. Applied calculus; principles and applications . epressure of water is h ft. in height and b ft. in breadth. Thepressure of the water varies as the depth; the intensity ata depth x is wx, w being the constant weight of a cubic unitof water. Required the total pressure on the face of thedam, and the location of the center of pressure. Let the area of pressure be divided into strips of width Axand length b, then wx -b is approximately the pressure onthe element of area — for wx is the intensity of pressure atthe top of the strip. The sum of a finite number of terms of the form wbx Axwould
. Applied calculus; principles and applications . epressure of water is h ft. in height and b ft. in breadth. Thepressure of the water varies as the depth; the intensity ata depth x is wx, w being the constant weight of a cubic unitof water. Required the total pressure on the face of thedam, and the location of the center of pressure. Let the area of pressure be divided into strips of width Axand length b, then wx -b is approximately the pressure onthe element of area — for wx is the intensity of pressure atthe top of the strip. The sum of a finite number of terms of the form wbx Axwould give a result for the total pressure less than the actualvalue; but the exact value is _. ,. ^^ , . 7 P J wbx^l wbh^ wh ,, .^. P= lim X wbx/!:ix = wb I xdx= —^^ =—^r-=-^bh. (1) 358 INTEGRAL CALCULUS The intensity of pressure is a uniformly varying force havingzero value at the surface of the water and value wh at thebottom. The center of pressure, being the point of applica-tion of the resultant pressure, is given by taking the moment ^=^A. of P about the surface line equal to the limit of the sum ofthe moments of the elementary pressures about that line: XP = I whx^ax = —5— = —5— ,Jo o Jo o £ whx^ dx i whxdx wb¥/S wbhy2 h- (2) In general, the center of pressure of a rectangle with aside at the surface is two-thirds the height of the rectanglebelow the surface. When the top of the area is hi below thesurface and the bottom is h below, the total pressure is tJhi ^ and M x^2 h hi x^dx 2hlj-hl3 hi - h/ (3) PRESSURE OF LIQUIDS 359 It may be noted that the second moment in the numeratoris the moment of inertia of the area, and the first moment inthe denominator is the statical moment. Note. — That P = l wbx dx in (1) is the reversal of a rate may be seen by considering the rate of change of thetotal pressure when the depth x is increased by Ax, for thenthe pressure P on the area is increased by AP= vox • h ^x,approximately, and AP/Ax= whx (nearly). Hence ,. AP dP
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