PENDLETON-PENDULUM. the portions of vaults introduced in the angles of rectangular compartments, in order to reduce them to a circular or other suitable form to receive a dome. PE'NDLETON, a township of Lancashire, with a station on the Lancashire and Yorkshire Railway, is a suburb of Manchester, and is 24 miles west-northwest of the town of that name. In 1851, it contained 14,224, and in 1861, 20,900 inhabitants. Since then, the population has slightly increased. P. is part of the parliamentary borough of Salford, and since 1852 it has been incorporated with the municipality of the is due to the immense industry of the locality. The same borough. The rapid increase of its population inhabitants are employed in the numerous cotton and flax mills, print and dye-works, iron founand flax mills, print and dye-works, iron foundries, soap, and chemical works, in operation here. Hundreds of the population are also employed in the well-known P. collieries, which are conducted with much enterprise by the lessees. P. is also the residence of a portion of the mercantile community from Manchester, whose large mansions, with their parks and gardens, are dotted at intervals along the two roads leading from the township westward to Eccles. PE'NDULUM, in its widest scientific sense, denotes a body of any form or material which, under the action of some force, vibrates about a position of stable equilibrium. In its more usual application, however, this term is restricted, in conformity with its etymology (Lat. pendeo, to hang), to bodies suspended from a point, or oscillating about an axis, under the action of gravity, so that, although the laws of their motion are the same, Rocking Stones (q. v.), Magnetic Needles, Tuning-forks, Balance wheel of a watch, &c., are not included in the definition. The simple pendulum consists (in theory) of a heavy point or particle, suspended by a flexible string without weight, and therefore constrained to move as if it were always on the inner surface of a smooth spherical bowl. If such a pendulum be drawn aside into a slightly-inclined position, and allowed to fall back, it evidently will oscillate from side to side of its position of equilibrium, the motion being confined to a vertical plane. If, instead of being allowed to fall back, it be projected horizontally in a direction perpendicular to that in which gravity tends to move it, the bob will revolve about its lowest position; and there is a particular velocity with which, if it be projected, it describes a circle about that point, and is then called a conical pendulum. As the theory of the simple pendulum can be very easily explained, by reference to that of the conical pendulum, we commence with the latter, which is extremely simple. To find the requisite velocity, we have only to notice that the (so-called) Centrifugal Force (q. v.) must balance the tendency towards the vertical. This tendency is not directly due to gravity, but to the tension of the susFig. 1. pending cord. In the fig. let O be the point of suspension, OA the pendulum in its lowest position, P the bob in any position in the (dotted) circle which it describes when revolving as a conical pendulum; PB, a radius of the dotted circle, is evidently perpendicular to OA. Now, the centrifugal force is directly as the radius PB of the circle, and inversely as the square of the time of revolution. Also the radius PB is PO sin. BOP, the length of the string multiplied by the sine of the angle it makes with the vertical; and the force attraction, and to the tangent of the above angletowards the vertical is proportional to the earth's the three forces acting on the bob at P are parallel as may be at once seen from the consideration that and therefore proportional, to the sides of the tri lution is directly as the length of the string and the and therefore proportional, to the sides of the tri angle OBP. Hence the square of the time of revo sine of the angle BOP, and inversely as the earth's attraction and the tangent of the same angle; or (what is easily seen to be equivalent) to the length of the string and the cosine of its inclination to the vertical directly, and to the earth's attraction inversely. Hence, in any given locality, all conical pendulums revolve in equal times, whatever be the lengths of their strings, so long as their heights are of the string by the cosine of its inclination to the equal; the height being the product of the length vertical. Also the squares of the times of revolu tion of conical pendulums are as their heights directly, and as the earth's attraction inversely. Now, so long as a conical pendulum is deflected only through a very small angle from the vertical, the motion of its bob may be considered as compounded of two equal simple pendulum oscillations in directions perpendicular to each other, such as it appears to make to an eye on a level with it and viewing it at some distance, first from one point, say on the north, and then from another 90° round, say on the east. And these motions_take place, by Newton's second law (see MOTION, LAWS OF), independently. Also the time of a (double) oscillation in either of these directions is evidently the same as that of the rotation of the conical pendulum. Hence, for small arcs of vibration, the square of the time of oscillation of a simple pendulum is directly as its length, and inversely as the earth's attraction. Thus, the length of the second's pendulum at London being 39-1393 inches, that of the half-second's pendulum is 9.7848 inches, or onefourth; that of the two seconds' pendulum 156 5572 inches, or four times that length. It follows from the principle now demonstrated, that so long as the arcs of vibration of a pendulum are all small relatively to the length of the string, they may differ considerably in length among themselves without differing appreciably in time. It is to this property of pendulum oscillations, known as Isochronism (q. v.), that they owe their value in measuring time. See HOROLOGY. That the times of vibration of different pendulums are as the square roots of their lengths, may be demonstrated to the eye by a very simple experiment. Suspend three musket balls on double threads as in the figure, so that the heights in the dotted line may be as 1, 4, and 9. When they are made to vibrate simultaneously, while the lowest ball makes one oscillation the highest will be found to make three, and the middle ball one and a half. Fig. 2 A pendulum of given length is a most delicate instrument for the measurement of the relative amounts of the earth's attraction at different PENDULUM. 24 hours At the plane of the pendulum's motion is sin. λ places. Practically, it gives the kinetic measure- to 1. Hence the time of the apparent rotation of ment of gravity, which is not only by far the most convenient, but also the true measure. By this application of the pendulum, the oblateness of the earth has been determined, in terms of the law of decrease of gravity from the poles to the equator. The instrument has also been employed to determine the mean density of the earth (from which its mass We have not yet alluded to the obvious fact, that is directly derivable), by the observation of its times a simple pendulum, such as we have described above, of vibration at the mouth and at the bottom of a exists in theory only, since we cannot procure either coal-pit. It was shewn by Newton, that the force of a single heavy particle, or a perfectly light and attraction at the bottom of a pit depends only flexible string. But it is easily shewn, although the upon the internal nucleus which remains when a process cannot be given here, that a rigid body of shell, everywhere of thickness equal to the depth of any form whatever vibrates about an axis under the the pit, has been supposed to be removed from the action of gravity, according to the same law as the whole surface of the earth. The latest observations hypothetical simple pendulum. The length of the by this method were made by Airy, the present astro- equivalent simple pendulum depends upon what is nomer-royal, in the Harton coal-pit, and gave for the called the Radius of Gyration (q. v.) of the pendumean density of the earth a result nearly equiva-lous body. Its property is simply this, that if the lent to that deduced by Cavendish and Maskelyne whole mass of the body were collected at a point from experiments of a totally different nature. See whose distance from the axis is the radius of gyration, the moment (q. v.) of inertia of this heavy point (about the axis) would be the same as that of the complex body. The square of the radius of gyration of a body about any axis, is greater than the square of the radius of gyration about a parallel axis through the centre of gravity, by the square of the distance between those lines. Now, the length of the simple pendulum equivalent to a body oscillating about any axis is directly as the square of the radius of gyration, and inversely as the distance of the centre of gravity from the axis. Hence, if k be the radius of gyration of a body about an axis through the centre of gravity, 2+ h2 is that about a parallel axis whose distance from the first is h; and the length, 1, of the equivalent simple pendulum is k2 + h2 EARTII. If the bob of the simple pendulum be slightly displaced in any manner, it describes an ellipse about its lowest position as centre. This ellipse may, of course, become a straight line or a circle, as in the cases already considered. The bob does not accurately describe the same curve in successive revolutions; in fact, the elliptic orbit just mentioned rotates in its own plane about its centre, in the same direction as the bob moves, with an angular velocity nearly proportional to the area of the ellipse. This is an interesting case of progression of the apse (Apsides, q. v.), which can be watched by any one who will attach a small bullet to a fine thread; or, still better, attach to the lower end of a long string fixed to the ceiling a funnel full of fine sand or ink which is allowed to escape from a small orifice. By this process, a more or less permanent trace of the motion of the pendulum is recorded, by which the elliptic form of the path and the phenomena of progression are well shewn. According to what is stated above, there ought to be no progression if the pendulum could be made to vibrate simply in a straight line, as then the area of its elliptic orbit vanishes. It is, however, found to be almost impossible in practice to render the path absolutely straight; so that there always is from this cause a slight rate of change in the position of the line of oscillation. But as the direction of this change depends on the direction of rotation in the ellipse, it is as likely to affect the motion in one way as in the opposite, and is thus easily separable from the very curious result obtained by Foucault, that on account of the earth's rotation, the plane of vibration of the pendulum appears to turn in the same direction as the sun, that is, in the opposite | direction to the earth's rotation about its axis. To illustrate this now well-known case, consider for a moment a simple pendulum vibrating at the pole of the earth. Here, if the pendulum vibrates in a straight line, the direction of that line remains absolutely axed in space, while the earth turns round below it once in 24 hours. To a spectator on the earth, it appears, of course, as if the plane of motion of the pendulum were turning once round in 24 hours, but in the opposite direction. To find the amount of the corresponding phenomenon in any other latitude, all that is required is to know the rate of the earth's rotation about the vertical in that latitude. This is easy, for velocities of rotation are resolved and compounded by the same process as forces, hence the rate at which the earth rotates about the vertical in latitude a is less than that of rotation about the polar axis in the ratio of sin. a l h This expression becomes infinitely great if h be very large, and also if h be very small (that is, a body vibrates very slowly about an axis either far from, or near to, its centre of gravity). It must therefore have a minimum value. By solving the equation above as a quadratic in h, we find that l cannot be less than 2k, which is, therefore, the length of the simple pendulum corresponding to the quickest vibrations which the body can execute about any axis parallel to the given one. In this case, the value of h is equal to . Hence, if a circular cylinder be described in a body, its axis passing through the centre of gravity, and its radius being the radius of gyration about the axis, the times of oscillation about all generating lines of this cylinder are equal, and less than the times of oscillation about any other axes parallel to the given one. Also, since the formula for l, above given, may be thus written, h(l – h) = k2, it is obvious that it is satisfied if l-h be put for h. Hence, if any value (of course not less than 2k) be assigned as the length of the equivalent simple pendulum, there are two values of h which will satisfy the conditions; that is, there are two concentric cylinders, about a generating line of either of which the time of oscillation is that of the assigned simple pendulum. When = 2k, these cylinders coincide, and form that above described. And, since the sum of the radii of these cylinders is l, it is obvious that if we can find experimentally two parallel axes about which a body oscillates in equal times, and if the centre of gravity of the body lie between these axes, and in their plane, the distance between these axes is the length of the equivalent simple pendulum. This result is of very great importance, because it enabled Kater (who was the first to employ it) to use the complex pendulum for the determination of the length of the simple PENELOPE-PENGUIN. second's pendulum in any locality. The simple pendulum is perfect in theory, but cannot be constructed; and thus the method which enables us to obtain its results by the help of such a pendulum as we can construct, is especially valuable. a D B C as the steel rod bc, being added to the instrument for the sake of symmetry, strength, and stiffness only. If the effective lengths of steel and brass be inversely as their respective dilatation coefficients, the position of the bob is A Compensation Pendulum.-As the length of a rod unaltered by temperature; and there or bar of any material depends on its temperature fore the pendulum will vibrate in the (see HEAT), a clock with an ordinary pendulum same period as before heating. This goes faster in cold, and slower in hot, weather. is on the supposition that the weight Various contrivances have been devised for the of the framework may be neglected purpose of diminishing, if not destroying, these in comparison with that of the bob; effects. The most perfect in theory, though per- if this weight must be taken into haps not the most available in practice, is that account, the requisite adjustments, of Sir D. Brewster (q. v.), founded upon the experi- though possible, are greatly more mental discovery of Mitscherlich, that some crystals complex, and can only be alluded to expand by heat in one direction, while contracting here. Practically, it is found that a c in the perpendicular one; and therefore that a strip of dry fir-wood, carefully varnrod may be cut out of the crystal in such a direc-ished, to prevent the absorption of tion as not to alter in length by any change of moisture, and consequent hygrometric temperature. In the method of correction usually alterations of its length, is very little Fig. 4. employed, and called compensation, advantage is affected by change of temperature; taken of the fact that different substances have dif- and, in many excellent clocks, this is used as a very ferent coefficients of linear dilatation; so that if the effective substitute for the more elaborate forms bob of the pendulum be so suspended as to be raised just described. To give an idea of the nicety which by the expansion of one substance, and depressed modern astronomy requires in the construction of by the expansion of another, the lengths of the an observing clock, we may mention that the Ruseffective portions of these substances may be so sian astronomers find the gridiron superior to the adjusted that the raising and depression, taking mercurial pendulum; because differences of templace simultaneously, may leave the perature at different parts of the clock case (though position of the bob unaffected. There almost imperceptible in a properly protected instruare two common methods of effecting ment), may heat the steel or the mercury unduly in this, differing a little in construction, the latter; while, in the former, the steel and brass but ultimately depending on the same bars run side by side through the greater part of the principle. Of these, the mercurial pen- length of the pendulum, and are thus simultaneously dulum is the more easily described. affected by any such alterations of temperature. The rod AC, and the framework CB, are of steel. Inside the framework is placed a cylindrical glass jar, nearly full of mercury, which can be raised or depressed by turning a nut at B. By 8 increase of temperature, the steel portion AB is lengthened by an amount Fig. 3. proportional to its length, its coefficient of linear dilatation, and the change of temperature, conjointly--and thus the jar of mercury is removed from the axis of suspension. But neglecting the expansion of the glass, which is very small, the mercury rises in the jar by an amount proportional to its bulk, its coefficient of cubical dilatation, and the change of temperature, conjointly. Now, by increasing or diminishing the quantity of mercury, it is obvious that we may so adjust the instru A C It would lead us into details of a character far too abstruse for the present work to treat of the effects of the hydrostatic pressure and viscosity of the air upon the motion of a pendulum. PENELOPÉ, in Homeric legend, the wife of Ulysses (Odysseus), and mother of Telemachus, who was still an infant when Ulysses went to the Trojan war. During his long wanderings after the fall of Troy, he was generally regarded as dead, and P. was vexed by the urgent suits of many lovers, whom she put off on the pretext that she must first weave a shroud for Laertes, undid by night the portion of the web which her aged father-in-law. To protract the time, she she had woven by day. When the suitors had discovered this device, her position became more difficult than before; but fortunately Ulysses returned in time to rescue his chaste spouse from their distasteful importunities. Later tradition represents P. in a very different light, asserting that by Hermes (Mercury), or by all her suitors together, she became the mother of Pan (q. v.), and that Ulysses, on his return, divorced her in consequence. But the older Homeric legend is the simpler and more genuine version of the story. The construction of the gridiron pendulum will PENGUIN (Aptenodytes), a genus of birds of the be easily understood from the cut. The black bars family Alcide (see AUK), or constituting the family are steel, the shaded ones are brass, copper, or some Aptenodida, regarded by many as a sub-family of substance whose coefficient of linear dilatation is Alcidæ, and divided into several genera or submore than double that of steel. It is obvious from genera. They have short wings, quite unfit for the figure that the horizontal bars are merely con- flight, but covered with short rigid scale-like nectors, and that their expansion has nothing to do feathers, admirably adapted for swimming, and with the vibration of the pendulum, so they may be much like the flippers of turtles. The legs are very made of any substance. It is easily seen that an short, and are placed very far back, so that on land increase of temperature lowers the bob by expand-penguins rest on the tarsus, which is widened like ing the steel rods, whose effective length consists of the sum of the lengths of Aa, BC, and the steel bar to which the bob is attached; while it raises the bob by expanding the brass bars, whose effective length is that of one of them only; the other, as well the sole of the foot of a quadruped, and maintain a perfectly erect posture. Their bones, unlike those of birds in general, are hard, compact, and heavy, and have no air-cavities; those of the extremities contain an oily marrow. The body is of an elliptical |