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Let G, R, o be the centres of gravity, gyration, and oscil, lation, as determined by the foregoing propositions: and let r be the point where the ball strikes the face of the pendulum the momentum of which, or the product of its weight and velocity, is expressed by the force ƒ, acting
at r, in the foregoing propositions. Now,
the whole weight of the pendul.
the weight of the ball,
so the dist. of the cen. of grav.
so the dist. of the cen. of oscilla.
u that of the point of impact P.
chord of the arc described by o.
By prop. 56, if the mass p be placed all at a, the pendulum will receive the same motion from the blow in the
point P; and as sp2; SR2 ; : P • por gap or
quantities of matter, namely, b and p, the former moving
with the velocity v, and striking the latter at rest; to deter-
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 down 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: 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
the velocity acquired by the
the velocity u, of the point e.
Then, by substituting this value for u, the velocity of the bii +5°p xc. So that the
ball before found, becomes v
velocity of the ball is directly as the chord of the arc de scribed by the pendulum in its vibration.
299. 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 treated at full length,
300. For an example in numbers of this method, suppose the weights and dimensions to be as follow: namely,
b = 180z. 11dr. bii+gop xc=
p = 570lb,
i = 94 inc. And
Therefore 656:56 X2.1337 or 1401 feet, is the velocity, per second, with which the ball moved when it struck the pen, dulum.
301. HYDROSTATICS is the science which treats of the pres sure, or weight, and equilibrium of water and other fluids, especially those that are non-elastic.
302. A fluid is elastic, when it can be reduced into a less volume 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 by such force; as water, &c.
303. 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.
306. 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.
307. 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.
308. 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.
309. Corol. 5. 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.
310. When a Fluid is Pressed by its own Weight, or by any other Force; at any Point it Presses Equally, in all Direc tions 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.
311. 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.
312. 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 HI the pressure at the depth AI, and so on; because the parallels 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 area of the triangle ADE, is as the pressure against all the points G, I, &C. 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 CD is as the rectangle ED. DC, while that against the side is as the triangle ADE or AD. DE; therefore the pressure on the horizontal line Dc, is to the pressure against the vertical line DA, as DC to DA. And hence, if the vessel be an upright rectangular one, the pressure 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.
313, 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 1bd2 =¦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 LAB. AD2, is 1440 feet; and the pressure is 1440000 ounces, or 90000 pounds, br 40,5 tons.
314. The pressure of a Fluid on a Surface any how immersed in it, either Perpendicular, or Horizontal, or Oblique; 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 below 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 dits distance or depth below the top surface of the fluid. Then, by art. 309, the pressure of the fluid on the section is equal to the weight of ds; consequently the total pressure on the whole surface is equal to all the weights ds. But, if b denote the whole surface pressed, and g the depth of its centre of gravity below the top of the fluid; then, by art. 256 or 259, bg is equal