EXAMPLES.-II. (1) IF the pressure at a depth of 32 feet be 15 lbs. to the square inch, what will the pressure be at a depth of 42 feet 8 inches? (2) If the pressure at a depth of 8 feet be 14 lbs. to the square inch, what will be the pressure at a depth of 20 ft. 6 in.? (3) In two uniform fluids the pressures are the same at the depths of 3 and 4 inches respectively compare the pressures at the depths of 7 and 8 inches respectively. : (4) In two uniform fluids the pressures are the same at the depths of 2 and 3 inches respectively compare the pressures at the depths of 9 and 12 inches respectively. (5) Find the height of a column standing in water 30 feet deep, when the pressure at the bottom is to the pressure at the top as 3 to 2. (6) If the pressure of a uniform fluid, not exposed to external pressure, be 15 lbs. to the square inch at a depth of 15 feet, what will be the pressure at a depth of 12 feet? (7) If the pressure of a uniform fluid, not exposed to external pressure, be 3 lbs. to the square inch at a depth of 4 feet, what will be the pressure on a square inch at a depth of 12 feet? (8) What is the pressure on the horizontal bottom of a vessel filled with water to the depth of 2 feet, the area of the base being 20 square feet, and the weight of a cubic foot of water 1000 oz. ? (9) A cubic foot of mercury weighs 13600 oz. Find the pressure on the horizontal base of a vessel containing mercury, the area of the base being 8 square inches, and the depth of the mercury 3 inches. (10) What is the pressure on the horizontal base of a vessel filled with water to the depth of 15 feet, the area of the base being 24 square feet, and the weight of a cubic foot of water 1000 oz. ? (11) A cistern shaped like an equilateral triangle of which one side is 6 feet is filled with water to the depth of two feet : find the pressure on the base, the weight of a cubic foot of water being 1000 oz. (12) The spout of a teapot springs from the middle point of one side, and its upper extremity is on a level with the lid. If the spout be broken off half-way, how high can the teapot be filled? (13) When bottles that have been sunk in deep water have been brought up, their corks have been found driven in. How do you explain this? (14) If a pipe, whose height above the bottom of a vessel is 112 feet, be inserted vertically in the vessel, and the whole be filled with water, find the pressure in tons on the bottom of the vessel, the area of the bottom being 4 square feet, and the weight of a cubic foot of water 1000 oz. (15) A hole, a square inch in area, is bored in the flat cover of a vessel full of water, and a smooth piston weighing 7 lbs. 13 oz. is fitted into it; a vertical tube is then fitted into another hole in the cover, and water is poured into it; find how high the water must be made to ascend in it in order that the piston may be driven out, a cubic foot of water weighing 1000 oz. CHAPTER III. On Specific Gravity. 40. SOME substances are from the nature of their composition more weighty than others. We call gold a heavier metal than lead, because we know by experience that a given volume of gold is more weighty than an equal volume of lead. 41. We make a distinction between the terms weight and weightiness. We speak of the weight of a particular lump of gold or iron. We speak of the weightiness of gold or iron, not referring to any particular lump, but to the special characteristics of the metals in question. Further we say that gold is heavier than iron, having no particular lump of the metals in view, but expressing our notions of the degree of weightiness that is peculiar to either substance. This degree of weightiness is known by the name Specific Gravity. DEF. The Specific Gravity of a substance is the degree of weightiness of that substance. 42. If of two substances, one of which is twice as weighty as the other, we take two lumps of equal volume, the weight of one lump is evidently twice that of the other: and, generally, if one substance be S times as weighty as the other, the weight of any volume of the first is S times the weight of an equal volume of the other. Now by a substance, the measure of the specific gravity of which is S, we mean a substance which is S times as weighty as the standard by which specific gravities are estimated. Therefore any volume of this substance will weigh S times as much as the equal volume of the standard. 43. The requisites for a Standard are that it should be definite and uniform, and these requisites are possessed by Pure Distilled Water at a certain temperature. This substance is therefore taken as the standard for estimating the specific gravities of solid bodies and inelastic fluids. 44. When we say that the specific gravity of gold is 19, we mean that the specific gravity of gold is 19 times that of Pure Distilled Water, and therefore a given volume of gold weighs 19 times as much as the same volume of distilled water. 45. To measure the Weight of a body we must have a unit of weight, and to measure the Volume of a body we must have a unit of volume. These units we may select in any way that may suit our purpose, and we connect them with the unit of specific gravity by the following convention : The unit of specific gravity is the specific gravity of that substance of which a unit of volume contains a unit of weight. 46. To find the numerical relation existing between the measure of the specific gravity of a substance and the measures of the weight and volume of any given quantity of the substance. Let W represent the measure of the weight of a substance, that is the number of times it contains the unit of weight. Also, let represent the measure of the volume of the substance, that is the number of times it contains the unit of volume. And let S represent the measure of the specific gravity of the substance, that is the number of times it contains the unit of specific gravity. Then one unit of volume of this substance will weigh S times as much as a unit of volume of the standard substance, (Art. 42) that is, its weight is S times the unit of weight. Therefore the weight of V units of volume is VS times the unit of weight; therefore the measure of the weight of V units of volume of the substance is VS; but this measure we have denoted by W; .. W= VS. 47. The equation W= VS gives us merely the relation between three numbers, and two of these must be given in order that we may determine the third. When we have found it we know the measure of the weight or volume or specific gravity, as the case may be, and we must have the unit of weight, or of volume, or of specific gravity also given to enable us to determine the weight or volume or specific gravity of a particular substance. So that we may put it thus: W specific gravity=times (unit of specific gravity). 48. A cubic foot of pure distilled water at a temperature of 62° Fahrenheit weighs about 998 oz., and for rough calculations it is assumed that the weight of a cubic foot of water is 1000 ounces. Then if we take 1 cubic foot as our unit of volume and pure distilled water as our standard of specific gravity, the unit of weight will be 1000 ounces. Or if we prefer to take 1lb. avoirdupois as our unit of weight and pure distilled water as our standard of specific 16 gravity, the unit of volume will be of a cubic foot, that is 1000 '016 cub. ft. |