A rectangular plane surface of width 2 m and height 3 m is placed vertically in water. What will be the location of center of pressure of the surface when its upper edge is horizontal and lies 2.5 m below the free surface of water?a)4.0755 mb)4.0125 mc)4.2525 md)4.1875 mCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Given data:
Width of surface (b) = 2 m
Height of surface (h) = 3 m
Distance of upper edge from free surface of water (h1) = 2.5 m

To calculate the location of center of pressure, we need to find the area of the surface (A) and the height of the center of pressure (h2) from the bottom edge of the surface.

Calculation:
The area of the rectangular surface can be calculated as:
A = b × h
A = 2 × 3
A = 6 m²

The height of the center of pressure can be calculated using the following formula:
h2 = (Iy / Ax) + h1

Where,
Iy = Moment of inertia of the area about the horizontal axis passing through the centroid of the area
Ax = Area of the water plane

The moment of inertia of the rectangular surface about the horizontal axis passing through the centroid of the surface can be calculated as:
Iy = (b × h³) / 12
Iy = (2 × 3³) / 12
Iy = 2.25 m⁴

The area of the water plane can be calculated as:
Ax = b × x
Where x is the distance of the centroid of the water plane from the bottom edge of the surface.

The centroid of the water plane will be at a distance of (2.5 + (3/2)) = 3.5 m from the bottom edge of the surface.

Ax = 2 × 3.5
Ax = 7 m²

Now, we can calculate the height of the center of pressure as:
h2 = (Iy / Ax) + h1
h2 = (2.25 / 7) + 2.5
h2 = 0.3214 + 2.5
h2 = 2.8214 m

Therefore, the location of the center of pressure is 2.8214 m from the bottom edge of the surface. The correct answer is option D.

Answer

Concept:
For a plane surface of arbitrary shape immersed in a liquid in such a way that the plane of the surface makes an angle θ with the free surface of the liquid:
where A = Total area of an inclined surface, h̅ = Depth of centre of gravity of inclined area from a free surface, h* =
Distance of centre of pressure from the free surface of a liquid.
Centre of pressure:

For vertical body θ = 90°

where I_G = Moment of Inertia of the body from its centroidal axis.
Calculation:
Given:
Width b = 2 m, Height h = 3 m
h̅ = 2.5 + h/2 = 2.5 + 3/2 = 4 m

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