505. In the above table, the specific gravity of rain water is represented by 1000; and, because a cubic foot of rain water weighs 1000 ounces avoirdupois, or 624lbs. the numbers against each substance represent the weight of a cubic foot of that substance in avoirdupois ounces. The scales made use of to weigh bodies, for the purpose of finding their specific gravities, are called the Hydrostatic Balance. PROPOSITION CXXIV. 506. To find the magnitude of a body, its weight being given: say, As the specific gravity of the body. Is to its weight in avoirdupois ounces; To the solid content of the body in feet, or inches. Ex. 1. If a piece of dry oak, whose specific gravity is 925, weigh 1720 ounces; what is its magnitude? -As 925: 1720: : 1 (foot): -1.8594 cubic feet. Er. 2. Find the content of an irregular block of common stone, which weighs 112 pounds? As 2520 (see the table): 112 × 16 :: 1728: 1228.8 cubic inches. PROPOSITION CXXV. 50%. To find the weight of a body, its magnitude being given. As one cubic foot, or 1728 inches, To its weight in avoirdupois ounces. E. 1. If the magnitude of a piece of oak be 1.8594 cubic feet; what is its weight? As 1: 1.8594: : 925: 1720 ounces, or 107 pounds. Ex, 2. If the magnitude of a piece of bar iron be 1.35 cubic feet; what is its weight? As 1: 1.35 :: 7788: 10513.8oz. or 657.1125 pounds. MISCELLANEOUS EXAMPLES. Ex. 1. The diameter of a globe of oak is 1 foot; how deep will it sink in common water? Answ. 9.9867 inches. Ex. 2. If a cube of wood floating in common water have 3 inches of its height dry above the water, and 48 inches dry when it floats in sea water; required the dimensions of the cube, and what kind of wood it is made of? Answ. The wood is oak, and each side of the cube is 40 inches. Ex. 3. A cube of fir, each side of which is one foot, sinks to the depth of 3 inches in common water; what is its weight? Answ. 250oz. or 15 lbs. Ex. 4. An irregular fragment of glass in the scale weighs 171 grains, and another of magnet 102 grains; but in water the first fetches up only 120 grains, and the other 79; what then is their specific gravities? Answ. Glass to magnet as 3933 to 5202. Ex. 5. A cubic inch of common glass weighs 1.4921oz. troy, the same quantity of sea water,59542oz. and of brandy, ,5368; now a seaman has a gallon of brandy in a glass bottle that, out of water, weighs 3.84lbs. and to conceal it from the officers of the customs throws it overboard: required, if it will sink what force will buoy it up again? Answ. 14.1496 oz. Ex. 6. Suppose, by measurement, it be found that a man of war with its ordnance, rigging, and appointments, sinks so deep as to displace 50000 cubic feet of fresh water; what is the whole weight of the vessel ? Answ. 1395.1 tons. Ex. 7. To find the thickness FH (fig. 228.) of a hollow globe FDME, made of any given substance, the specific gravity of which is known; so that it may swim when immersed wholly or in part in a homogenious fluid whose specific gravity is known. Put the diameter FM-D, the diameter HN=d, LM-x and 3.14159=p; then pD' content of the sphere FDE, and pd is the content of the sphere or concavity HIK; and their difference=1px(D3-d3), is the content of the whole spherical shell. Also 3 pDr_pa is the content of the segment DME, or of 2 3 the Buid displaced; therefore, if m and n denote the specific gravities of the globe and fluid respectively, by art. 480, we shall have mp× (D3—d3)=' 2 find d= 2nx3 3nDx2 m m D-d +D3), and is the thickness of the shell, the concavity being a vacuum, or entirely void of air. If we suppose the shell to swim in the fluid when wholly immersed, then Dr, and d=Dx3v(1- n Let the sphere be of copper, and its diameter 10 feet, and let the fluid medium be air; the specific gravity of copper(m) being 7788, and that of air (n) 1.2 we have d=Dx n (1. :10×3√/(1- )=9.9995, and 1.2 7788 D-d part of a foot, or the 0.003 part of an inch, is the thickness of the copper, when it swims in air. Ex. 8. If the inner axis of a globe of copper, exhausted of air, be 100 feet; of what thickness must the shell be that it may just float in air? (the specific gravity of the copper being 9000.) Answ.,02688 of an inch thick. Ex. 9. What must be the thickness of a hollow cube made of 4 solid inches of copper, that it may just sink to the depth of one inch in water? Answ.,0185 of an inch thick. Ex. 10. Suppose a hollow sphere of copper, one foot in of its diameter, when put into common water, to sink till superfices be immersed; what is the thickness of the copper? On the resistance of fluids. 508. When bodies move in fluid mediums, such as air, water, &c. their motions are retarded; and the retarding force evidently varies with the density of the fluid: thus water will impede the motion of bodies moving in it more than air. The force with which fluids retard the motions of bodies moving in them, is called "the resistance of fluids." The resistance which a body meets with, when moving in a fluid, depends on the velocity with which it moves, its form and magnitude, and likewise on the tenacity and inertia of the fluid. Fluids retard the motions of bodies moving in them by their inertia, by their tenacity, and by the friction of the particles of the fluid against the moving body in perfect fluids, the two latter causes of resistance are inconsiderable; but the first takes place equally the same in perfect as in imperfect fluids. Care must be taken to distinguish between resistance and retardation; resistance is the quantity of motion lost; retárdation is the quantity of velocity lost: hence the retardations are as the resistances applied to the quantities of matter, and in the same body they are equal. PROPOSITION CXXVI. 509. If a plane surface move in a direction perpendicular to its plane, in a fluid whose particles move freely and disturb not each other's motion; the resisting force varies, within certain limits of the velocity, as v2, or as the square of the velocity. |