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Unformatted text preview: 422 CHAPTER 3. EFFECTS OF GRAVITY ON PRESSURE 3.68 Chapter 3, Problem 68 Problem: Determine the force on the gate AB if the upper layer of fluid has density ρ1 = ρ and the lower layer has density ρ2 = 2ρ. The gate is rectangular and has width 3H out of the page. ......... ......... ......... ......... . ...... ......... .......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ....... ......... .......... ......... ......... .. ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... Air ρ 2ρ ¡ ¡ • ............ .. .... ............ ............ ........ ............ ............ .. .. ............ ............ ............ . ............ ............ .......... ............ ............ ............ ............ ............ ............ ............ ............ ............ ........ .. ............ ............. .... ........... ............ ....... ............ ............ ........ Air ρ H ¡B H ¡ ¡ •A H • ........... ............. ... .. ............ ............ ..... . ............ ............ ............ ............ ............ .. .. ............ ............ .. . ............ ............ ............ ......... H ¡B H ρ ¡ •A H ............. ............. . ... ......... .............. ............. ............. ............. ............. ....... ............................................................................................ ........................................................................... .... ... . . .. . . . . . .................................................................................................................................................................................................... ..... ... ...................................................................................................................................................................................................... .................................................................................................................................................................... ..... ..... .. .......................................................................................................................................................................................... .......... .......... .......... .......... .......... .......... .......... .......... .......... ................................................. ................................................................................................................................................................................... ................................................................ ....................................................................................................................................................................................... ............................................................................................................................................................ ............................................................................................................................................................ Original Problem ¡ Air Problem A = ¡ ¡ • H ¡B H ¡ •A H ............. ............. ............. ............. ............. ............. ............. ............. ........ . .. ..................................................................................................................................................................................................... ....................................................................................................................................................................................................... ........ ....................................................................................................................................................................................................... .......................................................................................... ...... .............................................................................. . ....................................................................................................................................................................................................... + Problem B Solution: To begin, use superposition as illustrated in the figure. For Problem A, the arc’s projection on a vertical plane is a rectangle of height H and width 3H . So, the centroid is located at z = 3 H and A = 3H 2 . Thus, 2 9 FxA = ρg zA = ρgH 3 2 1 9 and FzA = ρgH 2 (3H ) + ρgH 2 (3H ) = ρgH 3 2 2 Similarly, for Problem B, the centroid is located at z = 1 H and A = 3H 2 , wherefore 2 3 FxB = ρg zA = ρgH 3 2 and FzB = ρg 3 12 H (3H ) = ρgH 3 2 2 So, the solution to the original problem is 9 3 Fx = FxA + FxB = ρgH 3 + ρgH 3 = 6ρgH 3 2 2 9 3 Fz = FzA + FzB = ρgH 3 + ρgH 3 = 6ρgH 3 2 2 Therefore, the force vector is F = 6ρgH 3 (i + k) ...
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