Explanation 1) Concept: The pressure in any direction at a depth d in a fluid with mass density ρ is given by P = ρ g d = δ d The force in any direction with pressure P and area A is given by F = P A 2) Given: A trapezoid 100 f t wide at the top and 50 f t wide at the bottom and slant height is 70 f t The aquarium in witch water is 1 m deep 3) Calculation: Choose vertical x-axis as with origin at the surface of water and directed in downward direction Depth of the water is h f t that is, height of the dam is h f t From figure 1 ( b ) cos ⁡ 30 0 = h 70 2 - 25 2 3 2 = h 4900 - 625 2 h = 3 4275 h = 15 3 19 2 Divide interval [ 0 , h ] into subinterval of equal length x i and choose x i * ∈ [ x i - 1 , x i ] The i t h horizontal strip of a trapezoid is approximated by a rectangle with height ∆ x and width w i . From similar triangle in Figure 1(b) a h - x i * = 25 h a = 25 h - x i * h So w i = 2 25 + a = = 2 25 + 25 h - x i * h = 50 + 2 25 h h - 50 x i * h w i = 100 - 50 x i * h If A i is the area of the i t h strip, then A i ≈ w i ∆ x = 100 - 50 x i * h ∆ x Now , ∆ x From Figure 1 (b) C o s 30 0 = ∆ x z z = ∆ x C o s 30 0 z = ∆ x sec ⁡ 30 0 z = 2 ∆ x 3 By using the concept, pressure on a strip is P = δ x i * The hydrostatic force on a strip is F = P A = δ x i * 100 - 50 x i * h 2 ∆ x 3 Total hydrostatic force ≈ ∑ i = 1 n δ x 100 - 50 x h 2 ∆ x 3 Now expressing force as a Riemann sum as F = lim n → ∞ ⁡ ∑ i = 1 n δ x 100 -

8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 8.3, Problem 16E

To determine

To find:

The hydrostatic force on the dam when it is full of water

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Chapter 8.3, Problem 15E

Chapter 8.3, Problem 17E

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