Thus water presses in all directions upon things which are immersed in
it.
If a long wooden or metal pipe, placed vertically, has its lower end
stopped with a cork which does not fit very tightly, and water is poured
into the top of the tube, the water will at first fill the part of the
tube above the cork, and its weight will exert a certain =pressure= on
the cork. In fact, if the end of the tube is stopped by applying the
palm of the hand closely against it, the downward pressure of the water
will have to be overcome by a certain amount of effort. As the water
accumulates, this downward pressure will become greater and greater
until the hand is driven away, or the cork is forced out, and the water
falls to the ground. The pressure in this case is the same as the weight
of the water, and the cork would have been driven out equally well by a
rod of lead of the same weight.
Suppose the tube to be square, and that the inside of the square
measures exactly one inch each way. Then an inch of height of the tube
will hold exactly one cubic inch of water. Since one cubic inch of water
weighs 252 grains and a half, as much water as will fill the tube about
two feet three inches and a half high, will weigh a pound (7,000
grains), and fifteen pounds of water will fill such a tube between
thirty-three and thirty-four feet high. And these respective weights
measure the pressure of two columns of water, one twenty-seven and a
half inches high, and the other nearly thirty-four feet high, on a
square inch of the surface on which they rest.
The specific gravity (§ 24) of lead is 11·45; in other words it is about
eleven and a half times denser than water. Therefore if a bar of lead
cut square and one inch in the side, and rather less than 1/11th of the
height of a column of water, is slipped into the tube in place of the
water, it will exert the same pressure on the bottom.
And now comes a difference between the lead and the water, which depends
on the fluidity of the latter. The lead exerts no pressure on the sides
of the tube, but the water does. If a small hole is cut in the side of
the tube close to the bottom, and stopped with a cork, the lead will not
press upon the cork. But if the column of water is high enough the cork
will be driven out with as much force as before, so that the water
presses just as much sideways as downwards. It is easy to satisfy
oneself of this by inserting a long glass tube, with its lower end bent
at right angles and fitted with a cork, into the side of the wooden
pipe. The water will at once rise in the tube to the same height as it
has in the pipe. Whence it is obvious that the pressure of the water on
any point of the side is exactly equal to the vertical pressure at that
point; for the pressure outwards is exactly balanced by that of the
vertical column in the tube inwards. The water in a watering-pot always
stands at the same level in the can and in the spout.
If a glass tube is bent into the shape of a =U=, and water is poured
into it, the water will always stand at the same level in the two legs
of the tube, whatever the shape of the bend may be, or the relative
capacities of the two legs, or the inclination of the tube.
And this must needs be so, for the force with which the water tends to
flow out of the one half of the arrangement depends on the vertical
height[2] of the surface of the water above the aperture of exit; so
that any column of equal vertical height must balance it.
Footnote 2:
Vertical height is the height measured along a line drawn from the
surface of the water perpendicularly to the surface of the earth. A
plumb-line is a string to one end of which a weight is attached and
thus hangs suspended. If the other end of the line is brought opposite
the surface of the water the direction of the string answers to the
line of vertical height.
That a column of water will stand at exactly the same level as any other
with which it communicates, may be seen still more simply by placing a
glass tube, open at each end, in a basin of water. However the tube may
be inclined or bent, whether its lower end is wide or narrow, the column
of water inside it will be at exactly the same level as the water
outside it. Yet, of course, the rigid glass walls of the tube cut off
all communication between the column of water inside it and the rest,
except at the bottom.
In a well-ordered town, water is supplied to every house and can be
drawn from taps placed in the highest stories. These are fed by pipes
which lead from a cistern at the top of the house. This water is brought
from a large pipe, or =main=, in the street, by a smaller house-pipe,
which is often made to twist about in various directions before it
reaches the cistern at the top of the house into which it delivers the
water. If you followed the main, you would find that it took a long
course up and down, beneath the pavement of the streets, until at last
it reached the water-works. Here you would find that the main was
connected with a reservoir; and either this reservoir is at a greater
height than any of the cisterns into which the water is delivered, or
there is some means of pumping the water from it to that height on its
way to the main. Thus the reservoir, the main, and the house-pipe form
one immense =U=-tube, and the water in the house-pipe tends to rise to
the same level as that of the water in the reservoir, and hence flows
into the cistern when the supply-pipe is open.
28. =The Transference of Motion by Moving Water: the Momentum of Moving
Water.=
Suppose a wooden vat with a horizontal tap, the sectional area[3] of the
tube of which is one square inch, inserted close to the bottom, to be
filled with water up to 100 inches above the tap. Then supposing the tap
to be shut, the pressure upon its sectional area will be 25,250 grains,
or rather more than three pounds and a half—and there is the same
pressure on every square inch of the bottom of the vat.
Footnote 3:
The sectional area of a tube is the surface occupied by its cavity
when it is cut across. It would be represented by the surface of a
piece of cardboard, like the wad of a gun, just large enough to go
into the tube.
If the tap is now turned, the water nearest to it being unsupported on
its outer side, the pressure on the inner side sets it in motion, and it
flows out in a stream. At first the stream shoots out violently and the
water is carried to a long distance. That is to say, the weight of the
column of water 100 inches high acts as a force, or cause of motion,
upon the water nearest the tap, and this water is forced out with a
velocity depending on that force in a horizontal direction. Now suppose
that you take a common toy cup-and-ball and bring the ball into the way
of the stream of water. The stream will at once strike the ball and
drive it in the same direction as that which it is itself taking. The
power which the moving water has of transferring or communicating motion
to a body which is at rest, but free to move as the ball is, is due to
its =momentum=. The greater the mass of the stream and the more rapidly
it moves, the more motion will it communicate to the ball, or the
heavier the ball it will move. Close to the mouth of the tap the
direction of the stream is horizontal; but it very soon begins to bend
downwards, and describing a rapid curve, comes to the ground. It does
this for just the same reasons that a stone thrown horizontally
describes a curve; and at length strikes the ground; and, in fact, the
stream may be regarded as so much water thrown horizontally.
These reasons are two: firstly, as soon as the water has left the tap it
is an unsupported heavy body; and, as such, it begins to fall to the
ground. Secondly, the momentum of the water is continually being
diminished by the resistance of the air through which it passes. For,
although the air which surrounds us is so thin and movable a body that
we ordinarily take no notice of it—the fact that it offers resistance to
bodies which move through it is easily observed; as, for example, in
using a fan. The water has to overcome this resistance, and its momentum
is proportionally diminished.
If, when the water leaves the tap, the air and gravitation were alike
abolished, the water, keeping its momentum, would travel for ever in the
same direction.
As the water runs out, it will be observed that the velocity of the
stream becomes less and the curve which it describes sharper, so that it
comes to the ground sooner; and finally, when the vat is nearly empty,
the stream falls nearly vertically downwards. The reason of this is that
the level of the top of the water is gradually lowered; consequently,
the height of the column which presses on the water close to the tap is
gradually lessened, and therefore its weight is diminished. But this
weight or pressure is the cause of the motion of the water, and as the
cause diminishes the effect of that cause must diminish. Therefore the
momentum of the water is gradually lessened and it is carried less and
less far horizontally in the time which it takes to fall to the ground:
until finally, it acquires no appreciable horizontal motion at all, and
so falls vertically downwards from the mouth of the tap.