Acontrol gate in the form of a parabolic segment of base 12 and height 4 is submerged in water so that its base is 2 units below the surface of the water, as shown in examination figure i. find the horizontal force on the gate if the density of the water is w.

Answer-

The horizontal force on the gate is

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

The centroid of the parabola is at along the vertical center line.

As the height of the parabola is given as 4, so the centroid will be at, from the centre O.

As the gate is in the water, so the distance of the centroid from the surface of the water is

We know that,

Area of the parabola is,

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

Putting all the values,