393. Exam. 2. To find the altitude, when the state of the barometers and thermometers is as follows, viz. ON THE RESISTANCE OF FLUIDS, WITH THEIR FORCES AND ACTIONS ON BODIES. PROPOSITION LXXVII. 394. If any Body Move through a Fluid at Rest, or the Fluid Move against the Body at Rest; the Force or Resistance of the Fluid against the Body, will be as the Square of the Velocity and the Density of the Fluid. That is, R`¤ dv2. FOR, the force or resistance is as the quantity of matter or particles struck, and the velocity with which they are struck. But the quantity or number of particles, struck in any time, are as the velocity and the density of the fluid. Therefore the resistance, or force of the fluid, is as the density and square of the velocity. 395. Corol. 1. The resistance to any plane, is also more ór less, as the plane is greater or less; and therefore the resistance on any plane, is as the area of the plane a, the density of the medium, and the square of the velocity. That is, R ∞ adv2. 396. Corol. 2. If the motion be not perpendicular, but oblique to the plane, or to the face of the body; then the resistance, in the direction of motion, will be diminished in the triplicate ratio of radius to the sine of the angle of inclination of the plane to the direction of the motion, or as the cube of radius to the cube of the sine of that angle. So that R adv's3, putting 1 = radius, and s sine of the angle of inclination CAB. For, if AB be the plane, AC the direction of motion, and BC perpendicular to AC; then no more particles meet the plane than what meet the perpendicular BC, and therefore their number is diminished as AB to BC or as 1 to s. But the force of each par ticle, striking the plane obliquely in the direction CA, is also diminished as AB to BC, or as 1 to s; therefore the resistance, which is perpendicular to the face of the plane by art. 52, is as 12 to 2. But again, this resistance in the direction perpendicular to the face of the plane, is to that in the direction AC, by art. 51, as AB to BC, or as I to s. Consequently, on all these accounts, the resistance to the plane when moving perpendicular to its face, is to that when moving obliquely, as 13 to s3, or 1 to s3. That is, the resistance in the direction of the motion, is diminished as I to s3, or in the triplicate ratio of radius to the sine of inclination. PROPOSITION LXXVIII. 397. The Real Resistance to a Plane, by a Fluid acting in a Direction perpendicular to its Face, is equal to the Weight of a Column of the Fluid, whose Base is the Plane, and Altitude equal to that which is due to the Velocity of the Motion, or through which a Heavy Body must fall to acquire that Velocity. THE resistance to the plane moving through a fluid, is the same as the force of the fluid in motion with the same velocity, on the plane at rest. But the force of the fluid in motion, is equal to the weight or pressure which generates that motion; and this is equal to the weight or pressure of a column of the fluid, whose base is the area of the plane, and its altitude that which is due to the velocity. 398. Corol. 1. If a denote the area of the plane, v the velocity, n the density or specific gravity of the fluid, and g= 16 feet, or 193 inches. Then, the altitude due to the velocity v being therefore a x n x 22 be the whole resistance, or motive force R. anv2 will 4g 4g 399. Corol. 2. If the direction of motion be not perpendicular to the face of the plane, but oblique to it, in any angle, whose sine is s. Then the resistance to the plane will be 4g 400. Corol. 3. Also, if w denote the weight of the body, whose plane face a is resisted by the absolute force R; then 401. Corol. 4. And if the body be a cylinder, whose face VOL. II. S or or end is a, and radius r, moving in the direction of its axis, because then s = 1, and a = pr2, where p = 3.1416; then pnq2x2 402. Corol. 5. This is the value of the resistance when the end of the cylinder is a plane perpendicular to its axis, or to the direction of motion. But were its face an elliptic section, or a conical surface, or any other figure everywhere equally inclined to the axis, or direction of motion, the sine or inclination being s: then, the number of particles of the fluid striking the face being still the same, but the force of each, opposed to the direction of motion, diminished in the duplicate ratio of radius to the sine of inclination, the resistpnr222 ing force R would be 4g PROPOSITION LXXIX. 403. The Resistance to a Sphere moving through a Fluid, is but Half the Resistance to its Great Circle, or to the End of a Cylinder of the same Diameter, moving with an Equal Velocity. LET AFEB be half the sphere, moving in the direction CEG. Describe the paraboloid A!EKB on the same base. Let any particle of the medium meet the semicircle in F, to which draw the tangent FG, the radius FC, and the ordinate FIH. Then the force of any particle on the surface at F, is to its force on the base at H, as the square of the sine of the angle G, or its H equal the angle FCH, to the square of radius, that is, as HF to cr. Therefore the force of all the particles, or the whole fluid, on the whole surface, is to its force on the circle of the base, as all the HF to as many times cr2. But Cr2 is CA2 = AC, CB, and HF2 =AH. HB by the nature of the circle: also, AH. HB: AC. CB: HI: CE by the nature of the parabola; consequently the force on the spherical surface, is to the force on its circular base, as all the HI's to as many CE's, that is, as the content of the paraboloid to the content of its circumscribed cylinder, namely, as 1 to 2. 404. Corol. Hence, the resistance to the sphere is R = -, being the half of that of a cylinder of the same diameter: diameter. For example, a 9lb iron ball, whose diameter is 4 inches, when moving through the air with a velocity of 1600 feet per second, would meet a resistance which is equal to a weight of 1323lb, over and above the pressure of the atmosphere, for want of the counterpoise behind the ball. PRACTICAL EXERCISES IN MENSURATION. QUEST. 1. WHAT difference is there between a floor 28 feet long by 20 broad, and two others, each of half the dimensions; and what do all three come to at 45s. per square, or 100 square feet? Ans. dif. 280 sq. feet. Amount 18 guineas. QUEST. 2. An elm plank is 14 feet 3 inches long, and I would have just a square yard slit off it; at what distance from the edge must the line be struck? Ans. 7 inches. QUEST. 3. A cieling contains 114 yards 6 feet of plastering, and the room 28 feet broad; what is the length of it? Ans. 369 feet, QUEST. 4. A common joist is 7 inches deep, and 2 thick; but wanting a scantling just as big again, that shall be 3 inches thick; what will the other dimension be? Ans. 114 inches. QUEST. 5. A wooden cistern cost me 3s. 2d. painting within, at 6d. per yard; the length of it was 102 inches, and the depth 21 inches; what was the width? Ans. 274 inches. QUEST. 6. If my court-yard be 47 feet 9 inches square, and I have laid a foot-path with Purbeck stone, of 4 feet wide, along one side of it; what will paving the rest with flints come to, at 6d. per square yard? Ans. 5. 16s. Oid. QUEST. 7. A ladder, 263 feet long, may be so planted, that it shall reach a window 22 feet from the ground on one side of the street; and, by only turning it over, without moving the foot out of its place, it will do the same by a window 14 feet high on the other side; what is the breadth of the street? Ans. 37 feet 9 inches. QUEST. 8. The paving of a triangular court, at 18d. per foot, came to 100.; the longest of the three sides was 8 feet; required the sum of the other two equal sides? QUEST. 9. There are two columns in the ruins of Perse polis left standing upright: the one is 64 feet above the plain, and the other 50: in a straight line between these stands an ancient small statue, the head of which is 97 feet from the summit of the higher, and 86 feet from the top of the lower column, the base of which measures just 76 feet to the centre of the figure's base. Required the distance between the tops of the two columns? Ans. 157 feet nearly. QUEST. 10. The perambulator, or surveying wheel, is so contrived, as to turn just twice in the length of 1 pole, or 16 feet; required the diameter ? Ans. 2.626 feet. QUEST. 11. In turning a one-horse chaise within a ring of a certain diameter, it was observed that the outer wheel made two turns, while the inner made but one: the wheels were both 4 feet high; and supposing them fixed at the distance of 5 feet asunder on the axletree, what was the circumference of the track described by the outer wheel? Ans. 62.83 feet. QUEST. 12. What is the side of that equilateral triangle, whose area cost as much paving at 8d. a foot, as the pallisading the three sides did at a guinea a yard? Ans. 72.746 feet: QUEST. 13. In the trapezium ABCD, are given, AB = 13, BC = 313, CD = 24, and DA = 18, also в a right angle; required the area? Ans. 410 122. QUEST. 14. A roof which is 24 feet 8 inches by 14 feet 6 inches, is to be covered with lead at 8lb. per square foot: what will it come to at 18s per cwt. ? Ans. 221. 19s. 10d. QUEST. 15. Having a rectangular marble slab, 58 inches by 27, I would have a square foot cut off parallel to the shorter edge; I would then have the like quantity, divided from the remainder parallel to the longer side; and this alternately repeated, till there shall not be the quantity of a foot left: what will be the dimensions of the remaining piece? Ans. 20.7 inches by 6.086, QUEST. 16. Given two sides of an obtuse-angled triangle, which are 20 and 40 poles; required the third side, that the triangle may contain just an acre of land? Ans. 58-876 or 23.099. QUEST. 17. The end wall of a house is 24 feet 6 inches in breadth, and 40 feet to the eaves; of which is 2 bricks thick, more is 1 brick thick, and the rest I brick thick. Now the triangular gable rises 38 courses, of bricks, 4 of which usually make a foot in depth, and this is but 4 inches, |