201. Prol. 8. The range and elevation being given ; to greatest height to which the shell rises above the horizontal si Rule. As radius is to the tangent of elevation, so is of the to the height required. Ex. A shell, when discharged at an elevation of 40°, range feet, what is its greatest height during the flight? Rad : Tan 40' :: 750 : 629 feet. Ans. 202. Prob. 9. The range and elevation being given ; to find the petus. Rule. As the sine of twice the elevation is to the radius, so is balt the range to the impetus. Er. With what impetus must a shell be discharged at an elevation of 35° to strike an object at the distance of 3180 feet? Sin 70°: Rad :: 1590 : 1692 feet. Ans. Of Projectiles made on au Inclined Plane. 203 Prol. 10. The inclination of the plane, and the impetus and elevation of the piece being given, to find the range. Rule. Add together twice the log. secant of the plane's inclination, the log. sine of the elevation above the plane, the log. co-sine of the elevation above the horizon, and the log. of four times the impetus ; then will the sum be the log. of the range. Er. How far will a shot range on a plane which ascends 10°, and on another which descends 10°, the impetus being 2000 feet, and the elevation of the piece 32°30' ? The elevation above the plane, in the first case, is 220:30, and in the second 420-30. 1st, For the ascending plane. 2dly, for the descending plane. 10° 2 sec .. 0:013297 10°...... 2 sec ..0013297 22° 30'. 9:582840 420.30 ... 9 829683 32:30.... cosin 9.926029 32:30 .. cosin... · 9°926029 8000 log..... 3.903090 8000 log..... 3.903090 sin .... sin Range = 2662 feet = 3.425256 Range=4700 feet=3 672099 204. Prob. 11. The inclination of the plane, the range and elevation being given, to find the inipetus. Rule. Add together twice the log. co-s'ne of the plane's inclination, the log. co-secant of the elevation above ihe plane, the log. secant of the elevation above the horizon, and the log of of the range; then will the sum be the log. of the impetus. With what impetus must a shell be discharged to strike an object at the distance of 2662 feet on an inclined plane which ascends 10o, the elevation of the mortar being 320-30' ? Plane's inclination........ 10°...... 2 cos.... 19996703 0073971 2 823148 Impetus = 2000 feet, 120.971= 3 300982 1 HYDROSTATICS. sitions. -1. HYDROSTATICS treat of the nature, gravity, and motion, of duids in general, and of the methods of ng solids in them. And its mechanical practice, called ulics, relates particularly to the motion of water through 2. A fluid is a body, the parts of which yield to any impression, and are easily moved among each other. Fluids are either non-elastic and incompressible, as watcr, oil, Obs -1. Deat, or motion, is supposed to be the cause of Nuidity: for ater; and with more heat, an clastic fluid, in steam. In the first state, 2. Philosophers have usually assumed, that the particles of Nuids are round and smooth, since they are so easily moved among one another. This supposition will account for some circumstances belonging to them. If the particles are round, there must be vacant spaces between them, in the saine manner as there are vacuities between cannon-balls that are piled together; between these balls smaller shot may be placed, and between'these, others still smaller, or gravei, or sand, may be dillused. In a similar manner, a certain quantity of particles of sugar can be taken up in water without increasing the bulk, and « lien the water has dissolved the sugar, salt may be dissolved in it, and, yet the bulk remain the same; and, adınitting that the particles of water are round, this is easily accounted for. 3. Others have supposed that the cause of Audity is the mere want of cohesion of the particles of water, oil, &c. and froin this in perfect cohe. sion, fluids in small quantities, and under pecuiiar circumstances, arrange themselves in a spherical manner and form drops. Fluids are scbject to the same laws of gravity with solids ; but their want of coliesion occasions some peculiarities. The parts of a solid are so connected as to form a whole, and their weight is concentrated in a single point, called the centre of gravity: but the atoms of a fluid gravitate independently of each other. PROP. 1.-If one part of a fluid be higher than another, the higher parts will continually descend to the lower places, and will not be at rest till the surface of it is quite level. For the parts of a buid being movable every way, if any part is above the rest, it will descend by its own gravity as low as it can get. And afterwards other parts, that are now become higher, will descend as the otter did, till at last they will all be reduced to a lerel or horizontal plane. Cow 1. Ilence water that communicates by means of a channel or pipa The other water, will settle at the same height in both places. Cor. 2. For the same reason, if a fluid gravitates towards a centre ; it will dispose itself into a spherical figure, whose centre is ibe cintre of force; as the sca in respect of the earth. PROP. 2.- If a fluid be at rest in a vessel whose base is parallel to the horizon, equal parts of the base are equally pressed by the fluid. For upon every part of the base there is an equal column of the fluid supported by it. And, as all these columns are of equal weight, they must press the base equally; or equal parts of the base will sustaiu an equal pressure. Cor. 1. All parts of the fluid press equally at the same depth. For, imagine a plane drawn through the fluid parallel to the horizon. Then the pressure will be the same in any part of that plane, and therefore the parts of the fluid at the same depth sustain the same pressure. Cor. 2. The pressure of a fluid, at any depth, is as the depth of the fluid. For the pressure is as the weight, and the weight is as the height, of a column of the fluid. PROP. 3.- If a fluid is compressed by its weight or otherwise ; at any point it presses equally, in all manner of directions. This arises from the nature of fluidity ; which is, to yield to any force in any direction. If it cannot give way to any force applied, it will press against other parts of the fluid in direction of that force. And the pressure in all directions will be the same. For, if any one was less, the fluid would move that way, till the pressure be equal every way. Cor.-In any vessel containing a fluid, the pressure is the same against the bottom as against the sides, or even upwards, at the same depth. PROP. 4—The pressure of a fluid upon the base of the contain. ing vessel, is as the base and perpendicular altitude, whatever be the figure of the vessel that contains it. Let ABIC, EGKH, figs. I and 2, be two vessels. Then (Prop. 2. Cor. 2.) the pressure upon an inch on the base AB= height CDX1 inch; and the pressure upon an inch on the base HK is = height fH x I inch.' But (Prop 2.) equal parts of the bases are equally pressed, therefore the pressure on the base AB is CD X number of inches in AB; and pressure on the base HK is FH X number of inches in HK. That is, the pressure on AB is to the pressure on IIK, as base AB x height CD, to the base HK X height FH. Obs. From this proposition may be calculated, the pressure upon, and the strength required for, dams, cisterns, pipes, &c. Cor. 1. Hence, if the heights be equal, the pressures are as the bases. And, if both the heights and bases be equal, the pressures are equal in both; though their contents be ever so different. For, the reason that the wider vessel EK has no greater pressure at the bottom, is, because the oblique sides EH, GK, take off part of the weighit. And in the narrower vessel CB, the sides CA, IB, re-act against the pressure of the water, which is all alike at the same depth, and by this re-action the pressure is increased at the bottom, so as to besome the same every where. Cor. 2. The pressure against the base of any vessel, is the same as of a cylinder of an equal base and height, Čur: 3. Il there be a recurve tube ABF, fig. 3, in which are two different faids CD, EF; their heights in the two legs CD, EF, will be reciprocally as their specific gravities, when they are at rest. For, if the fluid EF be twice or thrice as ligli as CD; it must have twice or thrice the beight, to have an equal pressure, to counterbalance the other. Prop. 5.-If a body, of the same specific gravity of a fluid, be immersed in it, it will rest in any place of it. A body of greater density will sink, and one of a less density will swim. Let A, B, C, fig. 4, be three bodies ; whereof A is lighter bulk for bulk than the fluid ; B is cqual ; and C heavier. The body B, being of the same density, or equal in weight as so much of the fluid, it will press the Ouid under it just as much as if the space was filled with the luid. The pressure then will be the same all arouud it, as if the fluid was there, and consequently there is no force to put it out of its place. But, if the body be lighter, the pressure of it downwards will be less than before, and less than in other places at the same depth ; and consequently the lesser force will give way, and it will rise to the top. And, if the body be heavier, the pressure downwards will be greater than before; and the greater pressure will prevail and carry it to the bottom. Cor. 1.-Hence, if several bodies of different specific gravity be inimer. sed in a fluid, the heaviest will get the lowest. For the heaviest are impelled with a greater force and therefore will go fastest down. Cor. 2.-A borly, immersed in a Nuid, luses as much weight as an equal quantity of the fluid weighs; and the fluid gains it. For, if the body is of the same specific gravity as the timid, then it will lose all its weight. And, if it be lighter or heavier, there remains only the difference of the weights of the body and fluid to move the body. Cur. 3.- All bodies of equal magnitudes lose equal weights in the same Nuid. And bodies of different magnitudes lose weights propor. jonal to the magnitudes. Cor. 4.–The weights lost in different Puids, by immerging the same. body thercin, are as the specific gravities of the fluids. And bodies of equal weight, lose weights, in the same fluid, reciprocally as the specific gravities of the bodies. Cor 5.- The weight of a body swimming in a fluid is equal to the weiglit of as much of the fluid as the immersed part of the body takes up. For the pressure underneath the swimming body is just the same as so much of the immersed fluid ; and therefore the weights are the salllc. Cor. 6.Henco a body will sink deeper in a lighter Quid than ju a leavier. Cor. 7.-llence appears the reason why we dn not feel the whole weight of an immersed boily, till it be drawn quite out of the water SPECIFIC GRAVITIES. By the specific gravities of bodies, is meant the relative weights which equal bulks of different bodies have in regard to each other. Obs.-1. Thus a cubic foot of cork is not of egnal weight with a cnbic foot of water, or marble, or lead; but the water is four times heavier than the cork, the marble 11 times, and the lead 45 times; or, in other words, a cubic foot of lead would weigh as much as 45 of cork, &c. &c. 2. The terms absolnte gravity and specific gravity very frequently occur iu plıysics. The first is what we express in common life by the word weight, and signifies the whole of the power with which a body is carried to the earth. Every particle in every substance is heavy; that is, it has a tendency to fali toward the earth, or is attracied by the earth. Now, the greater the number of particles a substance has, the greater will be its momentum, and the more powerful will be its tendency toward the centre of the earth's motions. PROP. 10.--To find the specific grarity of solids or fluids. For a solid hearier than water.–Weigh the body separately, first out of water, and then suspended in water. And divide the weight cut of water by the difference of the weights, gives the specific gravity: reckoning the specific gravity of water ). For the dillisence of the weights is equal to the weight of as much water (by Cor. 2. Prop. 5); and the weights of equal magnitudes are as the specilic gravities; therefore, the difference of these weights, is to the weight of the body, as the specific gravity of water 1, to the specific gravity of the body. For a body lighier than water.-- Take a piece of any licary body, so big as, being tied to the light body, it may siok it in water. Weigh the heavy body in and out of water, and find the loss of weight. Also weiga the compound both in and out of water, and find also the loss of weight. Then divide the weight of the light body (out of water,) by the difference of these losses, gives the specific gravity; the specific gravity of water being 1. For the last loss is = weight of water equal in magnitude to the compound, And the first loss is = weight of water equal in magnitude to the beavy body, Whence the diff. losses is = weight of water equal in magnitude to the light body; and the weights of equal magnitudes being as the specific gravities; therefore the difference of the losses (or the weight of water equal to the light body) : weight of the light body :: specific gravity of Wa. ter 1 : specific gravity of the light body. For a fluid of any sort.-Take a piece of a body whose specific gravity you know; weigh it both in and out of the fluid ; take the difference of the weights, and multiply it by the specific gravity of the solid body, and divide the product by the weight of the body (out of water), for the specific gravity of the fluid. For the difference of the weights in and out of water, is the weight of so much of the fluid as equals the magnitude of the body. And the weight of equal magnitudes being as the specific gravities; therefore, weight of the solid : difference of the weights (or the weight of so much of the fluid) :: specific gravity of the solid : the specific gravity of the duid. |