g = √ { · √(DP2 — AP2). This will be the velocity with which the pendulum will describe an exceedingly minute portion of the arc, such as aa'. Draw, horizontally, de arc DP; with dp as radius de. scribe the quadrilateral arc dcc'q; make da = DA, aa' = AA', and draw ac, a'c', parallel to rq. Then, vel. at a = g g = √} • √(dp2—ap2) =√{ · √(dr2—ar2)=ac √/{. But, since time of describing a space as AA = aa', is in versely as the velocity, or t = S v aa' = = we have ✓ ac СР The same reasoning applies for every minuto successive portion, such as AA', of the semi-arc described by the pendulum and when the ball has descended from D to P, the corresponding arc to dr its equal is the quadrant dcc'a: the expression for the time, therefore, becomes, in that case, The time of ascending through PB PD is, manifestly, equal to the above: therefore, ultimately, the time of complete oscillation through DPB, is, Consequently, the times of oscillation are as the square roots of the lengths of the pendulums, the force of gravity remain. ing the same. 204. For the same reason that we have the above equa. when I is the length of the pendulum, and g the lineal mea. sure of the force of gravity, we have t↓· で " g in any other place where g' measures the force of gravity, and l' is the length of the pendulum. If the force of gravity be the same, we have . (2). If the same pendulum be actuated by different gravitating forces, we have g g When pendulums oscillate in equal times in different places, we have g:g': :l: l'. Other theorems may readily be deduced. 205. If either g or be determined by experiment, the equa. 1 for t will give the other. Thus, ifg, or the space fallen through by a heavy body in 1" of time, be found, then this theorem will give the length of the seconds pendulum. Or, if the length of the seconds pendulum be observed by experiment, which is the easier way; this theorem will give g. Now, in the latitude of London, the length of a pendu Jum which vibrates seconds, has been found to be 39 inches; and this being written for 1 in the theorem, it gives 391 g 193.07 = 1" : and hence is found g inches = 16 feet, for the descent of gravity in 1"; which it has also been found to be very exactly, by many accurate experiments,l=g X 20264; g1 x 4.9348. SCHOLIUM. = 206. Hence is found the length of a pendulum that shall make any number of vibrations in a given time. Or, the number of vibrations that shall be made by a pendulum of a given length. Thus, suppose it were required to find the length of a half-seconds pendulum, or a quarter-seconds pendulum; that is, a pendulum to vibrate twice in a second, or 4 times in a second. Then, since the time of vibration is as the square root of the length, therefore 1:39:1, or 1:1:: 391 391: = 9 inches nearly, the length of the half-seconds pendulum. And 1:391: 27 inches, the length of the quarterseconds pendulum. Again, if it were required to find how many vibrations a pendulum of 80 inches long will make in a minute. Here ✔80: √ 391 :: 60′′ or l′ : 60 √ 391 71⁄2 ✓ 31·3 = 41.95987, or almost 42 vibrations in a minute. 207. For military men it is a good practice 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, and the whole 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 is highly useful in experiments relative to falling bodies, the velocity of sound, &c. For the comparison of the times of oscillation in indefinitely small arcs of circles, in finite arcs of circles, and in cy. cloidal arcs, the student may turn to probs. 13 and 14, in Practical Exercises on Forces, and prob. 42, in Promiscuous Exercises near the end of this volume, CENTRAL FORCES. 208. Def. 1. Centripetal force is a force which tends constantly to solicit or to impel a body towards a certain fixed point or centre. 2. Centrifugal force is that by which it would recede from such a centre, were it not prevented by the centripetal force. 3. These two forces are, jointly, called central forces. 209. PROP. If a body, м, drawn continually towards a fixed point, c, by a constant force, o, and projected in a direction, MB, perpendicular to cм, describe the circumference of a circle about the centre c, the central force o, is to the weight of the body, as the altitude due to the velocity of projection, is to half the radius CM. Letv be the velocity of projection in the tangent MB, and r the radius CM. Independently of the action of the central force, the body would describe, along MB, during the very small time t, a space мN tv, and would recede from the point c by the quantity IN, which may, without C. GAM N error, be regarded as equal to GM, when the arc MI is exceedingly small. If, therefore, the body, instead of moving in the tangent, were kept in the circumference by the central force, its operation in the time t, would (art. 130) be equal to ¦❤ť3, and at the same time = MG. But by the nature of the Putting a for the altitude due to the velocity v, since (by Thus far, we have, in reality, considered only the unit of mass; but, if we multiply the first two terms of the above proportion by the mass of the body, the whole will still remain a correct proportion, and the general result may be thus enunciated: viz. The centripetal force of any body, if it be free, or its centrifugal force, if it be retained to the centre c, by a thread (or otherwise), is to the weight of that body, as the height due to the velocity v, is to the half of the radius CM. 210. Hence, it appears that, so long as q and r remain constant, the velocity will be constant. 211. If both members of the equation 1 be multiplied by the mass м of the body, and we put F to represent the cen. Mv2 trifugal force of that mass, we shall have F = In like manner, if ' is the centrifugal force of another body which revolves with the velocity v' in a circle whose radius is r', we shall have 212. If r and r' denote the times of revolution of the two 213. If the times of revolution are equal, we shall have FF; Tr VOL, II, 31 (4), 214. And, if we assume : T '2 :: 3:3, as in the plane. tary motions, the proportion (3) will become 215. The subject of central forces is too extensive and momentous to be adequately pursued here. The student may consult the treatises of mechanics by Gregory and Poisson, and those on fluxions by Simpson, Dealtry, &c. We shall simply present in this place, one example connected with practical mechanics. EXAM. Investigate the characteristic property of a conical pendulum applied as a regulator or governor to steam-engines, &c. G This contrivance will be readily comprehended from the marginal figure, where Aa is a vertical shaft capable of turning freely upon the sole a. CD, CF, are two bars which move freely upon the centre c, and carry at their lower extremities two equal weights, P, Q: the bars cn, CF, are united, by a proper articulation, to the bars G, B, which latter are attached to a ring, I, capable of sliding up and down the vertical shaft, aa. When this shaft and connected apparatus are made to revolve, in virtue of the centrifugal force, the balls P, Q, fly out more and more from sa, as the rotatory velocity increases: if, on the contrary, the rotatory velocity slackens, the balls descend and approach aɑ. ring I ascends in the former case, descends in the latter and a lever connected with 1 may be made to correct appropri ately, the energy of the moving power. Thus, in the steamengine, the ring may be made to act on the valve by which the steam is admitted into the cylinder; to augment its opening when the motion is slackening, and reciprocally diminish it when the motion is accelerated. The The construction is, often, so modified that the flying out of the balls causes the ring I to be depressed, and vice versa ; but the general principle is the same. If FQ=FI= DP =DI, then I, P, Q, are always in some one horizontal plane : but that is not essential to the construction. |