of inertia of the body I = Σ(*ma*²). If *k* be defined by the relation
Σ(*ma*²) = Σ(*m*) × *k*², then *k* is called the radius of gyration. If *k* be the
radius of gyration of a bob round a horizontal axis through its centre
of gravity, *h* the distance of its centre of gravity below its point of
suspension, and *k*′ the radius of gyration of the bob round the centre
of suspension, then *k*′² = *h*² + *k*². If *l* be the length of a simple pendulum
that oscillates in the same time, then *lh* = *k*′² = *h*² + *k*². Now
*k* can be calculated if we know the form of the bob, and *l* is the length
of the simple pendulum = 39.14 in.; hence *h*, the distance of the
centre of gravity of the bob below the point of suspension, can be
found.

In an ordinary pendulum, with a thin rod and a bob, this distance
*h* is not very different from the theoretical length, *l* = 39.14 in., of
a simple theoretical pendulum in which the rod has no weight and
the bob is only a single heavy point. For the effect of the weight
of the rod is to throw the centre of oscillation a little above the centre
of gravity of the bob, while the effect of the size of the bob is to
throw the centre of oscillation a little down. In ordinary practice
it is usual to make the pendulum so that the centre of gravity is
about 39 in. below the upper free end of the suspension spring and
leave the exact length to be determined by trial.

Fig. 7.—Section of Westminster Clock Pendulum. |

Since T = π√L/*g*, we have, by differentiating,
*d*L/L = 2*d*T/T, that is, any small percentage of increase in L will correspond to
double the percentage of increase in T. Therefore with a seconds pendulum,
in order to make a second’s difference in
a day, equivalent to 1/86,400 of the pendulum’s
Regulation.
rate of vibration, since there are 86,400 seconds
in 24 hours, we must have a difference of length
amounting to 2/86,400 = 1/43,200 of the length
of the rod. This is 39.138/43,200 = .000906 in.
Hence if under the pendulum bob be put a nut
working a screw of 32 threads to the inch and
having its head divided into 30 parts, a turn
of this nut through one division will alter the
length of the pendulum by .0009 in. and change
the rate of the clock by about a second a day.
To accelerate the clock the nut has always to
be turned to the right, or as you would drive
in a corkscrew and vice versa. But in astronomical
and in large turret clocks, it is desirable
to avoid stopping or in any way disturbing the
pendulum; and for the finer adjustments other
methods of regulation are adopted. The best
is that of fixing a collar, as shown in fig. 7 at C,
about midway down the rod, capable of having
very small weights laid upon it, this being the
place where the addition of any small weight
produces the greatest effect, and where, it may
be added, any moving of that weight up or
down on the rod produces the least effect. If
M is the weight of the pendulum and *l* its
length (down to the centre of oscillation), and
m a small weight added at the distance n
below the centre of suspension or above the
c.o. (since they are reciprocal), *t* the time of
vibration, and −*dt* the acceleration due to
adding *m*; then

−dt | = | m | ( | n | − | n^{2} |
) ; |

t | 2M | l | l^{2} |

from which it is evident that if *n* = *l*/2, then
= *dt*/*t* = *m*/8M. But as there are 86400 seconds
in a day, −*d*T, the daily acceleration, = 86400 *dt*, or 10800 *m*/M,
or if m is the 10800th of the weight of the pendulum it will
accelerate the clock a second a day, or 10 grains will do that
on a pendulum of 15 ℔ weight (7000 gr. being = 1 ℔.), or an
ounce on a pendulum of 6 cwt. In like manner if *n* = *l*/3 from
either top or bottom, m must = M/7200 to accelerate the clock a
second a day. The higher up the collar the less is the risk of disturbing
the pendulum in putting on or taking off the regulating weights,
but the bigger the weight required to produce the effect. The weights
should be made in a series, and marked ¼, ½, 1, 2, according to the
number of seconds a day by which they will accelerate; and the
pendulum adjusted at first to lose a little, perhaps a second a day,
when there are no weights on the collar, so that it may always have
some weight on, which can be diminished or increased from time
to time with certainty, as the rate may vary.

The length of pendulum rods is also affected by temperature and also, if they are made of wood, by damp. Hence, to ensure good time-keeping qualities in a clock, it is necessary (1) to make the rods of materials that are as little affected by such Compensation. influences as possible, and (2) to provide means of compensation by which the effective length of the rod is kept constant in spite of expansion or contraction in the material of which it is composed. Fairly good pendulums for ordinary use may be made out of very well dried wood, soaked in a thin solution of shellac in spirits of wine, or in melted paraffin wax; but wood shrinks in so uncertain a manner that such pendulums are not admissible for clocks of high exactitude. Steel is an excellent material for pendulum rods, for the metal is strong, is not stretched by the weight of the bob, and does not suffer great changes in molecular structure in the course of time. But a steel rod expands on the average lineally by .0000064 of its length for each degree F. by which its temperature rises; hence an expansion of .00009 in. on a pendulum rod of 39.14 in., that is .000023 of its length, will be caused by an increase of temperature of about 4° F., and that is sufficient to make the clock lose a second a day. Since the summer and winter temperatures of a room may differ by as much as 50° F., the going of a clock may thus be affected by an error of 12 seconds a day. With a pendulum rod of brass, which has a coefficient of expansion of .00001, a clock might gain one-third of a minute daily in winter as compared with its rate in summer. The coefficients of linear expansion per degree F. of some other materials used in making pendulums are as follows: white deal, .0000024; flint glass, .0000048; iron, .000007; lead, .000016; zinc, .000016; and mercury, .000033. The solid or cubical expansions of these bodies are three times the above quantities respectively.

The first method of compensating a pendulum was invented in
1722 by George Graham, who proposed to use a bob of mercury,
taking advantage of the high coefficient of expansion of that metal.
As now employed, the mercurial pendulum consists of a rod of steel
terminating in a stirrup of the same metal on which rests a glass
vessel full of mercury, having its centre of gravity about 39 in. below
the point of suspension of the pendulum. For each Fahrenheit
degree of temperature the centre of gravity of the bob is lowered
by the expansion of the rod about 1400 of an inch. The glass vessel
and the mercury in it have therefore to be so contrived, that their
centre of gravity will rise 1400 in. per degree F. The glass having a
small coefficient of expansion, the lateral expansion of the mercury
will be checked by it, and this will help to raise the column. For
the linear coefficient of expansion of glass is .0000048 per degree F.,
whence the sectional area of a glass vessel increases by .0000096 per
degree F., and therefore the coefficient of vertical expansion of a
column of mercury whose volumetric expansion coefficient is .0001
per degree F. is (.0001 − .0000096) = .0000904. Let *x* be the height
of the vessel necessary to compensate a steel rod upon the bottom
of which it rests. Then, the coefficient of expansion of steel being
.0000066 per degree F., we have

x | (.0000904 − .0000066) = .0000066 × 39.14, whence x = 6¼ in. |

2 |

It must, however, be remembered that the glass jar has some weight and that it does not rise by anything like the amount of the mercury. This tends to keep the centre of gravity down. So that the height of mercury of 6¼ in. will not be sufficient to effect the compensation, and about 6¾ to 7 in. will be required. Some authors specify 7 in.; this is when the diameter of the jar is small. A certain amount of negative compensation must also be deducted to allow for the changes of temperature in the air, as will presently be seen; this amounts in the case of mercury to about 15 in.

In consequence of the complication of all these calculations it is usual to allow about 6¾ to 7 in. of mercury in the glass vessel and to adjust the exact amount of mercury by trial.

Another very good form of mercurial pendulum was proposed by E. J. Dent; it consists of a cast-iron jar into the top of which the steel pendulum rod is screwed, having its end plunged into the mercury contained in the jar. By this means the mercury, jar and rod rapidly acquire the same temperature. This pendulum is less likely to break than the form just described. The depth of mercury required in an iron jar is stated by Lord Grimthorpe to be 8½ to 9 in. The reason why it is greater than it is when a glass jar is employed is that iron has a larger coefficient of expansion than glass, and that it is also heavier. In all cases, however, of mercury pendulums experiment seems to be the only ultimate test of the quantity of mercury required, for the results are so complicated by the behaviour of the oil and the barometric errors that at its best the regulation of a clock can only be ultimately a matter of scientifically guided compromise. A small amount of compensation of a purely experimental character is also allowed to compensate the changes which temperature effects on the suspension spring. This is sometimes made as much as 16 of the length correction.

As an alternative to the mercurial pendulum other systems have been employed. The “gridiron” pendulum consists of a group of alternate rods of steel and brass, so arranged that the expansion of the brass acts upwards and counteracts that of the steel downwards. It was invented in 1726 by John Harrison. Assuming that 9 rods are used—5 of steel and 4 of brass—their lengths may be as follows from pin to pin:—Centre steel rod 31.5 in.; 2 steel rods next the centre 24.5 in.; 2 steel rods farthest from centre 29.5 in.; from the lower end of outside steel rods to centre of bob 3 in.; total 89.5 in. Of the 4 brass rods the 2 outside ones are 26.87 in.; and the two inside ones 22.25 in.; total 49.12 in. Thus the expansion of 88½ in. of steel is counteracted by the expansion of 4918 in. of brass. Everything depends, however, on the expansion coefficient of the steel and brass employed, the requirement in every case being that of total lengths of the brass and iron should be in proportion to the linear coefficients of expansion of those metals. The above figures