PENDULUM 



25 



its circular path, it acquires energy enough to raise 

 it to an equal height on the opposite side. In 

 ordinary experiments the bullet will perform many 

 thousand oscillations by itself alone, before the 

 resistance of the air and other interferences cause 

 the movement to subside and at last cease, by 

 imperceptibly diminishing the length of the arc. 



This long-continued and self-sustaining action is 

 manifestly due to the attraction of the earth, the 

 force that causes a stone to fall to the ground, 

 because at the end of each swing of the bullet its 

 weight tends to pull it vertically downwards, and 

 the string constrains it to repeat its course along 

 the circular arc. A most interesting and valuable 

 application of the pendulum, therefore, is for 

 measuring the acceleration of velocities of falling 

 bodies. For that purpose it is much superior to 

 At wood's Machine (q.v. ) or any other method 

 which has yet been devised. 



If the circular path or swing is short not 

 exceeding, for example, that of a clock pendulum 

 which beats seconds there are two results to be 

 remembered. First, that so long as the length of 

 thread is unchanged, it matters not how far the 

 bullet may swing on each side, the time or dura- 

 tion of each oscillation is also unchanged. This 

 'pendulum-law' was discovered by Galileo in the 

 church of Pisa, as he watched a lamp swinging by 

 a chain. The quality that each swing occupies the 

 same time is so important in horology that the 

 introduction of the pendulum by Huygens as a 

 time-measurer formed the principal epoch in the 

 history of that science. The term isochronism 

 ( 'equal- timeness ') was invented to mark this 

 property of the pendulum. The second law of the 

 pendulum is that to make the bullet move faster 

 we must shorten the thread in the following pro- 

 portion : for twice as many oscillations take a 

 quarter the length of string ; for thrice as many 

 take one-ninth the length ; for four times as many 

 take one-sixteenth the length. That law is other- 

 wise expressed by saying the length of the thread 

 is inversely as the square of the number of oscilla- 

 tions made in a given time (see CENTRE OF OSCIL- 

 LATION ). 



These and other properties of the pendulum are 

 wrapped up in the formula : t* : w 3 :: / : g, which 

 mathematicians have established : where / time 

 in seconds of one oscillation, I = length (in inches) 

 of the thread, ir = 3-1415927, a well-known ratio; 

 and g = the accelerating fgrce of gravity, or 

 twice the sjiace through which a heavy body 

 falls in one second. When* = 1 in that formula 

 i.e. when our pendulum beats seconds, a result 

 easily attained at any part of the world then 

 immediately we have y = **l = 9'8696/. In 

 other words, multiply the length of the seconds 

 pendulum in any latitude or longitude by the 

 lixcd number 9'8696 to find the value of y. 

 By this valuable and simple result it has been 

 .shown that the force of gravity slightly and gradu- 

 ally increases as we travel from the equator to- 

 wards either pole, the length of the seconds 

 pendulum diminishing in the same proportion. 

 The poles are therefore nearer to the centre than 

 the equator is, which is an independent proof that 

 our planet is spheroidal, and resembles in shape 

 an orange rather than a lemon. 



The following table readily gives the length of 

 the seconds pendulum at any of the stations by 

 dividing the corresponding number in the third 

 column by the fixeu number 9-K696. At London, 

 for examp'h. 32-191 -= 9-8696 = 3-262 feet, length of 

 seconds pendulum. Dent's clock in the tower of 

 the House of Commons beats once in two seconds, 

 and most therefore have a pendulum 13-046 feet 



low. 



The table also shows the acceleration (feet 



per second) due to gravity, as ascertained from 

 observations made by means of the seconds pen- 

 dulum. The results" are arranged in the order of 

 their latitude. 



SUtion. ubMrver. Force of Gravitv. 



Fet. ' 



Rawak (between Jilolo and Sew Guinea .Freycinet. 32-088 



Sierra Leone Sabine. 32-093 



Ascension Sabine. 32-096 



Jamaica Sabine. 32-105 



Rio de Janeiro Freycinet. 32-112 



Cape of Good Hope Freycinet. 32-140 



Bordeaux Biot, Hathieu. 32-169 



Paris Borda. 32-182 



Dunkirk Blot, Mathieu. 32-190 



London Sabine. 32-191 



Edinburgh Kater. 32-204 



Unst, Shetland Biot, Mathieu. 32-217 



Spitzbergen Sabine. 32-263 



Since the length of the seconds pendulum is due 

 entirely to natural causes, and can always be easily 

 verified, it was chosen as a standard of the British 

 measures of length. Experience has taught, how- 

 ever, that these are more easily known by preserv- 

 ing an artificial standard. 



The universal application of the pendulum for 

 time-measurement and ascertaining the local value 

 of (j has been followed by some special uses of it 

 which are of interest. Thus, Sir G. B. Airy, the late 

 astronomer-royal, applied it to form an estimate of 

 the earth's mean density by observations taken at 

 a coal-pit, 1200 feet deep, near South Shields. One 

 pendulum being stationed at the surface and another 

 at the bottom of the pit, their oscillations were 

 exactly compared by means of an electric wire, 

 with the result that a clock 'at the mouth of the 

 pit would gain 2J seconds per day if removed to 

 the bottom. From these data (Phil. Trans. 1856, 

 p. 297 ) the density of the earth was estimated to 

 be 6-565. 



By the Foucault experiment the pendulum was 

 utilised in a striking manner to prove the perpetual 

 rotation of our planet round its axis. A globe of 

 metal is suspended by a long wire to a lofty roof, 

 the point of suspension being vertically over the 

 centre of a round table ; and after being drawn 

 aside from the position of rest this pendulum is 

 allowed to begin its vibrations, but so as to have 

 no tendency to right or left. Students of dynamics 

 know that it must continue swinging to and fro 

 in the same plane unless interfered with from 

 without. Owing to that the table beneath the 

 pendulum, when carefully observed, is seen to 

 revolve very slowly in a direction contrary to the 

 hands of a watch ; hut since the floor and whole 

 building revolve with the table, the observers 

 naturally refer the relative motion to the pen- 

 dulum, still swinging in its original plane. By 

 marking twenty-four equal divisions round the 

 edge of the table the spectators would be furnished 

 with a good clock, the pendulum pointing out the 

 hour at the point where it first began its oscilla- 

 tions, and apparently revolving in the usual direc- 

 tion. 



The pendulum, in Horology, is absolutely 

 accurate as a time-keeper, if only the proper 

 length is preserved. That is mainly done by 

 means of a screw turning on the rod, under the 

 ' boh ' or ball, so as to push it up and therefore 

 shorten the pendulum, or let it fall lower down 

 and lengthen the pendulum. It was found in 

 winter that clocks went too fast, and at mid- 

 summer too slow, because cold shortened the 

 metallic rod and heat lengthened it. A further 

 refinement was therefore devised to secure a 

 uniform length without the screw adjustment, the 

 result being what are known as ' compensation 

 pendulums. Both the common methods of these 

 depend on the same principle. ( A simple and prac- 

 tically accurate form of pendulum is made with 

 a wooden roil, which is less liable to expansion 



