the mass which being placed at P, the pendulum will still receive the same motion as before. go Here then are two quantities of matter, namely, b and P, the former moving with the velocity v, and striking the latter at rest; to determine their common velocity u, with which they will jointly proceed forward together after the stroke. In which case, by the law of the impact of non-elastic bodies, we have bii+gop u the ve bii go + b : b :: v : u, and therefore v = locity of the ball in terms of u, the velocity of the point P, and the known dimensions and weights of the bodies. But now to determine the value of u, we must have recourse to the angle through which the pendulum vibrates; for when the pendulum descends again to the vertical position, it will have acquired the same velocity with which it began to ascend, and by the laws of falling bodies, the velocity of the centre of oscillation is such as a heavy body would acquire by freely falling through the versed sine of the arc described by the same centre o. But the chord of that arc is c, and its radius is o; and, by the nature of the circle, the chord is a mean proportional between the versed sine and diameter, therefore 20: cc: CC " 20 the versed sine of the arc described by o. Then, by the laws of falling bodies point o in descending through the arc whose chord is c, where a 16 feet: and therefore o:i::c✔✅ 2a ci 2a which is the velocity u, of the point P. Then, by substituting this value for u, the velocity of the bii + ball, before found, becomes v = gop bio that the velocity of the ball is directly as the chord of the arc described by the pendulum in its vibration. SCHOLIUM. 241. In the foregoing solution, the change in the centre of oscillation is omitted, which is caused by the ball lodging in the point P. But the allowance for that small change, and that of some other small quantities, may be seen in my Tracts, where all the circumstances of this method are treat. ed at full length. For an example in numbers, suppose the weights and dimensions to be as follow: namely, P = 570lb. 18oz. 1drbii+gop Then = 1·131lb. bio g= 78 inc. 1·131×94·32+78×847×570 1-131 X94 X847 0 = 847 inc. =7.065 feet i = 2a c= 18.73 inc. Therefore 656·56 × 2·1337, or 1401 feet, is the velocity, per second, with which the ball moved when it struck the pendulum. 242. When the impact is made upon the centre of oscil. lation, the computation becomes simplified. In that case, since the whole mass, p, of the pendulum, may be regarded as concen. tered at o, and the ball, b, strikes that point, we shall have bus (b + p)v; v being the velocity of the ball before the impact, and v' that of the ball and pendulum together, after the impact. Now, if the centre of oscillation o, after the blow, describes the arc oo', before the motion is destroyed, the velocity will be equal to that acquired by falling through the versed sine vo, of the arc oo' or angle s to the radius so. But, if the time t of a very minute oscillation of the pendulum be known or inferred from that in an ascertained arc, we have (art. 233), so = 39 inches 33t feet. = Hence vo so nat. versin s, and (art. 154) v' = √(64 × 3.2604 = 14.48286t/versin s. (b+p)o' = b + p versin s) 14.48286t/versin s. This mode of computation, with a slight and obvious change, applics to qu. 48 of the Practical Exercises in Natural Philosophy. OF HYDROSTATICS. 243. HYDROSTATICS is the science which treats of the pressure, or weight, and equilibrium of water and other fluids, especially those that are non-elastic. 244. A fluid is elastic, when it can be reduced into a less bulk by compression, and which restores itself to its former bulk again when the pressure is removed; as air. And it is non-elastic, when it is not compressible or expansible, as water, &c. 245. PROP. If any part of a fluid be raised higher than the rest, by any force, and then left to itself; the higher parts will descend to the lower places, and the fluid will not rest, till its surface be quite even and level. For, the parts of a fluid being easily moveable every way, the higher parts will descend by their superior gravity, and raise the lower parts, till the whole come to rest in a level or horizontal plane. Corol. 1. Hence, water that communicates with other water, by means of a close canal or pipe, will stand at the same height in both places. Like as water in the two legs of a syphon. Corol. 2. For the same reason, if a fluid gravitate towards a centre; it will dispose itself into a spherical figure, the centre of which is the centre of force. Like the sea in respect of the earth. 246. PROP. When a fluid is at rest in a vessel, the base of which is parallel to the horizon; equal parts of the base are equally pressed by the fluid. For, on every equal part of this base there is an equal column of the fluid supported by it. And as all the columns are of equal height, by the last proposition they are of equal weight, and therefore they press the base equally.; that is, equal parts of the base sustain an equal pressure. Corol. 1. All parts of the fluid press equally at the same depth. For, if a plane parallel to the horizon be conceived to be drawn at that depth; then the pressure being the same in any part of that plane, by the proposition, therefore the parts of the fluid, instead of the plane, sustain the same pressure at the same depth. Corol. 2. The pressure of the fluid at any depth, is as the depth of the fluid. For the pressure is as the weight, and the weight is as the height of the fluid. Corol. 3. The pressure of the fluid on any horizontal surface or plane, is equal to the weight of a column of the fluid, whose base is equal to that plane, and altitude is its depth below the upper surface of the fluid. 247. PROP. When a fluid is pressed by its own weight, or by any other force; at any point it presses equally, in all directions whatever. This arises from the nature of fluidity, by which it yields to any force in any direction. If it cannot recede from any force applied, it will press against other parts of the fluid in the direction of that force. And the pressure in all directions will be the same: for if it were less in any part, the fluid would move that way, till the pressure be equal every way. Corol. 1. In a vessel containing a fluid; the pressure is the same against the bottom, as against the sides, or even upwards at the same depth. B Corol. 2. Hence, and from the last proposition, if ABCD be a vessel of water, and there be taken, in the base produced, DE, to represent the pressure at the bottom; joining AE, and drawing any parallels to the base, as FG, HI; then shall FG represent the pressure at the depth AG, and I the pressure at the depth ar, and so on; because the parallels E FG, HI, ED, by sim. triangles, are as the depths AG, AI, AD: which are as the pressures, by the proposition. And hence the sum of all the FG, HI, &c. or the area of the triangle ADE, is as the pressure against all the points G, I, e. that is, against the line AD. But as every point in the line CD is pressed with a force as DE, and that thence the pressure on the whole line co is as the rectangle ED. DC, while that against the side is as the triangle ADE or DA. DE; therefore the pressure on the horizontal line Dc, is to the pressure against the vertcal line DA, as DC to DA. And hence, if the vessel he an upright rectangular one, the pres sure on the bottom, or whole weight of the fluid, is to the pressure against one side, as the base is to half that side. Therefore the weight of the fluid is to the pressure against all the four upright sides, as the base is to half the upright surface. And the same holds true also in any upright vessel, whatever the sides be, or in a cylindrical vessel. Or, in the cylinder, the weight of the fluid is to the pressure against the upright surface, as the radius of the base is to double the altitude. Also, when the rectangular prism becomes a cube, it appears that the weight of the fluid on the base, is double the pressure against one of the upright sides, or half the pressure against the whole upright surface. Corol. 3. The pressure of a fluid against any upright surface, as the gate of a sluice or canal, is equal to half the weight of a column of the fluid whose base is equal to the surface pressed, and its altitude the same as the altitude of that surface. For the pressure on a horizontal base equal to the upright surface, is equal to that column; and the pressure on the upright surface, is but half that on the base, of the same area. So that, if b denote the breadth, and d the depth of such a gate or upright surface; then the pressure against it, is equal to the weight of the fluid whose magnitude is abd' = AB. AD2. Hence, if the fluid be water, a cubic foot of which weighs 1000 ounces, or 621 pounds; and if the depth AD be 12 feet, the breadth AB 20 feet; then the content, or AB. AD2, is 1440 feet; and the pressure is 1440000 ounces, or 90000 pounds, or 40 tons weight nearly. 248. PROP. The pressure of a fluid on a surface any way immersed in it, whether perpendicular, or horizontal, or ob lique, is equal to the weight of a column of the fluid, whose base is equal to the surface pressed, and its altitude equal to the depth of the centre of gravity of the surface pressed be. low the top or surface of the fluid. For, conceive the surface pressed to be divided into innumerable sections parallel to the horizon; and let s denote any one of those horizontal sections, also d its distance or depth below the top surface of the fluid. Then, by art. 246, YOL. II. 33 |