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.

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.

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.

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.

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).