in mechanics, denotes any heavy body, so suspended as that it may vibrate or swing, backwards and forwards, about some fixed point, by the force of gravity.
The vibrations of a pendulum are called its oscillations.
A pendulum, therefore, is any body, B, (fig. 2.) suspended upon, and moving about, a fixed point, A, as a centre.
The nature of a pendulum consists in the following particulars: 1. The times of the vibrations of a pendulum, in very small arches, are all equal. 2. The velocity of the bob, in the lowest point, will be nearly as the length of the chord of the arch which it describes in the descent. 3. The times of vibration in different pendulums, AB, AC, are as the square roots of the times of their vibrations. 4. The time of one vibration is to the time of the descent, through half the length of the pendulum, as the circumference of a circle to its diameter. 5. Whence the length of a pendulum, vibrating seconds, will be found 39.2 inches nearly; and that of an half-second pendulum 9.8 inches.
6. An uniform homogeneous body BG, Pendulum, (ibid. n° 2.) as a rod, staff, &c., which is one-third part longer than a pendulum AD, will vibrate in the same time with it.
From these properties of the pendulum we may discern its use as an universal chronometer, or regulator of time, as it is used in clocks, and such like machines. By this instrument also we can measure the distance of a ship, by measuring the interval of time between the fire and the sound of the gun; also the distance of a cloud, by numbering the seconds or half-seconds between the lightning and thunder. Thus, suppose between the lightning and thunder, we number 10 seconds; then, because sound passes through 1142 feet in one second, we have the distance of the cloud equal to 11420 feet. Again, the height of any room, or other object, may be measured by a pendulum vibrating from the top thereof. Thus, suppose a pendulum from the height of a room vibrates once in three seconds; then say, as 1 is to the square of 3, viz. 9, so is 39.2 to 352.8 feet, the height required. Lastly, by the pendulum we discover the different force of gravity on diverse parts of the earth's surface; and thence the true figure of the earth.
When pendulums were first applied to clocks, they were made very short: and, the arches of the circle being large, the time of vibration through different arches could not in that case be equal; to effect which, the pendulum was contrived to vibrate in the arch of a cycloid, by making it play between two semi-cycloids, CB, CD, (ibid. n° 3.) whereby it describes the cycloid BE, AD; the property of which curve is, that a body vibrating in it will describe all its arches, great or small, in equal times.
In all that has been hitherto said, the power of gravity has been supposed constantly the same. But if the said power varies, the lengths of pendulums must vary in the same proportion, in order that they may vibrate in equal times; for we have shewn, that the ratio of the times of vibration and descent through half the lengths is given, and consequently the times of vibration and descent through the whole length is given: But the times of vibration are supposed equal, therefore the times of descent through the lengths of the pendulum are equal. But bodies descending through unequal spaces, in equal times, are impelled by powers that are as the spaces described, that is, the powers of gravity are as the lengths of the pendulums.
The greatest inconvenience attending this most useful instrument is, that it is constantly liable to an alteration of its length, from the effects of heat and cold, which very sensibly expand and contract all metallic bodies.
To remedy this inconvenience, the common method is by applying the bob of the pendulum with a screw; so that it may be at any time made longer or shorter, according as the bob is screwed downwards or upwards, and thereby the time of its vibrations kept always the same. Again, if a glass or metallic tube, uniform throughout, filled with quicksilver, and 58.8 inches long, were applied to a clock, it would vibrate seconds for $39.2 = \frac{3}{4}$ of 58.8); and such a pendulum admits of a twofold expansion and contraction, viz. one of the metal and the other of the mercury; and these will be at the same time contrary, and therefore will Pendulum correct each other. For by what we have shewn, the metal will extend in length with heat, and so the pendulum will vibrate slower on that account. The mercury also will expand with heat; and since by this expansion it must extend the length of the column upward, and consequently raise the centre of oscillation; so that by this means its distance from the point of suspension will be shortened, and therefore the pendulum on this account will vibrate quicker; wherefore, if the circumstances of the tube and mercury are skillfully adjusted, the time of the clock might by this means, for a long course of time, continue the same, without any sensible gain or loss.
This is the invention of the late ingenious Mr Graham, in the year 1721, who made a clock of this sort, and compared it with one of the best of the common sort for three years together, and found the errors of the former but about \( \frac{1}{3} \) part of the latter; of which the reader may see a farther account in Phil. Trans. no. 393. It is what is now called Mr Graham's quicksilver pendulum.
In the 47th volume of the Phil. Trans. Mr Short gives us an account of other inventions to remedy the same inconvenience. Mr John Harrison of Barrow, in Lincolnshire, famous for his invention of a clock to find the difference of longitude at sea, without having the least knowledge of what Mr Graham had done before him, made several experiments upon wires of different metals, in order to find their different degrees of expansion and contraction. He thought, that by a proper combination of wires of two different metals, differing considerably in their expansion and contraction, he might be enabled to keep the centre of oscillation of a pendulum always at the same distance from the point of suspension. In consequence of these experiments, he made a pendulum consisting of one steel-wire, at the end of which is the bob or weight; and on each side of this wire, four wires, alternately brass and steel, so disposed and contrived as to raise the pendulum by the same quantity that it is lengthened by heat, and to let down the pendulum in the same proportion as it is raised by cold.
Mr Harrison, in his first machine for measuring time at sea, likewise applied this combination of wires of brass and steel, to prevent any alterations by heat and cold. And in the two machines or clocks he has since made for the same purpose, a like method of guarding against the irregularities arising from this cause is used.
Mr Graham also made a pendulum consisting of three bars, one of steel between two of brass; and the steel bar acted upon a lever, so as to raise the pendulum, when lengthened by heat, and to let it down, when shortened by cold; but he found this clock liable to sudden starts and jerks in its motion.
The ingenious Mr Ellicott, in the same volume of the Transactions, describes a pendulum of his invention, composed of brass and iron, with the method of applying it, so as to avoid the many jerks to which the machine might be liable.
But besides the irregularities arising from heat and cold, pendulum-clocks are liable to others from friction and foulness; to obviate which, Mr Harrison has several excellent contrivances, whereby his clocks are almost entirely free from friction, and never need to be cleaned.
PELEPOPE, in fabulous history, the daughter of Icarus, married Ulysses, by whom she had Telemachus. During the absence of Ulysses, who was gone to the siege of Troy, and who laid 20 years from his dominions, several princes, charmed with Penelope's beauty, told her that Ulysses was dead, offered to marry her, and pressed her to declare in their favour. She promised compliance on condition they would give her time to finish a piece of tapestry she was weaving; but at the same time she undid in the night what she had done in the day, and by this artifice eluded their importunity till Ulysses's return.