AVERAGE PRESSURE OVER A SURFACE.

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Corollary. If the boundary of BC vanishes, so that BC, instead of being a portion, is the whole surface of a body, immersed or partly immersed in liquid, which is in contact with the liquid, then the horizontal thrust on BC is zero; and the resultant thrust is vertical, and equal to the weight of the displaced liquid.

B

Fig. 35.

In other words "A body plunged into liquid is buoyed up by a force equal to the weight of the displaced liquid, acting vertically upwards through the C.G. of this liquid."

This Corollary is important as the first established Theorem of Hydrostatics, and it is called Archimedes' Principle, from the name of its discoverer; we shall return to this Theorem and its consequences in Chapter III. 46. The Average Pressure over a Surface.

To find the average pressure over a curved surface BC, we must find the sum of all the thrusts on every element of BC and then divide by the area of the surface.

Now if S denotes the area of the curved surface, and AS of an element of the surface at a depth z, at which the pressure is p=wz,

the liquid being homogeneous and at rest under gravity, then the sum of all the normal thrusts

=ΣPAS=ΣwzAS=wZzAS = wzS,

if z denotes the depth of the C.G. of the surface S; so that the average pressure is wz, the pressure at the depth of the C.G. of the surface.

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WHOLE NORMAL PRESSURE.

The sum of all the normal thrusts on a surface is sometimes called the WHOLE NORMAL PRESSURE, on the surface; but as it has no mechanical significance for a curved surface and employs the word pressure in the sense of thrust, we shall seldom employ this expression, but use the idea of AVERAGE PRESSURE instead.

47. In the case of a plane surface, the whole pressure and the RESULTANT THRUST are the same; and we find as before (§ 35) that the AVERAGE PRESSURE is the pressure at the C.G. of the plane area.

For instance, if a rectangle or parallelogram ABCD, with one side AB in the surface of the liquid, is to be divided up by horizontal straight lines into n parts on which the hydrostatic thrust is the same, then if PQ is the rth line of division below AB, the thrust on ABPQ must be r/n of the thrust on ABCD; so that if x, and h denote the depths of PQ and CD below AB, and AB=a,

so that the depths of the dividing lines below AB are as 1/2/3: ...: √r: √(n − 1);

and from each other are as

1:√2-1:√3- √2 : ... ;

like the dynamical formula required for the times of falling freely through equal vertical distances, deduced from s=1gt2, or t=√(2s/g).

Similarly it can be shown that for dividing up a triangle ABC, whose vertex A lies in the surface and base BC is horizontal, by horizontal lines into parts on which the hydrostatic thrust is the same, the formula is

EXAMPLES.

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Examples.

(1) A hollow sphere is filled with liquid. Prove that the average pressure on the zone of the surface between two given horizontal small circles is the pressure midway between these circles.

(2) Prove that the total normal pressure on a spherical surface, immersed to a given depth in water, is the same as that on a circumscribed cylinder.

(3) A hemispherical bowl is filled with water; show that the resultant vertical thrust is equal to two-thirds of the whole pressure on the bowl.

(4) A sphere is just immersed in a liquid; prove that the total normal pressure on the curved surface of a segment made by a plane passing through the highest point of a sphere is three times the weight of liquid which would fill the segment.

(5) A hollow hemisphere, filled with liquid, is suspended freely from a point in the rim of its base; prove

that the whole pressures on the curved surface and the base are in the ratio 19: 8.

(6) Into a vertical cylinder are put equal weights of two different liquids which do not mix; prove that the ratio of their whole pressures on the curved surface of the cylinder is equal to three times the ratio of their densities.

(7) A vertical cylinder contains equal volumes of two

liquids, the density of the lower liquid being three times that of the upper liquid. Find the whole pressure on the curved surface, and prove that, if the fluids be mixed together so as to become homogeneous, the whole pressure will be increased in the ratio of 4 to 3.

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EXAMPLES ON RESULTANT

(8) A cylindrical tumbler, containing water, is filled up with wine. After a time half the wine is floating on the top, half the water remains pure at the bottom, and the middle of the tumbler is occupied by wine and water completely mixed. If the weight of the wine be two-thirds of that of the water, and their densities be in the ratio of 11 to 12, prove that the whole normal pressure of the pure water on the curved surface of the tumbler is equal to the whole normal pressure of the remainder of the liquid on the tumbler. (9) Equal volumes of n fluids are disposed in layers in a vertical cylinder, the densities of the layers commencing with the highest being as 1:2:... : N ; find the average pressure on the cylinder, and deduce the corresponding expression for the case of a fluid in which the density varies as the depth. Also, if the n fluids be all mixed together, show that the average pressure on the curved surface of the cylinder will be increased in the ratio 3n: 2n+1.

(10) A vessel contains n different fluids resting in horizontal layers and of densities P1, P2, ..., pn respectively, starting from the highest fluid. A triangle is held with its base in the upper surface of the highest fluid, and with its vertex in the nth fluid. Prove that, if ▲ be the area of the triangle and h1, hq, hn be the depths of the vertex below the upper surfaces of the 1st, 2nd, ..., nth fluids. respectively, the thrust on the triangle is

HORIZONTAL AND VERTICAL THRUST.

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(11) A cylindrical vessel on a horizontal circular base of radius a is filled to a height h with liquid of density w. If now a sphere of radius c and density greater than w is suspended by a thread so that it is completely immersed, find the increase of pressure on the base of the vessel; and show that the increase of the whole pressure on the curved surface of the vessel is

(12) The shape of the interior of a vessel is a double cone, the ends being open, and the two portions being connected by a minute aperture at the common vertex. It is placed with one circular rim fitting close upon a horizontal plane, and is filled with water; find the whole pressure and the resultant thrust upon it, and prove that, if the latter be zero, the ratio of the axes of the two portions is 1:2.

If the water is on the point of escaping between the circular rim and the plane when this ratio of the axes is 2:1, prove that the weight of the vessel is three times the weight of the water.

(13) A circular disc moveable about its centre fits accurately into a vertical slit in the side of a vessel containing water, so that half the disc is in water and half in the air. The pressure of the water on the immersed portion acts vertically upward through the centre of gravity of that portion and will therefore tend to turn the disc about its centre. This has been proposed as a Perpetual Motion. Point out the fallacy.