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t; t . . v. l ; Vl. . . . . . (3). If the same pendulum be actuated by different gravitating forces, we have
When pendulums oscillate in equal times in different places, we have g: g’: ; l; l. . . . . . (5). Other theorems may readily be deduced. 205. If either g or l be determined by experiment, the equa. 1 for t will give the other. Thus, if #g, 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 Now, in the latitude of London, the length of a pendulum which vibrates seconds, has been found to be 394 inches; *QL and this being written for l in the theorem, it gives or V 39; = 1": and hence is found #g = 3rol =}* x 3 + = 193-07 inches = 16 to feet, for the descent of gravity in 1"; which it has also been found to be very exactly, by many accurate experiments, l = #g × 20264; $g = 1 × 4.9348.
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 : #: : V 39; : V l, or 1 ; ; ; ; 391 : * = 93 inches nearly, the
length of the half-seconds pendulum.
And 1 : P : : 39; : 2F, 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
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 usesul 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.
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, M, drawn continually towards a fixed point, c, by a constant force, p, and projected in a direction, Mb, perpendicular to cM, describe the circumference of a circle about the centre c, the central force p, is to the weight of the body, as the altitude due to the velocity of projection, is to half the radius cy.
Let v be the velocity of projection in the tangent MP, and r the radius cM. Independently TM of the action of the central force, the body would describe, along MB, during the very small time t, a space MN = tw. and would recede from the point c by the quantity IN, which may, without 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
Putting a for the altitude due to the velocity p, since (by 2 art. 154) v' = 2ag, we have p = * ; whence there re.
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 p and r remain constant, the velocity v will be constant.
211. If both members of the equation 1 be multiplied by
the mass M of the body, and we put F to represent the cen. - Mr.” trifugal force of that mass, we shall have F = ---. In like
manner, if F is the centrifugal force of another body which revolves with the velocity v in a circle whose radius is r", we shall have p” U'2 F: f' : : — : + - . . 2). r r , (2) 212. If T and T' denote the times of revolution of the two - 2mor , 2-r' bodies, because v = −, and v = 7, we have T
- * : * : . . . . . . . . . (3). 213. If the times of revolution are equal, we shall hays f : F : ; r ; r. . . . . . (4), Wol, II, 31
214. And, if we assume to : To : : r": r", as in the planetary 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, Dealiry, &c. We shall simply present in this place, one example connected with practical mechanics.
ExAM. Investigate the charaeteristic property of a conical pendulum applied as a regulator or governor to steam-engines, &c.
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, H, which latter are attached to a ring, 1, 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 Aa, as the rotatory velocity increases: if, on the contrary, the rotatory velocity slackens, the balls descend and approach Aa. The ring 1 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 construction is, often, so modified that the flying out of the balls causes the ring 1 to be depressed, and vice versa; but the general principle is the same. If Fq = FI = or = D1, then I, P, Q, are always in some one horizontal plane : but that is not essential to the construction.
Now, lett denote the time of one revolution of the shaft, x the variable horizontal distance of each ball from that shaft, or as usual = 3.141593: then will the velocity of each
ball be -*. and (art. 209.) its centrifugal force
- s - (#) -- r = *:: The balls being operated upon si
multaneously by the centrifugal force and the force of gravity, of which one operates horizontally, the other vertically, the resultant of the two forces is, evidently, always in the actual position of the handle cd, cf. It follows, therefore, that the ratio of the gravity to the centrifugal force, is that of cos. icq to sin. Ica, or that of the vertical distance of Q below c to its horizontal distance from Aa. Call the former d, the latter being x : 4tro or
Hence, the periodic time varies as the square root of the altitude of the conic pendulum, let the radius of the base be what it may.
Hence, also, when Icq = 10P = 45°, the centrifugal force of each ball is equal to its weight.
ON THE CENTRES OF PERCUSSION,
216. The Centre of Percussion of a body, or a system of bodies, revolving about a point, or axis, is that point, which striking an immoveable object, the whole mass shall not incline to either side, but rest, as it were, in equilibrio, without acting on the centre of suspension.
217. The Centre of Oscillation is that point, in a body vibrating by its gravity, in which if any body be placed, or if the whole mass be collected, it will perform its vibrations in the same time, and with the same angular velocity, as the whole body, about the same point or axis of suspension.
218. The Centre of Gyration is that point, in which if the whole mass be collected, the same angular velocity will