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 21⁄2 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 córks 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 weight of a cubic foot of water 1000 oz. square feet, and the (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. CHAPTER II. On the Pressure of a Fluid acted on by Gravity. 29. IN the preceding chapter we considered the consequences that result from the peculiar property, essential to all fluids, of transmitting equally in all directions the pressures applied to their surfaces. We have now to consider the effects produced by the action of gravity upon the substance of a fluid. 30. The student must mark carefully the distinction between force applied to a surface and force applied to each of the particles composing a body. As an example of these distinct forces consider the case of a book resting on a table. Force is applied to the surface of the book by the table, and thus is counterbalanced the force of gravity which acts upon each particle of which the book is composed. 31. All fluids are subject to the action of gravity in the same way as solid bodies. Each particle of a fluid has a tendency to fall to the surface of the earth, and in a mass of fluid at rest there is a particular point, called the centre of gravity, at which the resultant of all the forces exercised by the attraction of the Earth on the particles composing the fluid may be supposed to act. 32. The term density is applied to fluids, as it is to solid bodies, to denote the degree of closeness with which the parti cles are packed. When we speak of a fluid of uniform density, we mean that if from any part of the body of fluid a portion be taken, and if from any other part of the body of fluid a portion like in form and equal in volume to the former portion be taken, the weights of the two portions will be equal. 33. If a vessel be filled with a heavy fluid of uniform density the pressure at every point in the interior of the fluid will not be the same, because the pressure which results from the action of gravity will vary in magnitude according to the position of the point in the containing vessel. Consider a closed surface of small dimensions containing the point A, and suppose the fluid outside the closed surface to become solid. The fluid within the closed surface will exercise pressure against the surface at every point, and these pressures will be unequal, because the fluid is acted on by gravity. But we may conceive that, if the quantity of fluid within the surface be very small, the difference between the pressures at different points of the surface will be very small, and when the surface is indefinitely diminished the pressures exercised by the fluid at each point of the surface may be regarded as equal, and the weight of the fluid may be neglected. Thus we can consider it as the case of a weightless fluid and apply the conclusions of Art. 28. Hence all the planes of equal area which can be drawn, passing through the point A and not extending beyond the small surface, may be considered to be subject to equal pressures. |