Chapter III: Part 3

The mechanical properties of liquids are determined on the hypothesis that liquids are incompressible; according to known general principles this is found to be for all practical purposes true, yet liquids are more compressible than solids. If water be confined in a perfectly rigid cylindrical vessel, its compression would equal 1/300000 of its length for every pound per unit of area of the end pressure.

Water is nearly 100 times as compressible as steel, yet for almost all practical purposes, liquids may be considered as non-elastic bodies without involving sensible error.

_The pressure upon the horizontal base of any vessel containing a fluid, is equal to the weight of a column of the fluid, found by multiplying the area of the base into the perpendicular height of the column, whatever be the shape of the vessel._

This follows, since here the distance of the center of gravity of the base from the surface of the fluid, is the same as the perpendicular height of the column. With a given base and height, therefore, the pressure is the same whether the vessel is larger or smaller above, whether its figure is regular or irregular, whether it rises to the given height in a broad open funnel, or is carried up in a slender tube.

Hence, _any quantity of water, however small, may be made to balance any quantity, however great_. This is called the _hydrostatic paradox_. The experiment is usually performed by means of a water-bellows, as represented in Fig. 84. When the pipe AD is filled with water, the pressure upon the surface of the bellows, and consequently the force with which it raises the weights laid on it, will be equal to the weight of a cylinder of water, whose base is the surface of the bellows, and height that of the column AD. Therefore, by making the tube small, and the bellows large, the power of a given quantity of water, however small, may be increased indefinitely. The pressure of the column of water in this case corresponds to the force applied by the piston in the hydrostatic press.

We have already seen that the pressure on the bottom of a vessel depends neither on the form of the vessel nor on the quantity of the liquid, but simply on the height of the liquid above the bottom. But the pressure thus exerted must not be confounded with the pressure which the vessel itself exerts on the body which supports it. The latter is always equal to the combined weight of the liquid and the vessel in which it is contained, while the former may be either smaller or greater than this weight, according to the form of the vessel. This fact is often termed the _hydrostatic paradox_, because at first sight it appears paradoxical.

CD (Fig. 85) is a vessel composed of two cylindrical parts of unequal diameters, and filled with water to _a_. From what has been said before, the bottom of the vessel CD supports the same pressure as if its diameter were everywhere the same as that of its lower part; and it would at first sight seem that the scale MN of the balance, in which the vessel CD is placed, ought to show the same weight as if there had been placed in it a cylindrical vessel having the same weight of water, and having the diameter of the part D. But the pressure exerted on the bottom of the vessel is not all transmitted to the scale MN; for the _upward_ pressure upon the surface _n o_ of the vessel is precisely equal to the weight of the _extra_ quantity of water which a cylindrical vessel would contain, and balances an equal portion of the _downward_ pressure on _m_. Consequently the pressure on the plate MN is simply equal to the weight of the vessel CD and of the water which it contains.

_Pressure exerted anywhere upon a mass of liquid is transmitted
undiminished in all directions, and acts with the same force on all
equal surfaces, and in a direction at right angles to those surfaces._

To get a clearer idea of the truth of this principle, let us conceive a vessel of any given form in the sides of which are placed various cylindrical apertures, all of equal size, and closed by movable pistons. Let us, further, imagine this vessel to be filled with liquid and unaffected by the action of gravity; the pistons will, obviously, have no tendency to move. If now a weight of P pounds be placed upon the piston A (Fig. 86), which has a surface A, it will be pressed inwards, and the pressure will be transmitted to the internal faces of each of the pistons B, C, D, and E, which will each be forced outwards by a pressure P, their surfaces being equal to that of the first piston. Since each of the pistons undergoes a pressure, P, equal to that on A, let us suppose two of the pistons united so as to constitute a surface 2_a_; it will have to support a pressure 2P. Similarly, if the piston were equal to 3_a_, it would experience a pressure of 3P; and if its area were 100 or 1,000 times that of _a_, it would sustain a pressure of 100 or 1,000 times P. In other words, the pressure on any part of the internal walls of the vessel would be proportional to the surface.

The principle of the equality of pressure is assumed as a consequence of the constitution of fluids.

By the following experiment it can be shown that pressure is transmitted in all directions; a cylinder provided with a piston is fitted into a hollow sphere (Fig. 87). in which small cylindrical jets are placed perpendicular to the sides. The sphere and the cylinder being both filled with water, when the piston is moved the liquid spouts forth from all the orifices, and not merely from that which is opposite to the piston.

The reason why a satisfactory quantitative experimental demonstration of the principle of the equality of pressure cannot be given is that the influence of the weight of the liquid and of the friction of the pistons cannot be altogether eliminated.

NOTE.—_The influence of the weight (or gravity) of water and its
fractional resistance in practical use_ is so great upon all the
processes of numbers and of the application of the natural laws
governing the operation of fluids, as stated under the heading of
Hydraulic Data, that separate pages will hereafter be found devoted
to a more extended explanation of this subject of gravity and
friction of water.

Yet an approximate verification may be effected by the experiment represented in Fig. 88. Two cylinders of different diameters are joined by a tube and filled with water. On the surface of the liquid are two pistons, P and p, which hermetically close the cylinders, but move without friction. Let the area of the large piston, P, be, for instance, thirty times that of the smaller one, p. That being assumed, let a weight, say of two pounds, be placed upon the small piston; this pressure will be transmitted to the water and to the large piston, and as this pressure amounts to two pounds on each portion of its surface equal to that of the small piston, the large piston must be exposed to an upward pressure thirty times as much, or of sixty pounds. If now, this weight be placed upon the large piston, both will remain in equilibrium; but if the weight is greater or less, this is no longer the case.

It is important to observe that in speaking of the transmission of pressure to the sides of the containing vessel, these pressures must always be supposed to be perpendicular to the sides.

_Equilibrium or state of rest of superposed liquids._ In order that there should be equilibrium when several heterogeneous liquids are superposed in the same vessel, each of them must satisfy the conditions necessary for a single liquid, and further there will be a stable state of rest only when the liquids are arranged in the order of their decreasing densities from the bottom upwards.

The last condition is experimentally demonstrated by means of the _phial of four elements_. This consists of a long narrow bottle containing mercury, water, colored red, saturated with carbonate of potash, alcohol, and petroleum. When the phial is shaken the liquids mix, but when it is allowed to rest they separate; the mercury sinks to the bottom, then comes the water, then the alcohol, and then the petroleum. This is the order of the decreasing densities of the bodies. The water is saturated with carbonate of potash to prevent its mixing with the alcohol.

This separation of the liquids is due to the same cause as that which enables solid bodies to float on the surface of a liquid of greater density than their own. It is also on this account that fresh water, at the mouths of rivers, floats for a long time on the denser salt water of the sea; and it is for the same reason that cream, which is lighter than milk, rises to the surface.

_The pressure upon any particle of a fluid of uniform density is
proportioned to its depth below the surface._

_Example 1._ Let the column of fluid ABCD Fig. (1) be perpendicular to the horizon. Take any points, _x_ and _y_, at different depths, and conceive the column to be divided into a number of equal spaces by horizontal planes. Then, since the density of the fluid is uniform throughout, the pressure upon _x_ and _y_, respectively, must be in proportion to the number of equal spaces above them, and consequently in proportion to their depths.

_Example 2._ Let the column be of the same perpendicular height as before, but inclined as is Fig. (2); then its quantity, and of course its weight, is _increased_ in the same ratio as its length exceeds its height; but since the column is partly supported by the plane, like any other heavy body, the force of gravity acting upon it is _diminished_ on this account in the same ratio as its length exceeds its height; therefore as much as the pressure on the base would be augmented by the increased length of the column, just so much it is lessened by the action of the inclined plane; and the pressure on any part of C_c_ will be, as before, proportioned to its perpendicular depth; and the pressure of the inclined column AC_ac_ will be the same as that of the perpendicular column ABCD.

_Fluids rise to the same level in the opposite arms of a recurved
tube._

Let ABC, (Fig. 90) be a recurved tube: if water be poured into one arm of the tube, it will rise to the same height in the other arm. For, the pressure acting upon the lowest part at B, in opposite directions, is proportioned to its depth below the surface of the fluid. Therefore, these depths must be equal, that is, the height of the two columns must be equal, in order that the fluid at B may be at rest; and unless this part is at rest, the other parts of the column cannot be at rest. Moreover, since the equilibrium depends on nothing else than the _heights of_ the respective columns, therefore, the opposite columns may differ to any degree in quantity, shape, or inclination to the horizon. Thus, if vessels and tubes very diverse in shape and capacity, as in Fig. p. 84 be connected with a reservoir, and water be poured into any one of them, it will rise to the same level in them all.

The reason of this fact will be further understood from the application of the principle of _equal momenta_, for it will be seen that the velocity of the columns, when in motion, will be as much greater in the smaller than in the larger columns, as the quantity of matter is less; and hence the opposite momenta will be constantly equal.

Hence, water conveyed in aqueducts or running in natural channels, will rise just as high as its source. Between the place where the water of an aqueduct is delivered and the spring, the ground may rise into hills and descend into valleys, and the pipes which convey the water may follow all the undulations of the country, and the water will run freely, provided no pipe is laid higher than the spring.

_Pressure of water due to its weight._ The pressure on any particle of water is proportioned to its depth below the surface. The pressure of still water in pounds per square inch against the sides of any pipe, channel, or vessel of any shape whatever, is due solely to the “head” or height of the level surface of the water above the point at which the pressure is considered and is equal to ·43302 lbs. per square foot, every foot of head or 62·355 lbs. per square foot for every foot of head at 62° F.

The pressure per square inch is equal in all directions downwards, upwards or sideways and is independent of the shape or size of the containing vessel; for example, the pressure on a plug forced inward on a square inch of the surface of water is suddenly communicated to every square inch of the vessel’s surface, however great and to every inch of the surface of any body immersed in it.

It is this principle which operates with such astonishing effect in hydrostatic presses, of which familiar examples are found in the hydraulic pumps, by the use of which boilers are tested. By the mere weight of a man’s body when leaning on the extremity of a lever, a pressure may be produced of upwards of 20 tons; it is the simplest and most easily applicable of all contrivances for increasing human power, and it is only limited by want of materials of sufficient strength to utilize it.

UNIVERSAL GRAVITATION.

_Gravity or Gravitation_ is that species of attraction, or force by which, all bodies or particles of matter in the universe tend towards each other; it is also called _attraction of gravitation and universal gravity_; gravity, in a more limited, sense is the tendency of a mass of matter toward a center of attraction especially the tendency of a body toward the center of the earth.

This influence is conveyed from one body to another without any perceptible interval of time. If the action of gravitation is not instantaneous, it comes very nearly to it by moving more than fifty millions of times faster than light.

Gravity extends to all known bodies in the universe, from the smallest to the greatest; by it all bodies are drawn toward the center of the earth, _not because there is any peculiar property or power in the center_, but because the earth being a sphere, the _aggregate_ effect of the attractions exerted by all its parts upon any body exterior to it, is such as to influence that body toward the center.

This property manifests itself, not only in the motion of falling bodies, but in the _pressure_ exerted by one portion of matter upon another which sustains it; and bodies descending freely under its influence, whatever be their figure, dimensions or texture, all are _equally accelerated_ in right lines perpendicular to the plane of the horizon. The apparent _inequality_ of the action of gravity upon different species of matter near the surface of the earth arises entirely from the resistance which they meet with in their passage through the air. When this resistance is removed (as in the exhausted receiver of an air-pump), the inequality likewise disappears.

_The law of gravity_, discovered by Sir Isaac Newton, toward the end of the seventeenth century, may be stated as follows:

_Every particle of matter in the universe attracts every other
particle with a force whose direction is that of a line joining the
two particles considered, and whose magnitude is directly as the
product of the masses and inversely as the square of the distance
between them._

As a groundwork for this great generalization, Newton employed the results of two of the greatest astronomers who preceded him, Copernicus and Kepler. About 1500 A. D. Copernicus perceived and announced that the apparent rotation of the heavens about the earth could be explained by supposing the earth to rotate on an axis once in twenty-four hours. Previous to this time the earth had been regarded as the center of the universe. He also showed that nearly all the motions of the planets, including the earth, could be explained on the assumption that these revolved in circular orbits about the sun, whose position in the circle, however, was slightly eccentric.

Thus, building somewhat upon the labors of the two parties named, Newton was the first to prove the law of the forces which would account for the motions of all the bodies in the solar system.

LAWS OF FALLING BODIES.

Since a body falls to the ground in consequence of the earth’s attraction on _each_ of its molecules, it follows that, all other things being equal, all bodies, great and small, light and heavy, ought to fall with equal rapidity, and a lump of sand without cohesion should during its fall retain its original form as perfectly as if it were compact stone. The fact that a stone falls more rapidly than a feather is due solely to the unequal resistances opposed by the air to the descent of these bodies; _in a vacuum all bodies fall with equal velocity_.

_In a vacuum, however, liquids fall like solids without separation of their molecules._ The _water-hammer_, a model used in scientific schools, illustrates this: the instrument consists of a thick glass tube about a foot long, half filled with water, the air having been expelled by ebullition previous to closing one extremity with the blow-pipe. When such a tube is suddenly inverted, the water falls in one undivided mass against the other extremity of the tube, and produces a sharp metallic sound, resembling that which accompanies the shock of two solid bodies coming suddenly together.

NOTE.—The resistance opposed by the air to falling bodies is
especially remarkable in the case of liquids. The Staubbach in
Switzerland is a good illustration; an immense mass of water is seen
falling over a high precipice, but before reaching the bottom it is
shattered by the air into the finest mist. See Parker’s Philosophy,
pp. 69-70.

It has been ascertained, by experiment, that from rest, a body falling freely will descend 16-1/12 feet in the first second of time, and will then have acquired a velocity, which being continued uniformly, will carry it through 32-1/6 feet in the next second. Therefore if the first series of numbers be expressed in seconds, 1″, 2″, 3″, &c., the velocities in feet will be 32-1/6, 64-1/3, 96-1/2, &c.; the spaces passed through as 16-1/12, 64-1/3, 144-3/4, &c., and the spaces for each second, 16-1/2, 48-1/4, 80-5/12, &c.

TABLE.

_Showing the Relation of Time, Space and Velocity._

----------+-----------+--------+------------+------+-----------------
Time in | Velocity | |Space fallen| | Whole space
seconds of|acquired at|Squares.| through |Space.|fallen through in
the body’s|the end of | | in that | | the last second
fall. |that time. | | time. | | of the fall.
----------+-----------+--------+------------+------+-----------------
1 | 32·16 | 1 | 16·08 | 1 | 16·08
2 | 64·33 | 4 | 64·33 | 3 | 48·25
3 | 96·5 | 9 | 144·75 | 5 | 80·41
4 | 128·66 | 16 | 257·33 | 7 | 112·58
5 | 160·83 | 25 | 402·08 | 9 | 144·75
6 | 193· | 36 | 579· | 11 | 176·91
7 | 225·17 | 49 | 788·08 | 13 | 209·08
8 | 257·33 | 64 | 1029·33 | 15 | 241·25
9 | 289·5 | 81 | 1302·75 | 17 | 273·42
10 | 321·66 | 100 | 1946·08 | 19 | 305·58
| | | | |
----------+-----------+--------+------------+------+-----------------

Experience has shown that the measurement of all physical quantities may be expressed in terms of three fundamental magnitudes. Those commonly chosen for this purpose are _time_, _length and mass_ _or quantity of matter_. It may be assumed that our ideas of time and space are sufficiently exact for all practical purposes. The subject of matter, however, requires more particular consideration. Of the three magnitudes named, matter alone is directly cognizable by the senses, and invested with a variety of interesting properties.

For present purposes matter may be defined as anything that can be weighed, and the quantity of matter as proportional to its weight; _i. e._, its attraction towards the earth. The _weight_ of a body is the force it exerts in consequence of its gravity, and is measured by its mechanical effects, such as bending a spring. We weigh a body by ascertaining the force required _to hold it up_, or to keep it from descending. Hence, weights are nothing more than _measures of the force of gravity_ in different bodies.

Again, Gravitation, the most feeble of physical actions between
small masses, is almost imperceptible; yet it is an energy abundant
in proportion to the quantity of matter in the universe, and fully
competent, by its gradual condensing agency, to account for the
origination of planetary systems and their movements. It is not
strange, therefore, that by some physicists this energy is supposed
to be the beginning of that of which all other forms of force are
residues or metamorphoses. Gravity is the name especially given to
its terrestrial manifestations. _A particle or body without a sphere
or spheroid, solid or hollow, is attracted to the center of the mass
of such body; within a hollow sphere, it will remain at rest at any
point._ At different depths below the earth’s surface, a body will be
attracted with a force diminishing as the distance from its center
decreases. The slight variation in the gravitating force of the same
falling body at different heights is in practice usually disregarded.
The weight of a body, as the measure of its gravitating tendency,
must vary both with mass and with the force acting on it; hence, from
the form of the earth, the same body at the sea level _will weigh
less and less as it is removed from either pole toward the equator_.
An elevation above the sea level gives a like result. A stone falls
through a less distance in a given time on a mountain than in the
valley below, less at the equator than at either pole. The loss of
weight in these cases cannot be tested by lever scales, in which
this loss is equal on both sides; but it may be by the spring
balance, in which bodies are weighed by the pull they exert against
the elasticity of a coiled wire. The effect of centrifugal force,
increasing from the pole to the equator, co-operates with increasing
removal from the earth’s center to lessen weight; the result of the
combined action of these two causes is, that a body weighing 195 lbs.
at either pole will weigh but 194 over the equator. _The line of a
falling body, called also the line of direction_, is interesting as
being that direction in space at any point of the earth’s surface
with reference to which all other directions are named, and by which
they are to be determined.

A few points remain to be named. _The flow of water is the result of the force of gravity_; the importance of this fact and its wide influence cannot be over stated; the gently falling dew, the mighty currents in the unfathomable depths of the ocean, as well as the rivulet merrily falling over the rocks to a lower level are all subject to the laws of terrestrial gravity.

The upper surface of a liquid in a vessel exposed to the atmosphere is called _the free surface_ and is pressed downwards by the air under about 15 lbs. pressure per square inch. The free surface of a small body of a perfect liquid, at rest, is horizontal and perpendicular to the action of gravity although in large bodies of liquid, as lakes and ponds, the free surface is spherical, assuming the curvature of the earth’s surface.

RULES RELATING TO THE VELOCITY OF FALLING BODIES.

1.—_To find the Velocity a falling Body will acquire in any given time._

Multiply the time, in seconds, by 32-1/6, and it will give the velocity acquired in feet, per second.

_Example._ Required the velocity in seven seconds.

32-1/6 × 7 = 225-1/6 feet. Ans.

2.—_To find the Velocity a Body will acquire by falling from any given height._

Multiply the space, in feet, by 64-1/3, the square root of the product will be the velocity acquired, in feet, per second.

_Example._ Required the velocity which a ball has acquired in descending through 201 feet.

64-1/3 × 201 = 12931; √12931 = 113·7 feet. Ans.

3.—_To find the Space through which a Body will fall in any given time._

Multiply the square of the time, in seconds, by 16-1/12, and it will give the space in feet.

_Example._ Required the space fallen through in seven seconds.

16-1/12 × 7² = 788-1/12 feet. Ans.

4.—_To find the Time which a Body will occupy in falling through a given space._

Divide the _square root_ of the space fallen through by 4, and the quotient will be the time in which it was falling.

_Example._ Required the time a body will take in falling through 402.08 feet of space.

√402·08 = 20·049, and 20·049 ÷ 4 = 5·012. Ans.

5.—_The Velocity being given, to find the Space fallen through._

Divide the velocity by 8, and the square of the quotient will be the distance fallen through to acquire that velocity.

_Example._ If the velocity of a cannon ball be 660 feet per second, from what height must a body fall to acquire the same velocity?

660 ÷ 8 = 82·5² = 6806·25 feet. Ans.

6.—_To find the Time, the Velocity per second being given._

Divide the given velocity by 8, and one-fourth part of the quotient will be the answer.

_Example._ How long must a bullet be falling, to acquire a velocity of 480 feet per second?

480 ÷ 8 = 60 ÷ 4 = 15 seconds. Ans.

SPECIFIC GRAVITY.

_Specific Gravity_ is the proportion of the weight of a body to that of an equal volume of some other substance adopted as a standard of reference. For solids and liquids the standard is pure water, at a temperature of 60° F., the barometer being at 30 inches.

Aëriform bodies are referred to the air as their standard. A cubic foot of water weighs 1,000 ounces; if the same bulk of another substance, as for instance cast iron, is found to weigh 7,200 ounces, its proportional weight or specific gravity is 7·2. It is convenient to know the figures representing this proportion for every substance in common use, that the weight of any given bulk may be readily determined. For all substances the specific gravity is used in various tests for the purpose of distinguishing bodies from each other, the same substance being found, under the equal conditions of temperature, &c., to retain its peculiar proportional weight or density.

Hence tables of specific gravities of bodies are prepared for reference, and in every scientific description of substances the specific gravity is mentioned. _In practical use, the weight of a cubic foot is obtained from the figures representing the density by moving the decimal point three figures to the right, which obviously from the example above, gives the ounces, and these divided by 16 gives the pounds avoirdupois, in the cubic foot._

Different methods may be employed to ascertain the specific gravity of solids. That by measuring the bulk and weighing, is rarely practicable; as a body immersed in water must displace its own bulk of the fluid, the specific gravity may be ascertained by introducing a body, after weighing it, into a suitable vessel exactly filled with water, and then weighing the fluid which overflows. The proportional weight is thus at once obtained. Wax will cause its own weight of water to overflow; its specific gravity is then 1. Platinum, according to the condition it is in, will cause only from 1/21 to 1/21·5 of its weight of water to overflow, showing its specific gravity to be from 21 to 21·5. But a more exact method than this is commonly employed. The difference of weight of the same substance weighed in air and when immersed in water, is exactly that of the water it displaces, and may consequently be taken as the weight of its own bulk of water. See Fig. 93 and rule and example on page 95.

_The specific gravity is then obtained by weighing the body first in air, and then, suspended by a fibre of silk or a hair, in water, and dividing the weight in air by the difference._ It is hardly necessary to say that the substance examined must be free from mixture of foreign matters, and especially from cavities that may contain air.

NOTE.—_Hydrometers_ are instruments for determining the relative
density of fluids; _distilled water_ is usually referred to as the
standard of comparison. They consist usually of a bulb or float
weighted at bottom so as to float upright, and having an elongated
stem graduated to indicate the density of the liquid by the depth to
which they sink therein.

THE HYDROSTATIC BALANCE.

Every body immersed in a liquid is submitted to the action of two forces: gravity which tends to lower it, and the buoyancy of the liquid which tends to raise it with a force equal to the weight of the liquid displaced. The weight of the body is either totally or partially overcome by its buoyancy, by which it is concluded that a body immersed in a liquid loses a part of its weight equal to the weight of the displaced liquid.

This principle, which is _the basis of the theory of immersed and floating bodies_, is called the principle of Archimedes, after the discoverer. It may be shown experimentally by means of _the hydrostatic balance_ (Fig. 92). This is an ordinary balance, each pan of which is provided with a hook; the beam being raised, a hollow brass cylinder is suspended from one of the pans, and below this a solid cylinder whose volume is exactly equal to the capacity of the first cylinder; lastly, an equipoise is placed in the other pan. If now the hollow cylinder be filled with water, the equilibrium is disturbed; but if at the same time the beam is lowered so that the solid cylinder becomes immersed in a vessel of water placed beneath it, the equilibrium will be restored. By being immersed in water the solid cylinder loses a portion of its weight equal to that of the water in the hollow cylinder. Now, as the capacity of the hollow cylinder is exactly equal to the volume of the solid cylinder the principle which has been before laid down is proved.

Minerals, if suspected of containing spaces, should be coarsely pulverized, and then the second method may be conveniently applied to determine their density—thus prepared, a higher result will be obtained, and even metals when pulverized were found to give a greater specific gravity than when this is determined from samples in their ordinary state. Very fine powders may also be examined by the method in use for ascertaining the specific gravity of fluids, viz.: by comparing the weight of a measured quantity with that of the same quantity of water.

A glass vessel called a specific gravity bottle is commonly employed, which is furnished with a slender neck, upon which is a mark indicating the height reached by 1,000 grains of water. The substance to be examined is introduced till it reaches the same mark, and, the weight of the bottle being known, only one weighing is required to obtain the result.

The specific gravity of fluids is also taken by the instrument called a hydrometer or alcometer. Such instruments are much used for ascertaining the specific gravity of spirituous and other liquors, as an indication of their strength. _If the solid body to be tested is lighter than water, it must be attached to some heavy substance to cause it to sink._ Its specific gravity is then calculated by dividing its weight in the air by the sum of the weights of the attached body both in air and in water, first subtracting from this sum the weight of the two bodies together in the water.

Bodies soluble in water may be weighed in some other fluid, as alcohol, ether, olive oil, &c., and their proportional weight to that of this fluid being thus ascertained, their density compared with that of water is readily calculated or they may be enveloped in wax or other suitable substance to protect them, and then treated by the method just given for substances lighter than water. Gaseous bodies are weighed in a thin glass flask or other vessel made for the purpose, and provided with a stop-cock. The vessel is exhausted of air before the introduction of the gas.

RULE FOR FINDING THE SPECIFIC GRAVITY OF A SOLID BODY.

_Weigh the solid in air and then in pure water._

The difference is the weight of water displaced, whose specific gravity is 1.000.

Then, as the difference of weight is to 1·000, so is the weight in air to the specific gravity; or divide the weight of the body in air by the difference between the weights in air and in water.

EXAMPLE.

A lump of glass is found to weigh in air 577 grains; it is then suspended by a horse hair from the bottom of the scale pan, and immersed in a vessel of pure water, when it is found to weigh 399.4 grains. What is its specific gravity?

577.0 Then, as 177.6 : 1 :: 577.0 : sp. gravity.
399.4 1
----- -----
177·6 the difference 177·6)577·0(3·248, Ans.
532·8
-----
4420
3552
----
8680
7104
-----
15760
14208
-----
1552

NOTE.—The above figures are introduced to show more vividly the
comparison between _bulk and weight_, the size of the different
substances, of course, being merely approximate. A study of the Table
of Specific Gravities to be found in the next page is worthy of the
time and attention.

TABLE OF SPECIFIC GRAVITIES.

METALS.

Iron, (cast) 7·207
„ (wrought) 7·688
Steel (soft) 7·780
„ (tempered) 7·840
Lead (cast) 11·400
„ (sheet) 11·407
Brass (cast) 8·384
„ (wire drawn) 8·544
Copper (sheet) 8·767
„ (cast) 8·607
Gold (cast) 19·238
„ (hammered) 19·361
Gold (22 carats) 17·481
„ (20 „ ) 15·709
Silver (pure, cast) 10·474
„ (hammered) 10·511
Mercury (60°) 13·580
Pewter 7·248
Tin 7·293
Zinc (cast) 7·215
Platinum 21·500
Antimony 6·712
Arsenic 5·763
Bronze (gun metal) 8·700

STONES AND EARTH.

Coal (Bituminous) 1·256
„ (Anthracite) { 1·436
{ 1·640
Charcoal ·441
Brick 1·900
Clay 1·930
Common Soil 1·984
Emery 4·000
Glass (plalte) 3·248
Ivory 1·822
Grindstone 2·143
Diamond 3·521
Gypsum 2·168
Lime 2·720
Granite 2·625
Marble 2·708
Mica 2·800
Millstone 2·484
Nitre 1·900
Porcelain 2·385
Phosphorus 1·770
Pumice Stone ·915
Salt 2·130
Sand 1·800
Slate 2·672
Sulphur 2·033

WOODS.

Ash ·845
Beech ·852
Birch ·720
Oak 1·120
Pine (yellow) ·660
„ (white) ·554
Cherry ·715
Cork ·240
Elm ·671
Poplar ·383
Walnut ·671
Willow ·585

LIQUIDS.

Acid Sulphuric 1·851
„ Muriatic 1·200
Spirits of Wine ·917
Alcohol ·790
Oil (turpentine) ·870
„ (olive) ·915
Oil (linseed) ·932
„ (castor) ·961
Pure water 1·000
Vinegar 1·080
Milk 1·032
Sea water 1·029

TO FIND THE WEIGHT OF A CUBIC FOOT OF ANYTHING CONTAINED IN THESE TABLES.

_Rule._ Multiply 62·5 lbs. (the weight of a cubic foot of pure water) by the specific gravity of the given body.

_Example._ What is the weight of a cubic foot of sea water?

62·5 lbs.
1·029 sp. gravity.
-----
5625
1250
625
-------
Answer, 64·3125 lbs. is the actual weight: but 64
lbs. is taken in practice as the weight of a cubic foot of sea water.

_Example._ How many cubic feet of sea water will weigh a ton?

Divide 2240 lbs. (1 ton) by 64 lbs.

{ _8)2240
64 { ----
{ 8) 280
----
Ans. 35

_Example._ What is the weight of a cubic foot of wrought iron?

62·5 lbs.
7·69 sp. gravity.
-----
5625
3750
4375
-------
Answer, 480·625 lbs.

480 lbs. in practice is the weight of 1 cubic foot of wrought iron.

NOTE.—35 cubic feet of sea water is accounted to be a ton, as in sea
water ballast for steamers, and in calculating displacement of ships.

_Example._ What is the average weight of a cubic foot of Bituminous coal?

1·256 sp. gravity.
62·5 lbs.
-----
6280
2512
7536
-------
Answer, 78·5000 lbs.

This 78·5 lbs. is the weight of a cubic foot in a _solid block_, but loose, as used for fuel, a cubic foot weighs about 49.7 lbs. which is the average of 13 kinds.

_Example._ What is the weight of a solid cast cylinder of copper, 4 inches diameter and 6 inches high?

8·607 sp. gravity.
62·5 lbs.
-----
43035
17214
51642
--------
537·9375 lbs. per cub. ft.
--------
Say 538 lbs.

·7854
16 diam. squared.
-----
47124
7854
-----
12·5664 area of base.
6 high
-------
75·3984 cu. in. in volume
-------
Say 75·4 cubic inches.

cub. in. cub. in. lbs.
Then, as 1728 : 75·4 :: 538 : Answer.
538
----
6032
2262
3770
------
{ 12)40565·2
{---------
1728 { 12)3380·433
{---------
{ 12) 281·702
-------
23·475 lbs.

Answer, 23-1/2 lbs. nearly.

FRICTION AND VISCOSITY OF FLUIDS.

_Frictional Resistance._—The resistance with which bodies oppose the movement of one surface on another is termed _friction_. It depends on the nature, and the roughness of the surfaces in contact; at the commencement of the sliding, it is greater than when the motion is continued.

_Friction is in effect an equivalent force exerted in a direction opposite to that in which the sliding occurs._ Its whole amount is the product of two factors: the first of these, which sums up the effect of the nature and condition of the surfaces, is called _the coefficient of friction_; the second, which is the sum of all pressures, as weight strain, and the adhesion due to magnetism (when employed), which act to urge the two bodies together, _i. e._, perpendicularly to the surface of contact, is called _the normal pressure_. But this law holds only where, with dry surfaces, the pressure is not enough to indent or abrade either; or, with wet surfaces, not enough to force out the unguent. In either of these cases, the friction increases more rapidly than the ratio of normal pressure.

_No surfaces can be made absolutely hard or smooth_; when one surface is made to slide over another, the slight roughness of the one interlock with those of the other, so that the surfaces must be separated or the points abraded to allow of the motion; but if one surface roll upon another, the prominent points are successively raised, without the need of complete lifting of the body or wearing off those points. _Hence, there are two kinds of friction, the sliding and the rolling._ The former of these in amount greatly exceeds the latter; it is a leading element in the stability of structures and fabrics of all lands, and the most important resistance and source of waste in all machinery, and is therefore a chief object of regard in the arts of construction and the science of engineering.

Sliding friction increases with the roughness of the surfaces in contact; hence, it is in practice diminished as these surfaces become worn, also by polishing, and by the use of lubricants, which smooth the rubbing surfaces by filling their depressions. It increases, almost universally, in exact proportion with the entire pressure, owing to weight or other causes, with which the two surfaces are held together; but at very great pressures, somewhat less rapidly. Consequently, in all ordinary cases, so long as the entire weight or pressure remains the same, the friction is, in general, entirely independent of the extent of the surfaces in contact.

The exceptions are, some increase when the rubbing surfaces under the same total pressure are very greatly extended, or when either surface is comparatively soft; and considerable lessening of friction when the bodies are very small, as in the runners of skates upon ice. For ordinary rates of motion, the total friction within a given space or distance is in like manner entirely independent of the velocity with which one surface is caused to move over the other; but in very slow motions it is increased, and in very rapid motions perceptibly diminished.

_Friction is also increased in proportion to the tendency of the surfaces to adhere_; hence, it is usually found greater between bodies of the same kind (steel on steel proving almost an exception) than between those of different kinds; it is usually greater when the surfaces have been long in contact, and at the beginning of motion, and always so, unless corrected by lubricants, between metallic surfaces so highly polished that air may be excluded from between them.

The frictional resistance retarding the flow of _water_ is subject to three laws:

1. It is proportioned to the amount of surface in contact.

2. It is independent of the pressure.

3. It is proportional to the square of the velocity.

It should be remembered that the laws relating to friction, between _solid bodies_ operate quite differently from what they do when applied to liquids; hence, the large mass of data relating to the general subject of friction must be disregarded in the consideration of hydromechanics and allied subjects.

For all fluids, whether liquids or gaseous, the resistance is independent of the pressure between the masses in contact. This is in accordance with the second law as stated.

The friction for all fluids (liquid or gaseous) is in proportion to the area of the rubbing surface; this follows from the first law and as the sectional area of a circle is the least, _pipes_ from their circular form present the smallest resistance to the flow of water.

From the third law, in practice we desire the making of pipes as large as possible; experiment having proved that low speeds are preferable to moderate and still more so, as compared to high speeds in proportioning the piping of hydraulic apparatus.

Friction of fluids is also independent of the nature of the solid against which the stream may flow, but dependent to some extent upon their degree of roughness.

Friction _for all fluids is also proportional to the density of the fluid_, and related in some way to its adhesiveness.

_Water flowing through a pipe tends to drag the pipe along with it_ on account of friction; in all actual fluids there is viscosity or internal friction, but if the relative motion is only slow enough it makes little difference whether the fluid is viscous or not.

_Ordinary fluids will change in shape under the action of a force_, however small, if enough time is given for the change to take place, and the rate of change of shape is a measure of its viscosity.

_The laws of friction, both for solids and liquids, have been established from experiments endlessly varied._ In investigating these principles we first proceed on the supposition that the forces in question act without any impediments. Great simplicity is attained by first bringing the subject to this ideal standard of perfection, and afterwards making suitable allowances for all these causes which operate in any given case to prevent the perfect application of the law.

Several tables and other data relating to the friction of water will be found in the other sections of this work, reference to which is made in the Index.

_The term viscosity_ has been described in the Glossary at the beginning of this work; a perfect fluid is incapable of resisting—except by its weight or inertia—a change of shape. Such a substance does not actually exist for all fluids have _viscosity_ or internal friction.

This is defined as a resistance to a change of shape depending on the rate at which the change is effected, _but_, as the fluids which engineers have to deal with are water and vapors and gases, it simplifies nearly all the calculations to assume that they have no internal viscosity or friction.

A slow continuous change of the shape of solids or semi-solids under the action of gravity, or external force, is also by the extension of the term called _viscosity_, as the viscosity of ice, as observed in the slow movement of those rivers of ice, the glaciers.

The _viscosity_ of liquids arises from the mutual attractions of the molecules and is diminished by the effect of the wandering molecules (C. D.). The viscosity of gases increases while that of liquids diminishes as the temperature is raised.

The _viscosity_ of fluids presents a certain analogy with the malleability of solids.

_Vis Viva_ is equivalent to _active or living force_; temperature is the _Vis Viva_ of the smallest particles of a body; in bodies of the same temperature the atoms have the same _vis viva_, the smaller mass of the lighter atoms being compensated by their greater velocity.

This term, which is often met with in scientific treatises, was invented by Leibnitz. It is well known that water in rapid circulation will absorb more heat than when stagnant or moving slowly; this is caused by _Vis Viva_ of the atoms.

CAPILLARY ATTRACTION.

_Capillary attraction is the attraction which causes the ascent of fluids in small tubes._ This word is derived from the Latin _capillus_—a hair.

The tubes must be less than one-tenth of an inch in diameter in order to produce the most satisfactory results, and tubes whose bores are no larger than a hair present the phenomenon the most strikingly. But though the rise of water above its natural level, is most manifest in small tubes, it appears, in a degree, in vessels of all sizes and shapes, by a ring of water formed around the sides with a concavity upward.

The following are the leading _facts_ respecting capillary attraction.

(1.) _When small tubes, open at both ends, are immersed perpendicularly in any liquid, the liquid rises in them, to a height which is inversely as the diameter of the bore._ Though tubes of glass are usually employed in experiments on this subject, yet tubes made of any other material exhibit the same property. Nor does the thickness of the solid part of the tube, or its quantity of matter, make the least difference, the effect depending solely on the attraction of the surface, and consequently extending only to a very small distance.

(2.) _Different fluids are raised to unequal heights by the same tube._ Thus, according to experiment, a tube which will raise water 23 inches will raise alcohol only 9 inches.

(3.) A tube 1/100 of an inch in diameter raises water 5.3 inches; and since the height is reciprocally as the diameter, _the product of the diameter into the height is a constant quantity_, namely, the .053th part of an inch square.

(4.) _Fluids rise in a similar manner between plates of glass, metal, &c._, placed perpendicularly in the fluids, and near to one another. If the plates are parallel, the height to which a fluid will rise, is _inversely as the distance between the plates_; and the whole ascent is just _half that which takes place in a tube_ of the same diameter. If the plates be placed edge to edge, so as to form an angle, and they be immersed in water, with the line of their intersection vertical, the water will ascend between them in a _curve_ having its vertex at the angle of intersection. This curve is found to have the properties and form of the _hyperbola_.

In the adjustment of the stems of barometers and the thermometer an allowance is made to compensate for the influence of capillary attraction.

Such are the leading facts ascertained respecting capillary attraction. Various explanations of them have been attempted but that of La Place is most generally received. According to this high authority, the action of the sides of the tube draws up the film of fluid nearest to it, and that film draws along with it the film immediately below it, and so each film drags along with it the next below, until the weight of the volume of fluid raised exactly balances all the forces which act upon it. The fact that the elevation of the water between the parallel plates is exactly _half_ that in a tube of the same diameter, clearly indicates that the force resides in the surrounding body; and the additional fact that the thickness or quantity of that body makes no difference, proves that the force resides in the surface, and that the action extends only to a very small distance.

NOTE.—Several familiar examples of capillary attraction may be
added. A piece of sponge, or a lump of sugar, touching water at
its lowest corner, soon becomes moistened throughout. The wick of
a lamp lifts the oil to supply the flame, to the height of several
inches. A capillary glass tube, bent in the form of a syphon, and
having its shorter end inserted in a vessel of water, will fill
itself and deliver over the water in drops. A lock of thread or of
candle-wick, inserted in a vessel of water in a similar manner, with
one end hanging over the vessel, will exhibit the same result. An
immense weight or mass may be raised through a small space, by first
stretching a dry rope between it and a support, and then wetting the
rope.

The several figures above need no explanation to the attentive reader.

FLOW OF WATER UNDER PRESSURE.

Under this head three general cases are to be considered: 1, that of liquids _issuing from orifices_; 2, _their flow through tubes_, or in streams; and, 3, _the effects of the momentum and impact of liquids_. The principles governing the action of the two last will be introduced, 2, under that portion of the work relating to “piping,” and, 3, under the sections pertaining to the jet pump. Throughout the different portions of the book will the three cases be still further elucidated.

Now, as to the laws governing the escape of liquids under pressure through an opening, it may be understood that when the liquid escapes from a vessel, owing to the excess of the internal pressure, _the volume which escapes depends on the section of the orifice and the velocity with which the liquid molecules move at the moment of their escape from it_.

This velocity depends upon the density of the liquid, the excess of pressure at the opening, and the friction of the liquid, both at the opening and against the walls. When the aperture is made in a very thin wall of a large vessel, so as to remove, as much as possible, the causes tending to modify the motion of the escaping fluid, the laws of the escape are comprised in the following theorem, discovered by Torricelli, in 1643, as a consequence of the law of the fall of bodies discovered by Galileo: “Liquid molecules, flowing from an orifice, _have the same velocity, as if they fell freely in vacuo from a height equal to the vertical distance from the surface to the center of the orifice_.”

Deductions from the above:—1, _the velocity depends on the depth of the orifice from the surface, and is independent of the density of the liquid_. Water and mercury in vacuo would fall from the same height in the same time; and so escaping from an orifice at the same depth, below the surface, would pass out with equal velocity; but mercury, being 13·5 times as heavy as water, the pressure exerted at the aperture of a vessel filled with mercury, will be 13·5 times as great as the pressure exerted at the aperture of a vessel filled with water; 2, _the velocity of liquids is as the square roots of the depths of the orifices below the surfaces of the liquids_.

NOTE.—Torricelli discovered, in the early part of the 17th century,
the remarkable fact that a fluid issues from a small orifice with the
same velocity (friction and atmospheric resistance excluded) which it
would have acquired in falling through the depth from its surface.
This was one of a long series of discoveries leading toward the now
almost exact science of hydro-mechanics.

Thus stating, the velocity of a liquid escaping from an orifice one foot below the surface to be _one_; from a similar orifice four feet below the surface, it will be _two_, and at nine feet _three_, at sixteen feet _four_, and so on.

From comparative experiments made by a great number of observers, it is learned that the actual flow is only about two-thirds of the theoretical flow. See note.

The form and constitution of liquid veins have been studied and found to be:

1.—That the fluid issuing vertically from an orifice made in a plane and thin horizontal wall, is always composed of two distinct parts, Fig. 97, the portion nearest the orifice is calm and transparent, like a rod of glass, gradually decreasing in diameter. The lower part, on the contrary, is always agitated, and takes an irregular form, in which are regularly distributed elongated swellings, called _ventres_, whose maximum diameter is greater than that of the orifice.