The water side of the wall of a 60-m-long dam is a quarter-circle with a radius of 7 m. Determine the hydrostatic force on the dam and its line of action when the dam is filled to the rim. Take the density of water to be 1000 kg/m3.

Answer:

[tex]26852726.19\ \text{N}[/tex]

[tex]57.52^{\circ}[/tex]

Explanation:

r = Radius of circle = 7 m

w = Width of dam = 60 m

h = Height of the dam will be half the radius = [tex]\dfrac{r}{2}[/tex]

A = Area = [tex]rw[/tex]

V = Volume = [tex]w\dfrac{\pi r^2}{4}[/tex]

Horizontal force is given by

[tex]F_x=\rho ghA\\\Rightarrow F_x=1000\times 9.81\times \dfrac{7}{2}\times  7\times 60\\\Rightarrow F_x=14420700\ \text{N}[/tex]

Vertical force is given by

[tex]F_y=\rho gV\\\Rightarrow F_y=1000\times 9.81\times 60\times \dfrac{\pi 7^2}{4}\\\Rightarrow F_y=22651982.59\ \text{N}[/tex]

Resultant force is

[tex]F=\sqrt{F_x^2+F_y^2}\\\Rightarrow F=\sqrt{14420700^2+22651982.59^2}\\\Rightarrow F=26852726.19\ \text{N}[/tex]

The hydrostatic force on the dam is [tex]26852726.19\ \text{N}[/tex].

The direction is given by

[tex]\theta=\tan^{-1}\dfrac{F_y}{F_x}\\\Rightarrow \theta=\tan^{-1}\dfrac{22651982.59}{14420700}\\\Rightarrow \theta=57.52^{\circ}[/tex]

The line of action is [tex]57.52^{\circ}[/tex].

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