Question

A rectangular lamina of width b and depth d is submerged vertically in water, such that the

(a) upper edge of the lamina is at a depth h from the free surface. What will be the expression for the depth of the centre of pressure?

The question was posed to me during an online exam.

Question is taken from Hydrostatic Force on Plane Area topic in chapter Hydrostatic Forces on Surfaces of Fluid Mechanics

Answer 1

To find the expression for the depth of the center of pressure of a rectangular lamina submerged vertically in water, we need to follow the steps for calculating the center of pressure from hydrostatic principles.

Given:

• Width of the lamina = bbb
• Depth (height) of the lamina = ddd
• Depth of the upper edge of the lamina from the free surface = hhh
• The lamina is submerged vertically in water.

Center of Pressure Formula:

The depth of the center of pressure hcph_{\text{cp}}hcp​ is given by the following formula:

hcp=hcm+IGA⋅hcmh_{\text{cp}} = h_{\text{cm}} + \frac{I_G}{A \cdot h_{\text{cm}}}hcp​=hcm​+A⋅hcm​IG​​

Where:

• hcmh_{\text{cm}}hcm​ is the depth of the center of mass (centroid) of the lamina from the free surface,
• IGI_GIG​ is the second moment of area (moment of inertia) about the centroidal axis,
• AAA is the area of the lamina.

Step 1: Finding the center of mass hcmh_{\text{cm}}hcm​

For a vertical rectangular lamina submerged in water:

• The center of mass lies at the middle of the lamina, so its depth from the free surface is the midpoint of the total depth ddd.

Thus, the depth of the centroid (center of mass) hcmh_{\text{cm}}hcm​ is:

hcm=h+d2h_{\text{cm}} = h + \frac{d}{2}hcm​=h+2d​

Step 2: Moment of inertia IGI_GIG​

The moment of inertia IGI_GIG​ of the lamina about its centroid (centroidal axis) is given by:

IG=bd312I_G = \frac{b d^3}{12}IG​=12bd3​

This is the standard formula for the moment of inertia of a rectangle about its centroidal axis (parallel to the width bbb).

Step 3: Area of the lamina AAA

The area AAA of the lamina is:

A=b⋅dA = b \cdot dA=b⋅d

Step 4: Substituting into the center of pressure formula

Now, substituting all the values into the formula for the depth of the center of pressure:

hcp=h+d2+bd312bd⋅(h+d2)h_{\text{cp}} = h + \frac{d}{2} + \frac{\frac{b d^3}{12}}{b d \cdot \left( h + \frac{d}{2} \right)}hcp​=h+2d​+bd⋅(h+2d​)12bd3​​

Simplifying the expression:

hcp=h+d2+d212(h+d2)h_{\text{cp}} = h + \frac{d}{2} + \frac{d^2}{12 \left( h + \frac{d}{2} \right)}hcp​=h+2d​+12(h+2d​)d2​

Final Expression:

The depth of the center of pressure hcph_{\text{cp}}hcp​ is:

hcp=h+d2+d212(h+d2)h_{\text{cp}} = h + \frac{d}{2} + \frac{d^2}{12 \left( h + \frac{d}{2} \right)}hcp​=h+2d​+12(h+2d​)d2​

This is the expression for the depth of the center of pressure for a rectangular lamina submerged vertically in water with the upper edge at a depth hhh from the free surface.