What is the depth of the center of pressure for the rectangular lamina which is vertically inside the water of height h ?

Asked in SSC JE 29-01-2018 (Morning Shift)

Detailed Explanation

The determination of the centre of pressure for a vertical rectangular plate of height \(h\) immersed with its top edge in the water surface is one of the most frequently asked standard derivations in fluid mechanics.

1. Geometric and pressure parameters:

• Width of rectangle \(= b\)
• Height of rectangle \(= h\)
• Area \(A = b h\)
• Centroid depth \(\bar{h} = \frac{h}{2}\)
• Centroidal moment of inertia \(I_G = \frac{b h^3}{12}\)

2. Apply the centre of pressure equation for a vertical plane:

\[ h^* = \bar{h} + \frac{I_G}{A \bar{h}} \]

3. Substitute the values:

\[ h^* = \frac{h}{2} + \frac{\frac{b h^3}{12}}{b h \cdot \frac{h}{2}} = \frac{h}{2} + \frac{\frac{b h^3}{12}}{\frac{b h^2}{2}} = \frac{h}{2} + \frac{h}{6} = \frac{3h + h}{6} = \frac{4h}{6} = \frac{2}{3}h \]

Why “2h/3” Is Correct

The hydrostatic pressure increases from zero at the free surface to a maximum at depth \(h\), forming a triangular pressure distribution whose centroid is located at two-thirds of the depth from the top surface (\(2h/3\)).

Why the Other Options Are Wrong

• A. h/3: h/3 is the distance from the bottom edge to the centre of pressure.
• B. h/4: h/4 represents the centroid depth for a cubic parabola area.
• D. 3h/2: 3h/2 is greater than the total height of the lamina and is physically outside the plate.

Key Concepts for Students

• The centre of pressure coincides with the centroid of the pressure prism.
• For a uniform width plate with triangular pressure distribution, the resultant acts at 2/3 depth from apex.
• The moment of inertia I_O about the water surface axis is bh³/3, giving h* = I_O / (A h̄) = (bh³/3) / (bh · h/2) = 2h/3.

Practice more: this is question 20 in Hydrostatic Pressure MCQs – Part 2. Try the whole set and check every answer instantly.