p0364

# p0364 -

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online