A rectangular tank 4 m long, 1.5 m wide contains water up to a height of 2 m. Calculate the force due to water pressure on the base of the tank. Find also the depth of the centre of pressure from the free surface.

Solution

(i) Force on the Base of the Tank (\(F\))

The pressure on a horizontal surface is uniform across its entire area. First, we calculate this pressure at the base.

$$ P_{base} = \rho \times g \times h $$ $$ P_{base} = 1000 \times 9.81 \times 2 $$ $$ P_{base} = 19620 \, \text{N/m}^2 $$

Next, we calculate the area of the tank's base.

$$ A_{base} = L \times W $$ $$ A_{base} = 4 \times 1.5 = 6 \, \text{m}^2 $$

The total force is the pressure multiplied by the area.

$$ F = P_{base} \times A_{base} $$ $$ F = 19620 \times 6 $$ $$ F = 117720 \, \text{N} $$

(ii) Depth of Centre of Pressure (\(h^*\))

For any horizontal surface submerged in a fluid, the pressure is constant at all points. Therefore, the resultant force acts at the geometric center (centroid) of the area.

$$ \text{Depth of centroid, } \bar{h} = h = 2 \, \text{m} $$ $$ \text{Depth of Centre of Pressure, } h^* = \bar{h} $$ $$ h^* = 2 \, \text{m} $$

Explanation of Concepts

Pressure on a Horizontal Surface: The pressure exerted by a fluid at a certain depth is given by \(P = \rho g h\). For a flat, horizontal surface like the bottom of a tank, the depth \(h\) is the same for every point on that surface. This means the pressure is uniform over the entire base area.

Centre of Pressure on a Horizontal Surface: The centre of pressure is the point where the total resultant hydrostatic force is considered to act. Since the pressure on a horizontal surface is uniform, the resultant force acts through the geometric center, or centroid, of that surface. Therefore, for any horizontal plane, the centre of pressure and the centroid coincide. The depth to this point is simply the depth of the plane itself below the free surface.