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6. A cylinder, the axis of which is vertical, is filled with fluid, the density of which varies directly as the depth; to find the pressure on the concave surface of a portion of the cylinder included between two horizontal sections.
If h denote the length of the cylinder, c the circumference of a circular section, p the density at the lowest points of the fluid, and b, 6', the depths of the two horizontal sections, the required pressure will be equal to
h Bossut: Traité d' Hydrodynamique, tom. I. p. 40. 7. A semicircle, having its diameter in the surface of a fluid, is divided into three equal sectors; to compare the sum of the pressures on the outer sectors with the pressure on the middle sector, the density of the fluid being supposed to vary as the depth.
If P represent the sum of the pressures on the two outer sectors, and Q the pressure on the middle sector, then
8. A cycloidal area is immersed in fluid, with its axis vertical and its vertex at the surface; to determine the whole pressure on the cycloid, the density of the fluid varying directly as the depth.
If a denote the radius of the generating circle, and ß the density of the fluid at the middle point of the axis of the cycloid, the whole pressure on the cycloidal area will be equal to
imagß. 9. The density of the fluid being supposed to vary as the depth, to find the pressure upon a triangular plane, one angle of which is a right angle and the base of which coincides with the surface of the fluid, the inclination of the plane to the horizon being given.
Let h be the height of the triangle, 1 the length of its base, the inclination of its area to the horizon, and ß the density of the fluid at a depth a below its surface; then the required pressure will be equal to
SECTION III. Normal Pressure of Aeriform Fluids of Uniform Temperature on the
Surfaces of Solids.
In elastic or aeriform fluids of uniform temperature, the value of p is to be determined from the two equations
dp = gpdx, p = kp, where k is a constant quantity dependent upon the elasticity of the fluid ; the total normal pressure upon the surface of any solid immersed in the fluid being then determined, as in the preceding section, by the evaluation of the expression (KP).
Eliminating p between the two equations, we have
whence, C being an arbitrary constant, we obtain
Hence the total normal pressure is equal to
CΣ (Κ. ελε). In the aeriform fluids of nature, 1 is an extremely small quantity, and therefore, approximately, unless x be very large, as for instance in estimating the pressure of the earth's atmosphere at points very near the surface of the earth, we shall have p = C, and therefore the formula for the pressure will be reduced to CE (K).
The ancient philosophers were ignorant that air is a ponderable fluid. They conceived all substances as naturally arranging themselves into two classes, the one class comprehending heavy and the other light bodies, those bodies being regarded as essentially heavy which tend to fall towards the earth, and those as essentially light which tend to rise from it. This misconception of the mechanical properties of air resulted from their ignorance of the true theory of such phenomena as depending upon the displacement of heavy fluids or bodies by bodies or fluids more heavy, all substances in fact naturally tending to descend. The idea of the ponderable nature of air first presented itself to Galileo ; the first confirmation however of this notion by the test of experiment is due to his pupil Torricelli. In accordance with Galileo's idea, the pressure of the atmosphere, as indicated by the Torricellian tube or ordinary barometer, must be greater at the level of the sea than at the summits of mountains. The first independent proof of this conclusion was afforded by Pascal's experiments, in the year 1648, on the mountain of Puy-de-Dome, near Clermont in Auvergne : in ascending from the foot to an altitude of 3000 Paris feet, the barometric column subsided three inches and one eighth of an inch ; or, in English measure, in ascending to an altitude of 3204 feet, the height of the quicksilver was diminished by three inches and one third of an inch.
1. An upright cylinder, closed at both ends, is filled with an elastic fluid ; to determine the whole pressure on the concave surface of the vessel.
Let ß denote the elastic force of the fluid at the upper end of the vessel; then
p = C.elaB = C, and therefore Hence, c denoting the circumference of a horizontal section and h the altitude of the cylinder, the required pressure will be equal to
Supposing 1 to be very small and h not very large, we see that, expanding sdh by powers of wh, and neglecting terms of higher
powers than the first, the expression for the pressure will be reduced to cßh.
2. If the density of a fluid, supposed to consist of an indefinite number of mutually repellent particles at finite intervals, vary as the pressure, to determine the law of the repulsive force between the particles.
Conceive the fluid to be contained in the cubical space ACE (fig. 3), and then to be reduced by compression into the smaller cubical space ace; and the distances between the particles, occupying a similar position inter se in both the spaces, will be as the sides AB, ab, of the cubes, and the densities of the media reciprocally as the containing spaces, viz. the cube of AB and the cube of ab.
In the plane face of the greater cube ABCD take the square
: pressure of db on the included fluid
:: (ab)*: (AB).
: pressure of DP on the included fluid
: pressure of db on the included fluid
:: ab: AB. Conceive planes FGH, fgh, to be drawn through the two fluids parallel to their faces AE, ae; these planes will divide the fluids into two parts which will press against each other with the same forces with which the fluids are pressed by the planes AC, ac, that is, in the proportion of ab to AB; and accordingly the repulsive forces, by which these pressures are sustained, will be in the same ratio. But, the number of the
particles in the two cubes being the same and their positions being similar, the total mutual pressures of the two portions of the fluids at the planes FGH, fgh, are as the mutual repulsive forces of two similar particles in these two cubes. Hence
The repulsive force between two particles in the larger cube : the repulsive force between two particles similarly situated in
1 the smaller cube :: ab : AB :: r and r' being the dis
p tances between the two particles in the former and in the latter cube.
Cor. The converse proposition is also true. Suppose, in fact, that the mutual repulsions between two similarly situated
1 particles in the two cubes are as
Then the total pres
sures on the faces DB, db, will be as the sums of the forces of the separate particles, or, the number of the particles at these
1 faces being the same, as - : that is, as ab : AB. Also Pressure of DP : pressure of DB::(ab)? :(AB)”; hence Pressure of DP : pressure of db :: (ab)' : (AB);
:: density of ACE:density oface. Newton: Principia, Lib. 11. Sect. 5, Prop. 23. 3. A spherical envelope is filled with elastic fluid ; to determine the whole pressure on the envelope.
Let ß represent the unit of pressure of the fluid at the centre of the envelope, and let a denote its radius; then the required pressure will be equal to
If a be a very small quantity, then, supposing a not to be large, this expression, by puting 1 + da, 1 - la, for ela, respectively, is reduced to
4παβ, a value which might have been obtained at once by multiplying