An experimental method, invented by Pascal, of illustrating the preceding principles consists in taking a number of vessels of different shape, but all standing on equal horizontal bases, as in (i), (ii), (iii), (iv), fig. 34.

When filled with the same liquid to the same height h, the thrust on the base BC is found experimentally to be the same, namely the weight of superincumbent liquid contained in a vertical cylinder standing on the same base.

In (i), the vessel being a vertical cylinder, the thrust P on the base is equal to the weight of the liquid; and the resultant thrust of the liquid on the cylinder is zero.

In (ii), the vessel enlarges, and in (iii) the vessel contracts, but the thrust on the base is the same in each case as in (i); so that the resultant thrust of the liquid on the curved surface is the difference between W the weight of the liquid and P the thrust on the base, and acts vertically; it acts vertically downwards and is equal to W-P in (ii), but vertically upwards and is equal to P- W in (iii).

In (iv), the vessel is a slant cylinder or pipe, and the weight of liquid is the same as in (i), and equal to P the thrust on the base; and now the resultant thrust on the curved surface is a couple, of moment Ph tan a, if a is the inclination of the axis of the cylinder to the vertical; this is seen from the consideration of the equilibrium of the liquid in (iv).

The hydrostatic thrust on a piece of straight pipe, slanting at an angle a to the vertical, cut off by two horizontal planes at a vertical distance h is therefore equivalent to a couple of moment Wh tan a, if W denotes the weight of liquid in the pipe between these two horizontal planes.

THE LEVEL IN COMMUNICATING VESSELS.

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A similar theorem holds for the thrust exerted by spheres in a vertical or inclined cylinder.

44. A Liquid in Communicating Vessels maintains its Level.

Communication can be established by pipes between the vessels, and the equilibrium will not be disturbed, if the vertical depth of the liquid is the same in each vessel; but if the depths are originally different, then when the communication is made, liquid will flow from one vessel to the other, till it stands at the same level in each; that is till the free surfaces are in the same horizontal plane.

This principle of § 24 that "Liquids in Communicating Vessels maintain their Level" is employed in the design of water works; it is also seen exemplified in the case of the Ocean; but isolated bodies of water, like the Caspian Sea, the Dead Sea, or large inland lakes, not in direct communication with the Ocean, can have different levels. Thus from surveys it is found that the Caspian is about 83 feet below mean sea level, and the Dead Sea about 1300 feet below mean sea level; while the Aral Sea, fed by the Oxus, is 156 feet above sea level, and the Great Salt Lake in N. America is 4200 feet above sea level.

80 HORIZONTAL THRUST ON A CURVED SURFACE.

By this it is meant that if free communication was established with the Ocean, this number of feet would be the change in the depth.

A great part of Holland is below mean sea level, and would be covered by sea water if the banks and dykes were to give way.

45. Component Horizontal Thrust of a Liquid under Gravity against any Curved Surface.

Project the curved surface BC on any vertical plane By perpendicular to the given direction (figs. 25, 28, 29) by horizontal lines round BC; and consider the equilibrium of the liquid, real or fictitious, contained in BßYC.

By resolving perpendicularly to BC, we find that the component horizontal thrust on BC in this direction is equal to the thrust on the plane area By, which can be found by a preceding Theorem (§ 36).

Here, again, if the horizontal cylinder through the perimeter BC cuts the surface BC under consideration (fig. 35), we must draw the cylinder formed by the tangent planes of the surface perpendicular to the vertical plane By, touching along the curve EF; and consider separately the horizontal thrust perpendicular to the plane By on the two portions of the surface BC into which it is divided by the curve of contact EF.

But as the horizontal thrusts are in opposite directions, we see that the resultant horizontal thrust on the surface is always the same as that on By the projection of BC, however often the cylinder BCẞy may intersect the curved surface bounded by BC.

To find the resultant horizontal thrust on BC, we must find the component horizontal thrust in the direction perpendicular to that first found.

AVERAGE PRESSURE OVER A SURFACE.

81

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 and the resultant thrust is vertical, and equal to

the weight of the displaced liquid.

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=wΣzAS = 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.

82

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=√(28/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