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Obs.-1. The above table shews the specific weights of the various substances contained in it, and the absolute weight of a cubic foot of cach body is ascertained in avoirdupois ounces, by multiplying the nunber opposite to it by 1000, the weight of a cubic foot of water; thus the weight of a cubic foot of mercury is 14,019 ounces avoirdupois, or 876 lb,
2. If the ght of a body be known in avoirdupois ounces, its weight in Troy ounces will be found in multiplying it into .91145. And, if the weight be given in Troy ounces, it will be found in avoirdapois by multiplying it into 1.0971. MISCELLANEOUS COMPUTATIONS AND EXPERIMENTS.
The pendulum vibrating seconds of mean solar tiunc at London in a vacuum, and reduced to the level of the sea, is 39:1393 inches; consequently the descent of a heavy body from rest in one second of time in a vacuum, will be 193 145 inches. The logarithm 2.2858828.
A platina metre at the temperature of 320, siipposed to be the ten mil. Jionth part of the quadrant of the meridian, 39.3708 inches. The ratio to the imperial measure of three feet as 1.09363 to 1, the logarithm 0·0388717.
The five following standards, accurately measured, give these results :Gen. Lambton's scale, used in the Trig. Surv. of India, 35 99934 inches. Sir G. Shuckburgh's scale (which for all purposes may ?
35 99998 be considered as identical with the imperial standard) I Gen. Roy's scale ...
36.00088 Royal Society's standard
36.00135 Ramsden's bar.
36 00249 Weight of a cubic inch of distilled water in a vacuum
ai the temp. 620, as opposed to brass weights in a log. 2:4026430
vacuum also, 252 722 grains .. Consequently a cubic fool 62:3862 pounds avoirdupois.. log. 1.7950887 Weight of a cubic inch of distilled water in air at 620
of temperature with a mean height of the barometer log. 2:4021857
252-456 grains. Consequently a cubic fout 62 3206 pounds avoirdupois. .log. 1:7946314 And an ounce of water 1.73298 cubic inches.. .log. 0·2387924 Cubic inches in the imperial gallon, 277.276.. .log. 2: 4429124 Diameterof the cylinder containing a gallon at one inch log. 1-2739112
high, 18:78933 .. Specific gravity of water at different temperatures, ibat at 62° being
taken as uniiy. 70° 0 99913 620 1.
520 1 00076 44° 1:00107 08 099936 58 I 00035 50 1.00087 42 1.00111 60 009958 56 1.00050 48 1:0009.) 40 Twi13 04 0 99980 54 1.00061 46 1:00102 38 1:00113
The difference of temperatures between 620 and 390, where water at tains its greatest density, will vary the bulk of a gallon of water rather less than the third of a cubic inch.
And, assuming from the mean of numerous estimates the expansion of brass 0-00001044 for each degree of Fabrenheit's thermometer, the dir. ference of temperatures from 62° to -39° will vary the content of a brass gallou-measure just one filio of a cubic inch.
It appears that the specific gravity of clear water from the Thames prceeds that of distilled water, at the mean temperature, in the proportion of' 1.0006 to 1, making a diff. of about one-sixth of a cubic in. on a gallon.
Rain water does not differ from distilled water, so as to require any lowance for common purposes.
Definitions.-1. The science of Hydraulics teaches how to estimate the velocity and force of fluids in motion. Upon the principles of this science all machines worked by water are constructed, as engines, mills, pumps, fountains, &c.
2. Water can be sel in motion only by its own gravity; as when it is allowed to descend from a higher to a lower level : by an increased pressure of the air, or by removing the pressure of the at. mosphere, it will rise above its natural level.
Obs.- In the former case it will seek the lowest situation, in the latter it may be forced to almost any height.
Prop. 1.- If a fluid runs through a pipe, so as to leave no va. cuities; the velocity of the fluid in different parts of it, will be re. ciprocally as the transverse sections, in these parts.
Let AC, LB, fig. 5, be the sections at A and L. And let the part of the Quid ACBL come to the place acbl. Then will the solid ACBL= solid acbl ; take away the part acBL commion to both; and we have ACca=LBil. But in equal solids the bases and heights are recipro. cally proportional. But, if Df be the axis of the pipe, the heights Dd, Ff, passed through in equal times, are as the velocities. Therefore, section, AC : section LB :: velocity along Ff : velocity along Dd.
Prop. 2.-If AD, fig. 6, be a vessel of water or any other fluid; B a hole in the bottom or side. Then, if the vessel be always kept full ; in the time a heavy body falls through half the height of the water above the hole AB, a cylinder of water will flow out of the hole, whose height is AB, and base the area of the hole.
The pressure of the water against the hole B, by which the motion is generated, is equal to the weight of a column of water whose height is AB, and base the area B (See Hydrostatics, Prop. 4). But equal forces generate cqual motions; and, since a cylinder of water falling through JAB hy its gravity, acquires such a motion, as to pass through the whole beight AB in that time ; therefore in that time the water running out must acquire the same motion. And, that the effluent water may have the same motion, a cylinder must run out whose length is AB; and then the space described by the water in that time will also be AB, for that space is the length of the cylinder run out. Therefore this is the quantity run o'it in that time.
Cor. 1. Tbe quantity run out in any time is equal to a cylinder or prism, whose length is the space described in that time by the velocity acquired by falling through half the height, and whose base is the hole.
For the length of the cylinder is as the time of running out.
Cor. 2. The velocity, a little without the hole, is greater than in the hole; and is ncarly equal to the velocity of a body falling through the whole height AB.
For without the hole the stream is contracted by the water's converging from all sides to the centre of the hole, and this makes the velocity greater in about the ratio of 1 to v 2.
Cor. 3. The water spouts out with the same velocity, whether it be downwards, or sideways, or upwards. And therefore, if it be upwards, it asceuds nearly to the height above the hole.
Cur. 4. The velocities and likewise the quantities of the spouting ter, at different depths, will be as the square roots of the deptus.
SCHOLIUM.–From bence are derived the rules for the construction of fountains or jets. Let ABC, fig. 7, be a reservoir of water, CDE a pipe coming out from it, to bring water to the fountain which spouts up at E, to the heiglit £F, near to the level of the reservoir AB. ln order to have a fountaiu in perfection, the pipe CD must be wide, and covered with a thin plate at Ė with a hole in it, not above the fifth or sixth part of the diameter of the pipe CD. And this pipe must be curve, having no angles. If the reservoir be 50 feet high, the diameter of the hole at E may be ar inch, and the diameter of the pipe 6 inches. In general the diameter of the hole E, ought to be as the square root of the height of the reservois When the water runs through a great length of pipe, the jet will not rise so high.' A jet never rises to the fall height of ibe reservoir; in a 5-feet jet it wants an inch, and it falls short by lengths which are as the sqnares of the heights; and smaller jets lose more. No jet will rise 300 feet high.
A small fountain, (fig. 8.) is easily made by taking a strong botile A, and filling it half full of water; cement a tube BI very close in it, going near the bottom of the bottle. Then blow in at the top B, to compress the air within ; and the water will spout ont at B. If a fountain be placed in the sunshine and made to play, it will shiew all the colours of the rainbow, if a black cloth be placed beyond it.
A jet goes bigher if it is not exactly perpendicular; for then the apper part of the jet falls to one side without resisting the column below. The resistance of the air will also destroy a deal of its motion, and binder it from rising to the height of the reservoir. Also the friction of the tube or pipe of conduct has a great share in retarding the motion.
If there be an upright vessel, as AF (fig. 9.), full of water, and several holes be made in the side as B, C, D, then the distances the water will spout upon the horizontal plane EL, will be as the square roots of the rectangles of the segments ABE, ACE, and ADE. For the spaces will be as the velocitics and times. Bilt (Cor. 4.) the velocity of the water flowing out of B, will be as ✓ AB, and the time of its moving (which is the same as the time of its fall) will be as ✓ BE: therefore the distance EH is as VAB X BE; and the space EL as VACE. And bence, if two boles are made equidistant from top and bottom, they will project the water to the same distance; for, il AB = DE, then ABE=ADE, which makes EH the same for both; and hence also it follows, that the projection from the middle point C will be furthest, for ACE is the greatest rectangle. These are the proportions of the distances; but for the
mule distances it will be thus:- the velocity through any hole B, wil. carry it through AB in the time of falling through ŽAB; then to find how far it will move in the time of falling through BE. Since these times are as the square roots of the heights, it will be v AB : AB ::
BE V BE : EH = AB, = V 2ABE; and so the space EL=
GAB V2ACE. It is plain, these curves are parabolas. For the horizontal inotion being uniform, EH will be as the time ; that is, as v BE or BE will be as EH”, which is the property of a parabola.
If there be a broad vessel ABDC (fig. 10.) full of water, and the top AB fits exactly into it; and if the small pipe FE of a great lengti be soldered close into the top, and if water be poured into the top of the pipe F, till it be full; it will raise a great weight laid upon the top, witb the little quantity of water contained in the pipe; which weight will be nearly equal to a column of the finid, whose base is the top AB, and bright that of the pipe EF. For the pressure of the water against the. inp AB, is equal to the weight of that column of water, by Prop. 3 and Cor, and Prop. 4, Cor. 2, page 722.
But here the tube must not be too small. For in capillary tubes the attraction of the glass will take off its gravity. If a very small tube be immersed with one end in a vessel of water, the water will rise in the tube above the surface of the water; and the bigher, the smaller the tube is. But, in quicksilver, it descends in the tube below the external surface, from the repulsion of the glass.
To explain the operation of a syphon, (fig. 11.) which is a crooked pipe CDĖ, to draw liquors off. Set the syphon with the ends C, E, upwards, and fill it with water at the end E till it run out at C; to prevent it, clap the finger at C, and fill the other end to the top, and stop that with the finger. Then, keeping both ends stopt, invert the shorter end C into a vessel of water AB, and take off the fingers, and the water will ran out at E, till it be as low as C in the vessel ; provided the end E be always lower than C. Since E is always below C, the height of the coZumn of water DE is greater than that of CD, and therefore DE must outweigh CD and descend, and CD will follow after, being forced up by the pressure of the air, which acts upon the surface of the water in the vessel AB.
The surface of the carth falls below the horizontal level only an inch in 620 yards; and in other distances the descents are as the squares of the distances.
And, to find the nature of the curve DCG, (fig. 12 ) forming the jet IDG: Let AK be the height or top of the reservoir HF, and suppose the stream to ascend without any friction or resistance. By the laws of falling bodies the velocity in any place B, will be as ✓ AB. Put tho semidiameter of the bole at D=d, and AD=h. Then sivce the same Water passes through the sections at D and B; therefore (Prop. 1.) the
1 velocity will be reciprocally as the section ; w bence Vk: :: V AB
V AB : ; therefore
and dd v h= BC? V AB, wbence BC.
BC: da AB X BCʻ= HD*; which is a paraboliform figure, whose asymptote is AK, for the nature of the cataractic curve DCG. And, if the fluid was to descend through a boie, as IC, it would form itself into the same figure GCD in descending. Fig. 11. D
PROP. 3.-The resistance any body meets with in moving through 1 fluid is as the square of the velocity.
For, if any body moves with twice the velocity of another body equal to it, it will strike against twice as much of the fluid, and with twice the velocity, and therefore bas four times the resistance ; for that will be as the matter and velocity. And, if it moves with thrice the velocity, it strikes against thrice as much of the fluid in the same time, with thrice the velocity, and therefore has nine times the resistance. And so on for all other velocities.
Cor. If a stream of water, whose diameter is given, strike against an obstacle at rest; the force against it will be as the square of the velocity of the stream.
For the reason is the same; since with twice or thrice the velocity, twice or thrice as much of the fluid impinges upou it, in the same time.
PROP. 4.-The force of a stream of water against any plane obstacle at rest, is equal to the weight of a column of water, whose base is the section of the stream; and height the space descended through by a falling body, to acquire that velocity.
For let there be a reservoir wbose height is that space fallen through: then the water (by Cor. 2. Prop. 2), flowing out at the bottom of the reservatory, has the same motion as the stream; but this is generated hy the weight of that column of water, w bioh is the force producing it. And that same motion is destroyed by the obstacle; therefore the force against it is the very same: for there is required as much force lo destroy as to generate any motion.
Cor. The force of a stream of water flowing oat at a hole in the bottom of a reservatory, is equal to the weight of a column of the fluid of the same beight and wilosc base is the hole.