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remember the facts but that invention and the prediction of additional facts become possible.

22. Pressure and force. It is necessary to distinguish between two concepts, pressure and force. In the apparatus illustrated in Fig. 25, water exerts forces on two pistons. The pressure on each of these is numerically equal to the

force exerted on a unit area. If the force is expressed in pounds and the area in square inches, pressure is the number of pounds acting on each square inch of the piston.

An experiment made with this apparatus would show that the forces on the two pistons are not the same.

To water main

FIG. 25

From the numerical values obtained one could verify, within experimental errors,

Force on large piston

Area of large piston

force on small piston
area of small piston

But each of these fractions is equal to the pressure. Hence the result can be stated in either of two ways:

1. The force acting on a piston is proportional to its area. 2. The pressures on the two pistons are equal.

The second statement is simpler than the first. This is an example of what is generally true: that it is simpler to state facts and laws about fluids in terms of pressures than in terms of forces. 23. Some facts regarding pressures in liquids. Simple experiments show that

1. There is a pressure not only at the bottom but everywhere throughout a liquid.

2. Pressure increases with depth.

3. At any place in a liquid there is pressure in every direction. 4. At any place in a liquid the pressure is equal in all directions. 5. Pressure in a liquid at rest always acts perpendicularly to a surface.

6. Pressure in a liquid at rest is the same at all points on the same level.

One of the simplest ways of verifying most of these facts is by the use of the gauge illustrated in Fig. 26. A piece of thin sheet rubber is tied over the large open end of a thistle tube. The tube is cut off near the bulb, and a piece of rubber tubing is slipped over this shortened end. The other end of the rubber tubing is connected to a U-shaped glass tube partly filled with water colored with a little ink. When the thistle tube is immersed in water, pressure on the membrane is shown by the displacement of the colored water in the U-tube. The bulb with the membrane can be raised or lowered, turned sidewise, or inverted. The student should try to imagine what must be done with the gauge to verify facts 1, 2, 3, 4, and 6.

There are many well-known phenomena which when recalled serve to verify these facts. All who read the newspapers know that divers and submarine boats cannot descend to great depths on account of the enormous pressures at those depths. We are familiar with the disastrous results produced when the side pressure on a dam breaks it. There are abundant evidences of the

FIG. 26

existence of an upward pressure in a liquid. When a stick is forced end down into water, it is the upward pressure on the end that pushes the stick out. If a card is held over the smooth end of a glass tube while the tube is lowered into the water, the water pressure will hold the card tightly up against the tube.

That there is an upward pressure in the interior of a liquid should be expected as a result of very simple reasoning. Imagine a piece of thin tissue paper submerged in a liquid. If the paper is lying in a horizontal position, the weight of the liquid above it is pressing down on it. Since the paper is frail and not capable of sustaining a load, one is forced to the conclusion that there must be an upward pressure on the underside of the paper. If the paper is removed, these forces still exist. In a similar way it may be seen that there are forces all through the liquid.

Pressure is equal to the ratio of a force and an area taken perpendicularly to this force. But often, in referring to pressures in the interior of a liquid, nothing is said of an area. The area may be imagined, for example, by inserting an imaginary sheet of paper. It is in this sense that it is said that the weight of a liquid produces pressures at all places in the liquid.

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One of the statements in this section, fact 2, is that pressure increases with depth. This statement is incomplete. The increase of pressure is directly proportional to the increase in depth. This is more fully discussed in section 24, where it is proved that the difference in pressure between two levels in water is

p = 0.433 h pounds per square inch,

(2)

where h is the vertical distance in feet between the two levels. This means that for each foot of increase in depth, pressure increases 0.433 pound per square inch.

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Fig. 27 shows cases of pressure acting perpendicularly to the surface.

Fact 6 needs some further consideration. It should be obvious, in the case illustrated in Fig. 28, that the pressures at

A

FIG. 29

the points A, B, and C must be the same. If they were not, the side pressure at the point of greater pressure would cause a flow of the liquid toward the region of lower pressure. But in the case of some intricately connected system, such as that given in Fig. 29, it is not so evident that the pressures at the points C, D, and E on the same level are equal. The student should try to prove that the pressures are equal in this case (equation (2) may be used).

24. Derivation of the pressure-depth formulas. The principle underlying the theoretical proof of the pressure-depth formula, equation (2), is that pressure in a liquid is produced by the weight of that part of the liquid which lies above the place under consideration. Since pressure is numerically equal to the force acting on 1 square inch, the pressure on the bottom of a vessel is equal to the weight of a column 1 square inch in cross section extending up to the surface of the liquid. For example, the pressure of the water at the bottom of the vessel in Fig. 30 is equal to the weight of a column A of unit cross section; the pressure at the level MN is equal to the weight of a column B of unit cross section. The difference between the pressure at the bottom of the vessel and that at the level MN is equal to the weight of a column C, also of unit cross M section. In general, the difference in the downward pressures at two different levels is equal to the weight of a column of liquid of unit cross section and of a height equal to the difference of levels of

B

FIG. 30

C

N

the two places. Since the weight of a column of water 1 square inch in cross section and 1 foot high is 0.433 pound, the difference in pressure in pounds per square inch in water is given by (2),

p = 0.433 h pounds per square inch,

where h is the difference in the depth measured in feet.

In the C.G.S. system a pressure of 1 dyne per square centimeter is called a bar. The weight of a column of water 1 centimeter in cross section and 1 centimeter high is 980 dynes. Hence if pis the increase in pressure in dynes per square centimeter, or bars, and h the difference in depth in centimeters,

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If the liquid is one that does not weigh the same as water, its specific gravity must be taken into account. The specific gravity of a liquid is defined as the ratio of the weight of the liquid to the weight of an equal volume of water. If d is the specific gravity, the weight of a column of the liquid is d times the weight of a column of water of the same height and cross section. Hence for a liquid the specific gravity of which is d, the increase in pressure in pounds per square inch is given by

and in bars by

p = 0.433 hd,
p = 980 hd,

(4)

(5)

where h is measured in feet in (4) and in centimeters in (5). Equations (4) and (5) give the numerical value of the increase in pressure for a change in depth. If the liquid has a free surface (that is, if it is in an open vessel), these equations will give the total pressure due to the weight. In this case h is the depth below the free surface. For example, in Fig. 29 the pressure at Dor E is given by these equations if h is regarded as the depth of these points below the free surface AB of the first vessel.

Downward pressure was used in deriving the formulas for computing the numerical value of pressures; but since pressures are the same for all directions, these equations give the numerical value of pressure in any direction.

Since the pressure at any place depends on the depth below the level of the free surface, this distance is often used to express the pressure. When spoken of in terms of depth, the pressure is often called the head. For example, if the free surface of a reservoir is 100 feet above the given point, the head of water in the mains at that point is 100 feet (if the water is at rest). Equation (2) gives the numerical relation between head in feet (h) and pressure in pounds per square inch. For example, a head of 92 feet corresponds to a pressure of approximately 40 pounds per square inch. Often the pressure is given in terms of the head when no free surface exists. In that case the head is the height to which the pressure would raise water if an open tube were run up to that height.

25. The hydrostatic paradox. Consider the vessels shown in Fig. 31, which have the same area of base and which are filled wit' water to the same height. The pressures at the bottoms of the

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