the roots of the whole curves. Therefore, the whole times are in the same ratio of AC to ac. Corol. 1. Because the axes DC, Dc, of similar curves, are as the lengths of the similar parts AC, ac; therefore the times of descent in the curves AC, ac, are as DC to Dc, or the square roots of their axes. Corol. 2. As it is the same thing, whether the bodies run down the smooth concave side of the curves, or be made to describe those curves by vibrating like a pendulum, the lengths being DC, Dc; therefore the times of the vibration of pendulums, in similar arcs of any curves, are as the square roots of the lengths of the pendulums. SCHOLIUM. 201. Having, in the last corollary, mentioned the pendulum, it may not be improper here to add some remarks concerning it. B A simple pendulum consists of a small ball, or other heavy body B, hung by a fine string or thread, moveable about a centre A, and describing the arc CBD; by which vibration the same motions happen to this heavy body, as would happen to any body descending by its gravity along the spherical superficies CBD, if that superficies were perfectly hard and smooth. If the pendulum be carried to the situation AC, and then let fall, the ball in descending will describe the arc CB; and in the point B it will have that velocity which is acquired by descending through CB, or by a body falling freely through EB. This velocity will be sufficient to cause the ball to ascend through an equal arc BD, to the same height D from whence it fell at C; having there lost all its motion, it will again begin to descend by its own gravity; and in the lowest point B it will acquire the same velocity as before; which will cause it to re-ascend to C: and thus, by ascending and descending, it will perform continual vibrations in the circumference CBD. And if the motions of pendulums met with no resistance from the air, and if there were no friction at the centre of motion A, the vibrations of pendulums would never cease. But from these obstructions, though small, it happens, that the velocity of the ball in the point B is a little diminished in every vibration; and consequently it does not return precisely to the same points C or D, but the arcs described continually become shorter and shorter, till at length they are insensible; unless the motion be assisted by a mechanical contrivance, as that in clocks called a maintaining power. Our present investigations relate to the simple pendulum, above described : the consideration of compound pendulums requires the previous knowledge of the centre of oscillation. 202. PROP. When a pendulum vibrates in a circular arc, the velocities acquired in the lowest point, are as the chords of the semi-arcs described. For, the velocity at P of a body that has descended through any arc AP, is equal to the velocity at P of a body that has fallen freely through the versed-sine NP (art. 199, cor. 2.) Hence, velocity at P after descent through arc AP, is to velocity at P after descent through arc A ́P, as NP to /N/P, that is (Geom. th. 87) as chord AP to chord A'P. A' Corol. If, therefore, we would impart to a body a given velocity, v, we have v2 v2 and through 2g 64 feet' the point N draw the horizontal line NA; then AA'P an arc (of any circle passing through P) is one, through which when a body has fallen it will have acquired the proposed velocity. This is extremely useful in experiments on collision. only to compute the height NP, such that NP = = 203. PROP. To investigate the time of vibration of a pendulum of given length, in an indefinitely small arc. Now, in estimating the time of an oscillation in an indefinitely small circular arc, let it be recollected that the excess of such an arc above its chord, being incomparably less than itself, may be neglected; so that we may consider the square of such an arc (like that of its chord, Geom. th. 37) as equal to the rectangle under the versed sine and the diameter. Indeed, if instead of indefinitely small arcs we took arcs of 40' or 50', and compared the respective differences of their squares and those of their chords, we should find that the error would not exceed the 29000dth part of either result. Thus, arc2 50' arc2 40' while chord2 50' chord2 40' 1454442 1163552 S K IN d αν P 261799 × 29089, 1163542 261796 × 29088. Let. then, DPB represent such a very short oscillation of a pendulum whose length, 7, is SP, S being the point of suspension. must fall to acquire the velocity at A. Putting this value of the altitude in the usual expression for falling bodies, v=√(2gs), it becomes v= √(2g. g DP2-AP2) 21 =√ 3⁄41⁄2· √(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, dP = arc DP; with dP as radius describe the quadrantal arc dcc'Q; make da DA, aa AA', and draw ac, a'c', parallel to PQ. Then, vel. at A · · √(DP2 — AP2) = √ Z · √(dP2 — aP2) = acNo 2. But, since time of describing a space as AA'aa', is inversely as the velocity, The same reasoning applies for every minute 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 dP its equal is the quadrant dcc'Q: 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 remaining the same. 204. For the same reason that we have the above equa. when l is the length of the pendulum, and g the lineal measure of the force of gravity, we have t = ľ πν in any other place where g' measures the force of gravity, and l' is the g'' length of the pendulum. If the same pendulum be actuated by different gravitating forces, we have When pendulums oscillate in equal times in different places, we have 9 : g' : : l : l'. Other theorems may readily be deduced. (5). 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 g. Now, in the latitude of London, the length of a pendulum which vibrates seconds, has been found to be 39 inches; and this being written for 7 in the theorem, it gives π √ =1": and hence is found g = π2l = {π2 × 39} = 193·07 inches = 161⁄2 feet, for the descent of gravity in 1"; which it has also been found to be very exactly, by many accurate experiments. g x 20264; g = 1 X 4.9348. Hence / 391 9 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 four times in a second. Then, since the time of vibration is as the square root of the length, therefore 1 :: √39}: √l, or 1: seconds pendulum. 391 391: = 9 inches nearly, the length of the half And 139: 27 inches, the length of the quarter seconds pendulum. Again, if it were required to find how many vibrations a pendulum of 80 inches long will make in a minute. Here 80: 39: 60" or 1': 60 39 80 = 7√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 cycloidal arcs, the student may turn to some subsequent problems relating to forces, &c. EXERCISES ON THE DOCTRINE OF PENDULUMS. 1. How long will a pendulum 60 inches in length be in making one vibration? Ans. 1.2386 sec. 2. How many vibrations will a pendulum 36 inches long make in a minute? Ans. 62.55. 3. What is the length of the pendulum that vibrates 3 times in a second? Ans. 4 in. 4. Required the length of a pendulum that makes as many vibrations in a minute as it is inches in length. Ans. 52.03 in. 5. Find the length of a pendulum that will make 3 vibrations while a heavy body falls through a space of 500 feet. Ans. 135 096 in. 6. How often will a pendulum 25 inches in length vibrate in 2 minutes? Ans. 187.65 times. 7. What is the length of a pendulum that vibrates 20 times in a minute? Ans. 352 in. 8. If a clock lose a minute an hour, how much must the pendulum be shortened to make it keep true time? Ans. 1.337 in. 9. Suppose a clock was observed to lose 30 sec. in 12 hours; what must the pendulum be shortened to make it keep true time? Ans. 055 in. 10. Required the length of a pendulum that vibrates siderial seconds, the length of the siderial day being 23h. 56m. 4°. Ans. 38.911 in. OBS. I. The length of the seconds pendulum varies in different latitudes on account of the centrifugal force. Thus, it has been found that, for any latitude L, 1 = l' + d sin.2 L, where l' is the length of the seconds pendulum at the equator (= 39′0265 in.) and d the difference between that and the length at the poles (=1608 inches nearly); which gives the expression 7 39.0265 +1608 sin. L. 51° 30'. Ex. 1. Required the length of the seconds pendulum at London in latitude Ans. 39 124986 in. Ex. 2. Find the length of the seconds pendulum at St. Petersburgh in latitude 59° 56'. Ans. 39 1469 in. Ex. 3. What is the length of the pendulum at the Cape of Good Hope in latitude 34° 30'? Ans. 39-078 in. OBS. II. If ' denote the length of a seconds pendulum at any height h, above the earth's surface, r the radius of the earth, and 7 the length at the surface, we Ex. 1. Required the length of the pendulum that would vibrate seconds on the top of Chimborazo, 21,000 feet above the level of the sea, in lat. 1° 36', the radius of the earth being considered 3960 miles. Ans. 38 9486 in. 2. What would be the length of the seconds pendulum on the highest peak of the Himalaya mountains, supposed to be 27,000 feet high, and in latitude 20° ? Ans. 38 9634 in. 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, M, drawn continually towards a fixed point, C, by a constant force, o, and projected in a direction, MB, perpendicular to CM, describe the circumference of a circle about the centre C, the central force, is to the weight of the body, as the altitude due to the velocity of projection, is to half the radius CM. GAM N 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 MN = tv, 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 force o, its operation in the time t, would (art. 137) be equal to 1⁄2øt2, and at the same time MG. But by the nature of the circle MG= = MI2 MN2 2r 2r * The variation in the length of a pendulum oscillating seconds at any point upon or for three miles above the earth's surface, never amounts to three-tenths of an inch. |