CHAPTER XVI. HYDRAULICS. IN Chapter VIII. we stated that a body which resists a change of form when under the action of a distorting stress is termed a solid body, and if the body returns to its original form after the removal of the stress, the body is said to be an elastic solid (e.g. wrought iron, steel, etc., under small stresses); but if it retains the distorted form it assumed when under stress, it is said to be a plastic solid (e.g. putty, clay, etc.). If, on the other hand, the body does not resist a change of form when under the action of a distorting stress, it is said to be a fluid body; if the change of form takes place immediately it comes under the action of the distorting stress, the body is said to be a perfect fluid (e.g. alcohol, ether, water, etc., are very nearly so); if, however, the change of form takes place gradually after it has come under the action of the distorting stress, the body is said to be a viscous fluid (e.g. tar, treacle, etc.). The viscosity is measured by the rate of change of form under a given distorting stress. In nearly all that follows in this chapter, we shall assume that water is a perfect fluid; in some instances, however, we shall have to carefully consider some points depending upon its viscosity. Weight of Water. The weight of water for all practical purposes is taken at 625 lbs. per cubic foot, or o'036 lb. per cubic inch. It varies slightly with the temperature, as shown in the table on the following page, which is for pure distilled water. The volume corresponding to any temperature can be found very closely by the following empirical formula : = Volume at absolute temperature T, taking) T2 + 250,000 the volume at 39'2° Fahr. or 500° absolute as I 1000T Pressure due to a Given Head.-If a cube of water of I foot side be imagined to be composed of a series of vertical columns, each of I square inch section, and I foot high, each =0'434 lb. Hence a column of water 1 foot high produces a pressure of o'434 lb. per square inch. will weigh 62'5 57 2 62 417 62 425 62 409 6200 61 20 60*14 59'84 ... The height of the column of water above the point in question is termed the head. Let the head of water in feet above any surface; = = the pressure in pounds per square inch on that surface; = the weight of a column of water 1 foot high and I square inch section; = 0'434 lb. Thus a head of 2.31 feet of water produces a pressure of 1 lb. per square inch. Taking the pressure due to the atmosphere as 14'7 lbs. per square inch, we have the head of water corresponding to the pressure of the atmosphere— 14'7 X 2'31 = 34 feet (nearly) This pressure is the same in all directions, and is entirely independent of the shape of the containing vessel. Thus in Fig. 518 The pressure over any unit area of surface at a = p = 0'434ha b = p1 = 0'434h, and so on. The horizontal width of the triangular diagram at the side shows the pressure per square inch at any depth below the surface. Thus, if the height of the triangle be made to a scale of 1 inch to the foot, and the width of the base 0'434h, the width of the triangle measured in inches will give the pressure in pounds per square inch at any point, at the same depth below the surface. FIG. 518. Compressibility of Water.The popular notion that water is incompressible is erroneous; the alteration of volume under such pressures as are usually used is, however, very small. Experiments show that the alteration. in volume is proportional to the pressure, hence the relation between the change of volume when under pressure may be expressed in the same form as we used for Young's modulus on p. 320. Let the diminution of volume under any given pressure V = the original volume (corresponding to / on p. 320); = the pressure in pounds per square inch. Then = or K = pV ข K = from 320,000 to 300,000 lbs. per square inch. Thus water is reduced in bulk or increased in density by I per cent. when under a pressure of 3000 lbs. per square inch. This is quite apart from the stretch of the containing vessel. Total Pressure on an Immersed Surface. If, for any purpose, we require the total normal pressure acting on an immersed surface, we must find the mean pressure acting on the surface, and multiply it by the area of the surface. We shall show that the mean pressure acting on a surface is the pressure due to the head of water above the centre of gravity of the surface. Let Fig. 519 represent an immersed surface. Let it be divided up into a large number of horizontal strips of length But the sum of all the areas a, a, etc., make up the whole area of the surface A, and by the principle of the centre of gravity (p. 58) we have where H, is the depth of the centre of gravity of the immersed surface from the surface of the water, or Thus the total pressure in pounds on the immersed surface is the area of the surface in square units × the pressure in pounds per square unit due to the head of water above the centre of gravity of the surface. Centre of Pressure. The centre of pressure of a plane immersed surface is the point in the surface through which the resultant of all the pressure on the surface acts. It can be found thus = Let H. the head of water above the centre of pressure; the head of water above the centre of gravity of Ho = the surface; = the angle the immersed surface makes with the surface of the water; I1 = the second moment, or moment of inertia of the surface about a line lying on the surface of the water and passing through o; I = the second moment of the surface about a line parallel to the above-mentioned line, and passing through the centre of gravity of the surface; R1 = the perpendicular distance between the two axes; = the square of the radius of gyration of the surface about a horizontal axis passing through the c. of g. of the surface; a1, a2, etc. = small areas at depths h1, h, etc., respectively below the surface and at distances x1, x2, etc. On p. 76 we have shown that the quantity in brackets on the left-hand side of the equation is the second moment, or moment of inertia, of the surface about an axis on the surface of the water passing through O. Then we have The lateral position of the centre of pressure is found by obtaining the c. of g. of the modulus surface, which is obtained in the manner described in the next paragraph. The depth of the centre of pressure from the surface of the water is given for a few cases in the following table : |