Question

A 3-m-high, 7-m-wide rectangular gate is hinged at the top edge and is restrained by a fixed ridge. Determine the hydrostatic force exerted on the gate by the 2-m-high water and the location of the pressure center. (Round the final answers to the nearest whole number and to two decimal places, respectively.) Take the density of water to be 1000 kg/m^3 throughout.
The hydrostatic force exerted on the gate by the water is_________ kN.

The vertical distance of the pressure center from the free surface of water is______ m.

Answer 1

Answer:

The Hydrostatic force is   $F = 137.2 kN$

The location of pressure center is  $Z = 1.333 \ m$  

Explanation:

From the question we are told that

   The height of the gate is  $h = 3 \ m$

     The weight of the gate is  $w = 7 \ m$

      The height of the water is  $h_w = 2 \ m$

       The density of water is $\rho_w = 1000 \ kg/m^3$

Note used $h_w$ for height of water and height of gate immersed by water since both have the same value

The area of the gate immersed in water  is mathematically represented as

         $A = h_w * w$

substituting values

         $A = 2* 7$

         $A = 14 \ m^2$

The hydrostatic force is mathematically represented as

          $F = \rho_w * g * h_f * A$

Where

            $h_f =h- h_w$

           $h_f =3 -2$

           $h_f = 1\ m$  

So  

              $F = 1000 * 9.8 * 1 * 14$

            $F = 137.2 kN$

The center of pressure is mathematically represented as

        $Z = h_f + \frac{I_g}{h_f * A}$

Where $I_g$ is the moment of inertia of the gate which mathematically represented as

            $I_g = \frac{w * h_w^2}{12}$

The $h_w$ is the height of gate immersed in water

            $I_g = \frac{7 * 2^2 }{12}$

             $I_g = 4.667\ kg m^2$

Thus  

        $Z = 1 + \frac{4.66}{1 * 14}$

        $Z = 1.333 \ m$