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 each body is ascertained in avoirdupois ounces, by multiplying the number 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 weight 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 avoirdupois by multiplying it into 1.0971. MISCELLANEOUS COMPUTATIONS AND EXPERIMENTS. The pendulum vibrating seconds of mean solar time 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 32°, supposed to be the ten mil. lionth 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 be considered as identical with the imperial standard) § Gen. Roy's scale.... 35.99998 36.00088 Royal Society's standard 36.00135 Ramsden's bar 36 00249 Weight of a cubic inch of distilled water in a vacuum at the temp. 62°, as opposed to brass weights in a vacuum also, 252 722 grains ...... log. 2.4026430 Consequently a cubic foot 62 3862 pounds avoirdupois..log. 17950887 Consequently a cubic foot 62 3206 pounds avoirdupois. log. 2.4021857 .log. 1·7946314 ..log. 0-2387924 ..log. 24429124 log. 1.2739112 Specific gravity of water at different temperatures, that at 62° being taken as unity. The difference of temperatures between 62° and 39°, where water attains 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 difference of temperatures from 62° to 39° will vary the content of a brass gallon-measure just one-fifth of a cubic inch. It appears that the specific gravity of clear water from the Thames exceeds 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. al HYDRAULICS. 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 set 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 atmosphere, 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 vacuities; the velocity of the fluid in different parts of it, will be reciprocally 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 fluid ACBL come to the place acbl. Then will the solid ACBL= solid acbl; take away the part acBL common to both; and we have ACca LBbl. But in equal solids the bases and heights are reciprocally 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. B PROP. 2.-If AD, fig. 6, be a vessel of water or any other fluid; 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 equal motions; and, since a cylinder of water falling through AB by its gravity, acquires such a motion, as to pass through the whole height 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 Out in that time. Cor. 1. The 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 nearly 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 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 ascends nearly to the height above the hole. Cor. 4. The velocities and likewise the quantities of the spouting water, at different depths, will be as the square roots of the depths. SCHOLIUM. From hence 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 height EF, near to the level of the reservoir AB. In order to have a fountain in perfection, the pipe CD must be wide, and covered with a thin plate at E 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 an 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 reservoir When the water runs through a great length of pipe, the jet will not rise so high. A jet never rises to the full height of the reservoir; in a 5-feet jet it wants an inch, and it falls short by lengths which are as the squares 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 bottle 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 If a fountain be the air within; and the water will spout out at B. placed in the sunshine and made to play, it will shew all the colours of the rainbow, if a black cloth be placed beyond it. A jet goes higher if it is not exactly perpendicular; for then the upper 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 velocities and times. But (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 AB X BE; and the space EL as ACE. And hence, if two holes are made equidistant from top and bottom, they will project the water to the same distance; for, if 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 nute distances it will be thus:-the velocity through any hole B, will. carry it through AB in the time of falling through AB; 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 AB: AB :: BE ✓ BE EH = AB, :: = √2ABE; and so the space EL= AB V/2ACE. It is plain, these curves are parabolas. For the horizontal inotion being uniform, EH will be as the time; that is, as will be as EH2, which is the property of a parabola. Fig. 8. BE or BE 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 length 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, with the little quantity of water contained in the pipe; which weight will be nearly equal to a column of the fluid, whose base is the top AB, and height that of the pipe EF. For the pressure of the water against the. Sop 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 higher, 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 CDE, 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 run 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 coJumn 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 the semidiameter of the hole at D=d, and AD=h. Then since the same water passes through the sections at D and B; therefore (Prop. 1.) the velocity will be reciprocally as the section; whence v/h: :: VAB Vh VAB BC da : 1 BC 1 dd ; therefore and ddh BC2 V AB, whence AB X BC 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 hole, as IC, it would form itself into the same figure GCD in descending. 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 has 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 upon 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 destended through by a falling body, to acquire that velocity. For let there be a reservoir whose 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 by the weight of that column of water, which 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 to destroy as to generate any motion. Cor. The force of a stream of water flowing out at a bole in the bottom of a reservatory, is equal to the weight of a column of the fluid of the same height and whose base is the hole. |