Explanation Given information: The arbitrary depth is d = y sin θ Calculation: Calculate the moment of inertia about x axis as follows: I x = ∫ y 2 d A Calculate the moment of inertia about x axis as follows: I y = ∫ x y d A Find the pressure ( p ) at an arbitrary depth as shown below: p = γ ( y sin θ ) Find the hydrostatic force ( d F ) exerted on infinitesimal area as follows: d F = ( γ y sin θ ) d A Find the force P on the system as follows. P = ∫ d F Substitute ( γ y sin θ ) d A for d F . P = ∫ ( γ y sin θ ) d A = γ y ¯ sin θ ∫ d A = γ A y ¯ sin θ Apply the Equivalence of the force and couple and the system of infinitesimal hydrostatic force requires as follows. Summation of moment about x axis. ∑ M x = 0 − y ¯ P − M x ′ = ∫ ( − y d F ) Substitute ( γ y sin θ ) d A for dF . y ¯ P − M x ′ = − ∫ y ( γ y sin θ ) d A = − γ sin θ ∫ y 2 d A = − ( γ sin θ ) I x Substitute γ A y ¯ sin θ for P and I x for y 2 d A

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 9.2, Problem 9.65P

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Show that the system of hydrostatic force acting on P is perpendicular to the area and magnitude and couple force of the system.

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Chapter 9.2, Problem 9.64P

Chapter 9.2, Problem 9.66P

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The uniform lamina of density 2.2 kg/m2. for a=2.2 m,b=4.8 m,c=8 m,d=8.9 m 1-The x-co-ordinate of total area centroid is: (a.8.27 - b.13.17 - c.7.05 - d.8.88 - e.6.13) 2- The y-co-ordinate of total area centroid is: (a.1.96 - b.2.68 - c.3.04 - d.1.61 - e.3.57) 3- The total mass is : ( a. 163.00 - b. 326.00 - c. 244.50 - d. 81.50 - e. 40.75 )

The uniform lamina of density 2.2 kg/m2. for a=2.2 m,b=4.8 m,c=8 m,d=8.9 m, 1- x-co-ordinate of shape (1) is: (a. 1.78 - b. 2.23 - c. 4.45 - d.8.90 - e. 2.97 ) 2- y-co-ordinate of centroid of shape (2) is: (a. 3.80 - b. 4.52 - c.2.33 - d. 5.28 - e. 4.60 ) 3- The total area is: (a.81.50 - b. 150.78 - c. 52.98 - d. 69.27 - e. 110.03) 4- The total mass is: ( a. 163.00 - b. 326.00 - c. 244.50 - d. 81.50 - e. 40.75 )

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Vector Mechanics for Engineers: Statics and Dynamics

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Show that the system of hydrostatic force acting on P is perpendicular to the area and magnitude and couple force of the system.

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Show that the system of hydrostatic force acting on P is perpendicular to the area and magnitude and couple force of the system.

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