes the vertical
wire. Taking my seat with the stop-watch in my hand, I write down the
position of the hands of the stop-watch, and then look through the
telescope. I see the pendulum slowly moving to and fro, crossing the
vertical wire at every vibration; on one occasion, just as it passes
the wire, I touch the stud and start the watch. I allow the pendulum
to make 300 vibrations, and as the silk arrives at the vertical wire
for the 300th time, I promptly stop the watch; on reference I find that
241·6 seconds have elapsed since the time the watch was started. To
avoid error, I repeat this experiment, with precisely the same result:
241·6 seconds are again required for the completion of 300 vibrations.
614. It is desirable to reckon the vibrations from the instant when the
pendulum is at the middle of its stroke, rather than when it arrives
at the end of the swing. In the former case the pendulum is moving
with the greatest rapidity, and therefore the time of coincidence
between the thread and the vertical wire can be observed with especial
definiteness.
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).