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YourDreamWatch.com Blog
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PENDULUM SPRINGS
There should always be a degree plate, with a pointer to it on the pendulum
wherever it can be most conveniently seen. The length of 4 (degrees) on the
plate is always .07× its distance from the top of the pendulum; and as the
only use of the degree plate is to see that the pendulum keeps the same arc,
or to see how it varies, no very great accuracy is required in the degrees.
Pendulum springs.
—Various opinions have been propounded on the
proper strength for them, but none on any solid grounds, and some on
altogether mistaken ideas of the duty of the spring. Probably the thinner it is
the better, provided it is thick enough not to be bent too sharply or strained
by the weight of the pendulum.
Every now and then a clock is troubled with
the disease of spring-breaking over and over again. Sometimes this proceeds
from the lower edges of the upper chops being left sharp, which in time cut
the spring; and perhaps more frequently from some unseen inequality in the
fixing; and it is better to let the pendulum hang at first with the lower chops
a little loose, and only to screw them up after the pendulum has got a square
bearing, and to take care not to put it out in doing so.
It is wonderful how
much more trouble people will take to do things wrong than would serve to
do them right, and it is almost incredible that some clock-makers send out
their best clocks with the lower edges of the upper chops not horizontal but
rounded into a circular arc.
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CONNECTION WITH CRUTCH
A perfectly firm suspension is essential to good
performance of any clock, and the heavier the pendulum
the firmer its support must be. It is well
known that clocks will influence each other when
fixed against a wooden wall, however firm it may
appear, and that one pendulum will thus actually
set another of equal length vibrating.
Consequently
a free pendulum can be kept swinging
from an apparently immovable support which has
an invisible vibration or twist imparted to it by a
clock below beating in the same time. All good
regulators have the pendulum cock screwed to the
back of the case, and that firmly screwed to a
strong wall; or, better still, the cock is cast with
a cast iron back or bracket which also carries the
movement, and that is screwed directly to the wall,
the whole front of the case lifting off.
In the Westminster
clock the pendulum cock is a large iron
frame built in right through the wall with a flange
behind, and the cocks of many other large clocks
are fixed with bolts through the wall. With a firm
suspension the pendulum swings farther than with
a weak one, and we shall see afterwards that (other things being equal) the
errors of clocks vary inversely as the square and sometimes as the cube of
the arc.
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SUSPENSION OF PENDULUMS
It is true that a lens moves through the air with (See an amusing pamphlet on it by Mr. F. D. Wackerbarth, and the remarks on the
Pyramid in my ‘Book on Building.’ (Lockwood & Co.) less resistance than any other shape; but then it must be very large to be
of the same weight as a sphere or a thick cylinder with the axis vertical,
and we shall see afterwards that nearly all clock errors are inversely as the
weight (and length) of the pendulum.
But there is a shape which would
probably be found to combine some of the advantages of both though Baily’s
experiments (in Phil. Trans. for 1932) showed some results which no one
could have foreseen; I mean a cylinder of elliptic section, or one having the
same horizontal section as a lens, but thicker; or of the section formed by
two arcs of a circle of 120, which would make the horizontal length and
thickness as 121
8 to 7 inches—a very convenient size for a large bob 13 in.
high; which would weigh about 188 lbs. in cast iron and 288 in lead, and is
equal to a round cylinder of 81
4 inches diameter.
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SHORT AND SLOW PENDULUMS
Probably the authors of the new 25 inch to a mile Ordnance survey of England little thought that
they were reverting to that ancient standard. At the same time, I do not
accept the theory of the Scotch Astronomer Royal,3 that the 25 inch cubit
was embodied in the Pyramid dimensions, besides the one of 20.7 inches,
or 20 of some of the old continental inches. A variation of one thousandth
is practically nothing in unscientific ages, and is less than the variation of
many foot-rules now, and it is singular that if our inch were a 1000th longer,
the earth’s polar axis would be just 500 million inches.
Short and slow pendulums.—There is a kind of pendulum which is
properly enough used in the instrument called a metronome, for counting
the time in music lessons in a way that requires no particular accuracy, and
occasionally, but very improperly, used for small clocks. It follows from the
propositions I have been explaining, that if a pendulum with a heavy rod is
set vibrating on an axis very little above its c.g. it will vibrate slowly, like
a scalebeam; and consequently you may have a 2 seconds pendulum in the
compass of a few inches.
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BADNESS OF THE FRENCH METRE
That Act however has been repealed, and some standard weights and
measures are deposited in various public places under an Act of 18 & 19
Vic. c. 72. The late Astronomer Royal provided a set, open to anybody,
at the Observatory gate. But if they all happened to be destroyed, new
ones could be made from the old pendulum ratio. Indeed Mr. Johnson of
Wilmington Square had one ready for that calamity. A simple pendulum
36 in. long would vibrate in a time which is to 1s. as 362 to 39.142, or
.846 of a second, or 70.92 times in a minute; and he has made a clock with
a reversible pendulum vibrating in that time, and consequently 36 inches
between the two knife edges.
The length of a seconds pendulum so nearly resembles the French metre
of 39.371 in., that some persons may fancy that that most ridiculous and
mischievous revolutionary measure had an origin even as rational as being
the length of a seconds pendulum in some latitude. But it has not. It was
intended to be the 40 millionth part of a meridian of the earth—about as
rational a standard as if we enacted that the yard should be the 420 millionth
of the mean distance of the moon, which it is very nearly; and astronomers
know the moon’s distance within a less fraction than the difference of the
metre from what it pretends to be, but is not.
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CENTRE OF OSCILLATION
—It must be remembered that these are only the
theoretical lengths of simple pendulums with all the weight concentrated in
the centre of the bob, and that this theoretical length by no means coincides
with the actual length down to the centre of gravity of the pendulum, but is
always longer. This length may properly be called the radius of oscillation,
as the lower end of it is always called the centre of oscillation; which is
not a fixed point in the body, like the centre of gravity, but a relative one,
every axis of suspension having a radius and centre of oscillation of its own.
It is not always a simple process to calculate this length (which we may
as well call l) for any given pendulum as it requires either the integral
calculus or some rules deduced from it; but it will be easy to explain the
nature and meaning of the quantities on which it depends.
Let m be the
mass of each particle of the pendulum, which in these calculations must not
be confounded with the weight, which is written mg (g being the force of
gravity), r the distance of m from the axis of suspension, and M the mass
of the whole pendulum; then the radius of oscillation = the sum of each
particle multiplied into the square of its distance from the axis, divided by
the sum of each particle multiplied into its distance simply; that is to say,
l =
P
mr2 P
mr
(
P
being used to indicate this kind of summation, which can
only be performed by integration).
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