IN A HOMOGENEOUS LIQUID.

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If z ft is the depth below the free surface, then Po denotes the atmospheric pressure on the surface; and if this atmospheric pressure is absent, then

p=wz,

obtained as before from the consideration of the equilibrium of a cylinder of liquid with zero pressure at the upper end.

To verify this experimentally, take a glass cylinder, with the lower edge ground smooth and greased, and a metal disc of given weight and thickness and of the same diameter as the cylinder, placed on the bottom of the cylinder so as to fit watertight, and held in position by means of a thread (fig. 18).

On submerging the cylinder vertically in liquid, it will be found that the thread may be left slack and the metal disc will be supported by the pressure of the liquid, when the depth of the bottom of the disc is to the thickness of the disc in a ratio equal to or greater than the density of the disc to the density of the liquid; or algebraically, if w denotes the density of the liquid, w' of the metal disc, and a the thickness of the disc,

w'aAwzA, or z/aww.

32

PRESSURE IN A LIQUID.

The effect of the atmospheric pressure will not sensibly modify the result, provided the thickness of the glass is inconsiderable; however, in the general case, with atmospheric pressure Po, and A denoting the area of the disc, B and C the external and internal horizontal sections of the glass cylinder, then

w'aA=(Po+wz)A-(Po+wz-wa)(A-B)-p。C,

reducing, when A =B, to

w'aA=wzA+Po(A — C).

In the words of one of Boyle's Hydrostatical Paradoxes, "a solid body as ponderous as any yet known (that is 20 times denser than water, such as gold or platinum), though near the top of the water, can be supported by the upward thrust of the water."

(Cotes, Hydrostatical and Pneumatical Lectures, p. 14.) If we suppose the atmospheric p。 is the pressure due to an increase of depth h in the liquid, then

so that now the pressure in the liquid is the same as if the free surface of zero pressure was at a height of h feet above the horizontal plane where the pressure

Po=wh.

Again Po might be the pressure due to liquid of depth h' and density w', so that

and

So also for any

which do not mix;

Po=w'h', p=w'h'+wz.

number of superincumbent fluids

their surfaces of separation must be

horizontal planes, for instance with air or steam on water, water on mercury, and oil on water, etc.

THE HEAD OF A LIQUID.

22. The Head of Water or Liquid.

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The pressure wz at a depth z ft in liquid is called the pressure due to a head of z feet of the liquid.

Thus a head of z feet of water, of density or heaviness w lb/ft3 produces a pressure of wz lb/ft2 or wz÷144 lb/in2; and a head of z inches of water produces a pressure of wz÷1728 lb/in2; and on the average, w=62.4.

In round numbers a cubic foot of water weighs 1000 oz, and then w=1000+16=62.5.

In the Metric System, taking a cubic metre of water as weighing a tonne of 1000 kilogrammes, or a cubic decimetre as weighing a kilogramme, or a cubic centimetre as weighing a gramme, a head of z metres of water gives a pressure of z tonnes per square metre (t/m2) or 1000 z kg/m2, or z/10 kg/cm2, or 100 z grammes per square centimetre (g/cm2), and a head of z centimetres of water gives a pressure of z g/cm2; thus a great simplification in practical calculations is introduced by the Metric System of Units.

The pressure of the atmosphere, as measured by the barometer, was taken in §8 as about 143 lb/in2, or 2112 pounds (say 19 cwt, or nearly a ton) per square foot; with Metric Units the atmosphere was taken as one kg/cm2, or 10 t/m2); and an atmosphere is thus due, in round numbers, to a head of 30 inches or 76 centimetres of mercury, of specific gravity 136; a head of 10 metres or 33 to 34 feet of water; or a head of 26,400 feet or 5 miles, or 8500 metres of homogeneous air of normal density, occupying about 12.5 ft3 to the lb, or 754 cm3 to the g, or 0.754 m3 to the kg.

Any discrepancy in these results is due to taking the nearest round number in each system of units.

G.H.

C

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THE CORNISH PUMPING ENGINE.

Regnault worked with a standard barometric height of 76 cm of mercury, and we may call this pressure due to a head of 76 cm of mercury a Regnault atmosphere; but it is more convenient to take 75 cm; thus a head of 300 m of mercury, in a tube up the Eiffel Tower, gives a pressure of 400 atmospheres.

The density of sea water is generally taken as 64 lb/ft3, so that an atmosphere of 14 lb/in2 is equivalent to a head of 33 ft of sea water; thus a diver at a depth in the sea of 27 fathoms or 165 ft experiences a pressure of 5 atmospheres over the atmospheric pressure, in all a pressure of 6 atmospheres or 88 lb/in2.

In the previous discussions of the Hydraulic Press and Machines working by the Transmission of Pressure we supposed the pressure uniform and neglected the variations due to gravity and difference of level; but these variations are so slight compared with the great pressures employed as to be practically insensible.

Thus a pressure of 750 lb/in2 is due to a head of 1728 feet of water, compared with which an alteration of 10 feet, or even 100 feet, is insensible.

23. The Cornish Pumping Engine.

Suppose M lb is added to W in fig. 7, the equilibrium is destroyed: the piston A will descend say x feet, and the piston B will be raised y feet, such that

Ax=By;

and now the pressure under the piston A will become (M+W)/A lb/ft2,

while under the piston B it will still be P/B lb/ft2; and the difference between those pressures being due to a head of x+y feet of the liquid,

w(x+y)=(M+W)/A−P/B=M/A,

A LIQUID MAINTAINS ITS LEVEL.

35

since W/AP/B, when P and W balance at the same level; so that

This principle is employed in the Cornish Pumping Engine; the plunger or piston A of the pump at the bottom of the mine is weighted by M sufficiently to raise the column of water and B to the surface of the ground; the action of the steam being employed merely to raise the piston A and the weight M at the end of a stroke so as to make the next stroke.

24. A Liquid maintains its Level.

By alternate horizontal and vertical steps of appropriate magnitude, we can make the preceding theorems apply to homogeneous liquid contained in a vessel of any irregular shape, so as to be independent of the form of the containing vessel; and thus we prove that the surfaces of equal pressure are horizontal planes and that the pressure increases uniformly with the depth, even when the liquid is divided up into irregular channels, as in water mains; and that if left to itself the water will regain its original level, the principle applied in waterworks.

It was not from ignorance of these hydrostatical principles, but of the art of making strong waterpipes that the Romans constructed high stone aqueducts to carry water to cities on the level; where nowadays iron pipes would be employed, laid in the ground, at great economy and with the additional advantage of escaping long continued frost.