ELEMENTARY HYDRAULICS.

ELEMENTARYHYDRAULICS

There are three physical states or conditions of matter, which are thesolid,liquidandgaseous, which in this connection apply to Ice, Water and Steam.A solidoffers resistance both to change of shape and to change of bulk.

A Fluidoffers no resistance to change of shape. Again fluids can be divided intoliquidsand vapors or gases. Water is the most familiar example of a liquid. A liquid can be poured out in drops while a gas or vapor flows in a stream or streams.

Gasis a term at first used as meaning the same as the nameair, but is now restricted to fluids supposed to be permanently elastic, as oxygen, hydrogen, etc., in distinction from vapor such as steam which become liquid upon a reduction of temperature.

It is important to note that experiment proves that every vapor becomes a gas at a sufficiently high temperature or low pressure, while, on the other hand, every gas becomes a vapor at sufficiently low and high pressures. In present popular usage the term gas applies to any substance in the aeriform elastic condition.

Hydraulicsis that branch of science or of engineering which treats of the motion of liquids, especially of water and of the laws by which it is regulated.

As a science, hydraulics includeshydrodynamicsor the principles of mechanics applicable to the motion of water.

As a branch of engineering, hydraulics consists in the practical application of the mechanics of fluids, to the controland management of water, with reference to the wants of man, including water works, hydraulic machines, pumps, water wheels, etc.

The term hydraulics, so familiar in daily use, is formed from two Greek words meaning: 1, water; 2, a pipe; hence, it will be observed with interest how close the original meaning follows the development of the science in its practical adaptation; there is always the “pipe” or holding vessel and the “water” or its equivalent.

From the same elementary word meaning water, in the Greek language, has been formed very many other words in common use, for example hydrophobia, hydrogen, hydrant, hygrometer, etc., as well as the following:

Hydromechanicsis that branch of natural philosophy which treats of the mechanics of liquid bodies, or in other words, of their laws of equilibrium and motion. Hydromechanics comprises properly those phenomena of liquids by which these bodies differ from solids or from bodies at large; hence, its foundation is laid in the properties that distinguish the liquid from other states of bodies, viz.: the presence of cohesion, with great mobility of parts, and perfect elasticity.

Hydrostaticsis that branch of science which relates to the pressure and equilibrium of non-elastic fluids, as water, mercury, etc.; thus, the hydrostatic press is a machine in which great force with slow motion is action communicated to a large plunger by means of water forced into the cylinder in which it moves, by a forcing pump.Staticstreats of forces that keep bodies at rest or in equilibrium, the water through which the force operates in the hydrostatic press always remaining at rest serves as a good illustration.

Pneumaticsis that branch of science, which relates to air, or gases in general or their properties; also of employing (compressed) air or other gas as a motive power. The use of pneumatic pumping machinery is constantly increasing, especially of the direct pressure types; under the section ofthis work relating to Air Pumps additional data will be presented.

Hydropneumaticsis defined as involving the combined action of water and air, or gas, as shown, for example, in the hydropneumatic accumulator. The word is a compound formed of the Greek words meaning water and air.

Semi-liquids.All the results stated in reference to water are further modified in those semi-liquids which have greater or less viscidity, as pitch, syrup, fixed oils, etc. Viscosity may be defined as the quality of flowing slowly, thus the viscosity of such liquids as have been named is very great as compared with that of a mobile liquid like alcohol.

Water, considered from a chemical standpoint, is a compound substance consisting of hydrogen and oxygen, in the proportion of twovolumesof the former gas to one volume of the latter; orby weightit is composed of two parts of hydrogen united with sixteen parts of oxygen. It should be noted that the union of these two gases is effected bychemical actionand not bymechanical mixture. Pure water is transparent, inodorous and tasteless.

Under ordinary conditions water passes the liquid form only at temperatures lying between 32° F. and 212° F.; it assumes a solid form, that of ice or snow at 32° F., and it takes the form of vapor or steam at 212° F.

There are four notable temperatures for water, namely:

32° F.,or0° C.= the freezing point under one atmosphere.39°·1or4°= the point of maximum density.62°or16°·66= the standard temperature.212°or100°= the boiling point, under one atmosphere.

The temperature 62° F. is the temperature of water used in calculating the specific gravity of bodies, with respect to the gravity or density of water as a basis, or as unity.

Weight of one cubic foot of Pure Water.

At 32° F.= 62·418pounds.At 39°·1.= 62·425„At 62°(Standard temperature)= 62·355„At 212°= 59·640„

The weight of a cubic foot of water is, it may be added, about 1000 ounces (exactly 998·8 ounces), at the temperature of maximum density.

The weight of a cylindrical foot of water at 62° F. is 48·973 pounds.

Weight of one cubic inch of Pure Water.

At 32° F.= ·03612pound,or0·5779ounce.At 37°·1= ·036125„„0·5780„At 62°= ·03608„„0·5773„or 252·595 grains.At 212°= ·03451„„0·5522„

The weight of one cylindrical inch of pure water at 62° F. is ·02833 pounds, or 0·4533 ounce.

Volume of one pound of Pure Water.

At 32° F.= ·016021cubic foot,or27·684cubic inches.At 39°·1= ·016019„„27·680„At 62°= ·016037„„27·712„At 212°= ·016770„„28·978„

The volume of one ounce of pure water at 62° F. is 1·732 cubic inches.

The weight of water is usually taken in round numbers, for ordinary calculations, at 62·4 lbs. per cubic foot, which is the weight at 52°·3 F.; or it is taken at 621⁄2lbs. per cubic foot.

Salt water boils at a higher temperature than fresh water owing to its greater density, and because the boiling point of water is increased by any substance that enters into chemical combination with it. The density of water decreases as the temperature increases, since heat destroys cohesion and expands the particles, causing them to occupy greater space,where precision is not required; the pressure on a square foot at different ocean depths are approximate, in the following

TABLE.

Depth in feet.Pressure on sq. foot.8500lbs.161000„241500„322000„402500„483000„563500„644000„724500„805000„885500„966000„1 mile, or 5,280 feet,330,000 lbs.5 miles,1,650,000 „

This table is based upon an allowance of 621⁄2lbs. of water to the cubic foot, thus 8 feet × 621⁄2= 500, etc.

Water is practically non-elastic.A pressure of 30,000 lbs. to the square inch has been applied and its contraction has been found to be less than one-twelfth. Experiment appears to show that for each atmosphere of pressure it is condensed 471⁄2millionth of its bulk.

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 equal1⁄300000of its length for every pound per unit of area of the end pressure.

Fig. 84.

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 thehydrostatic 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.

Fig. 85.

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 thehydrostatic 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 toa. 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 theupwardpressure upon the surfacen oof the vessel is precisely equal to the weight of theextraquantity of water which a cylindrical vessel would contain, and balances an equal portion of thedownwardpressure onm. 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 2a; it will have to support a pressure 2P. Similarly, if the piston were equal to 3a, it would experience a pressure of 3P; and if its area were 100 or 1,000 times that ofa, 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.

Fig. 86.

Fig. 87.

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 useis 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.

Fig. 88.

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 thephial 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 torest 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.

Fig. 89.

Example 1.Let the column of fluid ABCD Fig. (1) be perpendicular to the horizon. Take any points,xandy, 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 uponxandy, 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, isincreasedin 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 isdiminishedon 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 actionof the inclined plane; and the pressure on any part of Ccwill be, as before, proportioned to its perpendicular depth; and the pressure of the inclined column ACacwill be the same as that of the perpendicular column ABCD.

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

Fig. 90.

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 theheights ofthe 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 ofequal 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 undulationsof 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.

various shapes of vessel

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.

Gravity or Gravitationis that species of attraction, or force by which, all bodies or particles of matter in the universe tend towards each other; it is also calledattraction 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, theaggregateeffect 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 thepressureexerted 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 areequally acceleratedin right lines perpendicular to the plane of the horizon. The apparentinequalityof 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.

Since a body falls to the ground in consequence of the earth’s attraction oneachof 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.Thewater-hammer, a model used in scientific schools, illustrates this: the instrument consists of athick 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 161⁄12feet in the first second of time, and will then have acquired a velocity, which being continued uniformly, will carry it through 321⁄6feet 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 321⁄6, 641⁄3, 961⁄2, &c.; the spaces passed through as 161⁄12, 641⁄3, 1443⁄4, &c., and the spaces for each second, 161⁄2, 481⁄4, 805⁄12, &c.

TABLE.

Showing the Relation of Time, Space and Velocity.

Time inseconds ofthe body’sfall.Velocityacquired atthe end ofthat time.Squares.Space fallenthroughin thattime.Space.Whole Spacefallen through inthe last secondof the fall.132·16116·08116·08264·33464·33348·25396·59144·75580·414128·6616257·337112·585160·8325402·089144·756193·36579·11176·917225·1749788·0813209·088257·33641029·3315241·259289·5811302·7517273·4210321·661001946·0819305·58

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 aretime,length and massor 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, requiresmore 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. Theweightof 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 requiredto hold it up, or to keep it from descending. Hence, weights are nothing more thanmeasures of the force of gravityin 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 levelwill 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 calledthe free surfaceand 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.

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

Multiply the time, in seconds, by 321⁄6, and it will give the velocity acquired in feet, per second.

Example.Required the velocity in seven seconds.

321⁄6× 7 = 2251⁄6feet. Ans.

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

Multiply the space, in feet, by 641⁄3, 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.

641⁄3× 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 161⁄12, and it will give the space in feet.

Example.Required the space fallen through in seven seconds.

161⁄12× 7² = 7881⁄12feet. Ans.

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

Divide thesquare rootof 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 Gravityis 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 from1⁄21to1⁄21·5of itsweight 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. 93and rule and example onpage 95.

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