p0364 - 3.64. CHAPTER 3, PROBLEM 64 417 3.64 Chapter 3,...

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Unformatted text preview: 3.64. CHAPTER 3, PROBLEM 64 417 3.64 Chapter 3, Problem 64 Problem: A swimming pool has a set of n steps at one end. Each step has a horizontal and vertical length of h/n, where h is the total depth. The width of the steps out of the page is 6h. Compute the hydrostatic force on the set of steps. Ar x h/n z h ρ h/n ...... . . . .. . .. . . . . . .. ........................ . ..... . ...... ....................... . . ... . .............................................................................................................................................................. . . . . .. . .. . . . . . ................................................................................................................................................................................................................................... . .. ... . . ............................................................................................................................................................................... .... ......................... .... .... .... ................................. ......... ........................ .. .. . . . .... .... .... .... ..... . . . .. .. . . . . ............................................................................................................................................................. .......... ................................................... ...... .................................................................................................... .. .. ... . .............. ..................................................................................................................................................................................... .. .. ......... .... . ............... .................. ............................................................................................................................................................................................................................................................................................................................ .. . .. .. . .. .. . . .. . . . . . . . . . . ......................................... .. . ...... . . ................................................................. ................................................................................................................................ ..... ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... . . . .... . . .......................... ....................................... . . ... .. ... . ... ... .. ... ... ... . . ... ... ... .. ... ... .. ... .. ... . ... ... ................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................................................................................................. ........ ... . . .. .... ..... ............................................. ...... ......................................................................................................................................................... Solution: The projection of the steps onto a vertical plane is a rectangle of height h and width 6h. So, the centroid and area are 1 A = 6h2 z = h, 2 The horizontal component of the force on the steps is 1 h 6h2 = 3ρgh3 Fx = ρg zA = ρg 2 Turning to the vertical component, the cross-sectional area of the fluid above the first step in the xz plane is h2 /n2 The area above the second step is 2h2 /n2 . Generalizing, the area above the ith step is ih2 /n2 . Thus, the total cross-sectional area for the complete set of steps is h2 n h2 n(n + 1) 1 12 Acs = 2 i= 2 = 1+ h n =1 n 2 2 n So the volume of the fluid above the steps is 13 h n V = 6hAcs = 3 1 + Therefore, the vertical force on the steps is Fz = ρgV = 3ρg 1 + 13 h n Thus, in vector form, the force on the steps is F = 3ρgh3 i + 1 + 1 k n ...
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