after passing through A B D, will be to the velocity at D after descending through the portion в D only, as A D to B D. 11. Farther, velocity after describing A B D, is to velocity after describing B D, as ✓F D to √E D. If, therefore, we would impart to a body a given velocity v, we have only to compute the height F D, such that v2 FD= = 2 g 64 feet and through the point F draw the hori zontal line FA; then, letting the body descend as a pendulum through the arch A B D, when arrived at D, it will have acquired the proposed velocity. This is extremely useful in experiments on the collision of bodies. 12. The oscillations of pendulums in any arcs of a cycloid are isochronal, or performed in equal times. 13. Oscillations in small portions of a circular arc are isochronal. 14. The numbers of oscillations of two different pendulums, in the same time, and at the same place, are in the inverse ratio of the square roots of the length of these pendulums. = 15. If be the length of a single pendulum, or the distance from the point of suspension to the centre of oscillation in a compound pendulum, g the measure of the force of gravity (324 feet, or 386 inches at the level of St. Paul's in the latitude of London), t the time of one oscillation in an indefinitely small circular arc, and then 3·141593 : * At the level of the sea, in the latitude of London, g is 386-289 inches, and the corresponding length of the second pendulum is 39-1393 inches, according to the determination of Major Kater. Conformably with this result are the numbers in the table following art. 30 of this section, computed at the expense of Messrs. Bramahs and Donkin, and obligingly communicated by them for this work. It has been suspected by M. Bessel, and demonstrated by Mr. Francis Bailey, that, in the refined computations relative to the pendulum, the formulæ for the reduction to a vacuum are inaccurate, and that, in consequence, we do not yet precisely know the length of a second pendulum. See Phil. Transac. 1832. In other words, whatever be the force of gravity, the length of a second pendulum, and the space descended freely by a falling body in 1 second, are in a constant ratio. 18. If ' be the length of a pendulum, g' the force of gravity, and t' the time of oscillation at any other place, then If the same pendulum be actuated by different gravitating forces, we have When pendulums oscillate in equal times in different places, we have gg'l: l'. For the variations of gravity in different latitudes, see art. 9, pa. 238. 18. If the arcs are not indefinitely short, let v denote the versed sine of the semi-arc of vibration; then In which, when the semi-arc of vibration does not exceed 4 or 5 degrees, the third term of the series may be omitted. If the time of an oscillation in an indefinitely small arc be 1 second, the augmentation of the time will be So that for oscillations of 21° on each side of the vertical, the augmentation would not occasion more than 2" difference in a day. 19. If D denote the degrees in the semi-arc of an oscillating pendulum, the time lost in each second by vibrating in a circle instead of the cycloid, is D2 52524 ; and consequently the time lost in a whole day of 24 hours, or 24 x 60 x 60 seconds, is Dnearly. In like manner, the seconds lost per day by vibrating in the arc of A degree, is A. Therefore, if the pendulum keep true time in one of these arcs, the seconds lost or gained per day, by vibrating in the other, will be (D2 A). So, for example, if a pendulum measure true time in an arc of 3 degrees, it will lose 11 seconds a day by vibrating 4 degrees; and 26 seconds a day by vibrating 5 degrees and so on. 20. If a clock keep true time very nearly, the variation in the length of the pendulum, necessary to correct the error will be equal to twice the product of the length of the pendulum, and the error in time divided by the time of observation in which that error is accumulated. If the pendulum be one that should beat seconds, and t' the daily variation be given in minutes, and n be the number of threads in an inch of the screw which raises and depresses 2 × 39 × n t' the bob of the pendulum, then 24 × 60 ·05434 n 1'= n t' nearly, for the number of threads which the bob must be raised or lowered, to make the pendulum vibrate truly. 21. For civil and military engineers, and other practical men, it is highly useful to have a portable pendulum, made of painted tape, with a brass bob at the end, so that the whole, except the bob, may be rolled up within a box, which may be enclosed in a shagreen case. The tape is marked 200, 190, 180, 170, 160, &c. 80, 75, 70, 65, 60, at points which being assumed respectively as points of suspension, the pendulum will make 200, 190, &c. down to 60 vibrations in a minute. Such a portable pendulum may be readily employed in experiments relative to falling bodies, the velocity of sound, &c. The pendulum and its box may go in a waistcoat pocket. 22. If the momentum of inertia (Def. 6) of a pendulum, whether simple or compound, be divided by the product of the pendulum's weight or mass into the distance of its centre of gravity from the point of suspension (or axis of motion), the quotient will express the distance of the centre of oscillation from the same point (or axis.) 23. Whatever the number of separate masses or bodies which constitute a pendulum, it may be considered as a single pendulum, whose centre of gravity is at the distance d from the axis of suspension, or of rotation: then if K denote the momentum of inertia of that body divided by its mass, the distance so from the axis of rotation to the centre of oscillation or the length of an equivalent simple pendulum, will be 24. To find the distance of the centre of oscillation from the point or axis of suspension, experimentally. Count the number, n, of oscillations of the body in a very short arc in a minute ; then Thus, if a body so oscillating, made 50 vibrations in a minute : = 56.34 inches. then so = 140850 2500 Or, s o=39 t, in inches, t being the time of one oscillation in a very small arc. If the arc be of finite appreciable magnitude, the time of oscillation must be reduced in the ratio 8+ versin of semi-arc to 8, before the rule is applied. 25. From the foregoing principles are derived the following expressions for the distances of the centres of oscillation for the several figures, suspended by their vertices and vibrating.flatwise, viz. (1.) Right line, or very thin cylinder, s o= of its length. (2.) Isosceles triangle, s o=3 of its altitude. (3.) Circle, s o= radius. (4.) Common parabola, s o= of its altitude. (5.) Any parabola, s o = 2 m +1 3 m + 1 xits altitude. Bodies vibrating laterally or sideways, or in their own plane : (6.) In a circle, s o= of diameter. (7.) In a rectangle suspended by one angle, so = of diagonal. axis+para (S.) Parabola suspended by its vertex, s o= axis + para meter. (9.) Parabola suspended by middle of its base, so= axis+ parameter. (10.) In a sector of a circle, s o= (11.) In a cone, s o = axis+ 3 arc X radius 4. chord 5 axis * For some curious and valuable theorems, by Professor Airy, for the reduction of vibrations in the air to those in a vacuum, see Mr. F. Baily's paper referred to in the preceding note. (12.) In a sphere, so rad. + d + 2 rad. Where d is the length of the thread by which it is suspended. (13.) If the weight of the thread is to be taken into the account, we have the following distance between the centre of the ball, and that of oscillation, where B is the weight of the ball, d the distance between the point of suspension and its centre, r the radius of the ball, and w the weight of the thread or wire, GO= († 2 + B) 4 r2 — } w (2 d r + d3) ; or, if в be expressed in terms of w considered as a unit, then GO = B+ d (14.) If two weights w, w', be fixed at the extremities of a rod of given length w w', s being the centre of motion between w and w'; then, if d = s w, D =s w', and m the weight of an unit in length of the rod, we shall have SO= M D3 + 3 w' D3 + M D3 + 3 w D3 m D2 + 2 w'D- m D2- 2 W D the radii of the balls being supposed very small in comparison with the length of the rod. (15.) In the bob of a clock pendulum, supposing it two equal spheric segments joined at their bases, if the radii of those bases be each, the height of each segment v, and d the distance from the point of suspension to G the centre of the bob, then is 52 + 152 v2 + vs G = d. 10 -; which shows the distance of the centre of oscillation below the centre of the bob. If r the radius of the sphere be known, the latter expression z l v ž r v becomes Go= d (r-v) (16.) Let the length of a rectangle be denoted by 7, its breadth by 2 w, the distance (along the middle of the rectangle) from one end to the point of suspension by D, then the distance so, from the point of suspension to the centre of oscillation, d2 — d l + § 1o + } w2 73 12+3 202 l-d l-d will be s o= whether the figure be a mere geometrical rectangle, or a prismatic metallic plate of uniform density. It follows from this theorem that a plate of 1 foot long, and of a foot broad, and suspended at a fourth of a foot from either end, would vibrate as a half second pendulum. Also, that a plate a foot long, of a foot wide, and suspended at of a foot from the middle, would vibrate 36.469 times in 5 hours. |