662. The actual length of the pendulum used, depends upon the purposes
for which the clock is intended, but it is essential for correct
performance that the pendulum should vibrate at a constant rate;
a small irregularity in this respect may appreciably affect the
indications of the clock. If the pendulum vibrates in 1·001 seconds
instead of in one second, the clock loses one thousandth of a second
at each beat; and, since there are 86,400 seconds in a day, it follows
that the pendulum will make only 86,400 - 86·3 vibrations in a day, and
therefore the clock will lose 86·3 seconds, or nearly a minute and a
half daily.
663. For accurate time-keeping it is therefore essential that the time
of vibration shall remain constant. Now the time of vibration depends
upon the length, and therefore it is necessary that the length of the
pendulum be absolutely unalterable. If the length of the pendulum be
changed even by one-tenth of an inch, the clock will lose or gain
nearly two minutes daily, according to whether the pendulum has been
made longer or shorter. In general we may say that, if the alteration
in the length amount to _k_ thousandths of an inch, the number of
seconds gained or lost per day is 1·103 × _k_ with a seconds pendulum.
664. This explains the practice of raising the bob of the pendulum when
the clock is going too slow or lowering it when going too fast. If the
thread of the screw used in doing this have twenty threads to the inch;
then one complete revolution of the screw will raise the bob through
50 thousandths of an inch, and therefore the effect on the rate will
be 1·103 × 50 = 55 nearly. Thus, the rate of the clock will be altered
by about 55 seconds daily. Whatever be the screw, its effect can be
calculated by the simple rule expressed as follows. Divide 1103 by the
number of threads to the inch; the quotient is the number of seconds
that the clock can be made to gain or lose daily by one revolution of
the screw on the bob of the pendulum.
665. Let us suppose that the length of the pendulum has been properly
adjusted so that the clock keeps accurate time. It is necessary that
the pendulum should not alter in length. But there is an ever-present
cause tending to change it. That cause is the variation of temperature.
We shall first illustrate by actual experiment the well known law that
bodies expand under the action of heat; then we shall consider the
irregularities thus introduced into the motion of the pendulum; and,
finally, we shall point out means by which these irregularities may be
effectually counteracted.
[Illustration: FIG. 97.]
666. We have here a brass bar a yard long; it is at present at the
temperature of the room. If we heat the bar over a lamp, it becomes
longer; but upon cooling, it returns to its original dimensions. These
alterations of length are very small, indeed too small to be perceived
except by careful measurement; but we shall be able to demonstrate in a
simple way that elongation is the consequence of increased temperature.
I place the bar A D in the supports shown in Fig. 97. It is firmly
secured at B by means of a binding screw, and passes quite freely
through C; if the bar elongate when it is heated by the lamp, the point
D must approach nearer to E. At H is an electric battery, and G is a
bell rung by an electric current. One wire of the battery connects H
and G, another connects G with E, and a third connects H with the end
of the brass rod B. Until the electric current becomes completed, the
bell remains dumb, the current is not closed until the point touches
E: when this is the case, the current rushes from the battery along
the bar, then from D to E, from that through the bell, and so back to
the battery. At present the point is not touching E, though extremely
close thereto. Indeed if I press E towards the point, you hear the
bell, showing that the circuit is complete; removing my finger, the
bell again becomes silent, because E springs back, and the current is
interrupted.
667. I place the lamp under the bar: which begins to heat and to
elongate; and as it is firmly held at B, the point gradually approaches
E: it has now touched E; the circuit is complete, and the bell rings.
If I withdraw the lamp, the bar cools. I can accelerate the cooling
by touching the bar with a damp sponge; the bar contracts, breaks the
circuit, and the bell stops: heating the bar again with the lamp, the
bell again rings, to be again stopped by an application of the sponge.
Though you have not been able to see the process, your ears have
informed you that heat must have elongated the bar, and that cold has
produced contraction.
668. What we have proved with respect to a bar of brass, is true for
a bar of any material; and thus, whatever be the substance of which
a pendulum is made, a simple uncompensated rod must be longer in hot
weather than in cold weather: hence a clock will generally have a
tendency to go faster in winter than in summer.
669. The amount of change thus produced is, it is true, very small.
For a pendulum with a steel rod, the difference of temperature between
summer and winter would cause a variation in the rate of five seconds
daily, or about half a minute in the week. The amount of error thus
introduced is of no great consequence in clocks which are only intended
for ordinary use; but in astronomical clocks, where seconds or even
portions of a second are of importance, inaccuracies of this magnitude
would be quite inadmissible.
[Illustration: FIG. 98.]
670. There are, it is true, some substances—for example, ordinary
timber—in which the rate of expansion is less than that of steel;
consequently, the irregularities introduced by employing a pendulum
with a wooden rod are less than those of the steel pendulum we have
mentioned; but no substance is known which would not originate greater
variations than are admissible in the performance of an astronomical
clock.
We must, therefore, devise some means by which the effect of
temperature on the length of a pendulum can be avoided. Various means
have been proposed, and we shall describe one of the best and simplest.
671. The mercurial pendulum (Fig. 98) is frequently used in clocks
intended to serve as standard time-keepers. The rod by which the
pendulum is suspended is made of steel; and the bob consists of a
glass jar of mercury. The distance of the centre of gravity of the
mercury from the point of suspension may practically be considered
as the length of the pendulum. The rate of expansion of mercury is
about sixteen times that of steel: hence, if the bob be formed of a
column of mercury one-eighth part of the length of the steel rod, the
compensation would be complete. For, suppose the temperature of the
pendulum be raised, the steel rod would be lengthened, and therefore
the vase of mercury would be lowered; on the other hand, the column of
mercury would expand by an amount double that of the steel rod: thus
the centre of the column of mercury would be elevated as much as the
steel was elongated; hence the centre of the mercury is raised by its
own expansion as much as it is lowered by the expansion of the steel,
and therefore the effective length of the pendulum remains unaltered.
By this contrivance the time of oscillation of the pendulum is rendered
independent of the temperature. The bob of the mercurial pendulum is
shown in Fig. 98. The screw is for the purpose of raising or lowering
the entire vessel of mercury in order to make the rate correct in the
first instance.