Answer-
The horizontal force on the gate is [tex]\dfrac{576wg}{5}[/tex]
Solution-
From hydrostatics we know that,
Total force on a submerged gate is the product of pressure at the centroid of the gate and the area of the parabolic gate.
i.e [tex]F=P_{Centroid}\times A[/tex]
The centroid of the parabola is at [tex]\dfrac{3h}{5}[/tex] along the vertical center line.
As the height of the parabola is given as 4, so the centroid will be at, [tex]\dfrac{12}{5}[/tex] from the centre O.
As the gate is in the water, so the distance of the centroid from the surface of the water is
[tex]=2+(4-\dfrac{12}{5})=\dfrac{18}{5}[/tex]
We know that,
[tex]P_{Centroid}=h_{Centroid}\times \rho \times g[/tex]
[tex]=\dfrac{18}{5}\times w \times g[/tex]
Area of the parabola is,
[tex]A=\dfrac{4ah}{3}[/tex]
where,
a is the half distance, i.e from centre to the extreme point.
Here a = 6-0 = 6
So, Area of the parabola is [tex]\dfrac{4\times 6\times 4}{3}=32[/tex]
Putting all the values,
[tex]F=\dfrac{18}{5}\times w \times g\times 32=\dfrac{576wg}{5}[/tex]
