A vertical rectangular plane surface is submerged in water such that its top and bottom surfaces are 1.5 m and 6.0 m res-pectively below the free surface. The position of center of pressure below the free surface will be at a distance of

Solution (By Examveda Team)

Understanding the Problem:
We need to find the center of pressure on a rectangular surface submerged in water.
The center of pressure is the point where the total hydrostatic force acts on the surface.
Key Concepts:
For a vertically submerged rectangle, the center of pressure (h_cp) is located below the centroid (center of area).
The formula to calculate h_cp is: h_cp = h_c + (I_xx / (A * h_c))
Where:
h_c is the depth of the centroid from the free surface.
I_xx is the second moment of area (moment of inertia) about the centroidal axis.
A is the area of the submerged surface.
Step-by-Step Solution:
1. Find the depth of the centroid (h_c):
The rectangle's top is at 1.5 m and the bottom is at 6.0 m.
The height of the rectangle is 6.0 m - 1.5 m = 4.5 m.
The centroid is located at the middle of the rectangle's height.
So, h_c = 1.5 m + (4.5 m / 2) = 1.5 m + 2.25 m = 3.75 m
2. Calculate the Area (A):
We are not given the width (b) of the rectangle so we consider it as b.
Height of the rectangle h = 4.5 m
A = b * h = b * 4.5 m
3. Calculate the Moment of Inertia (I_xx):
For a rectangle about its centroidal axis, I_xx = (b * h³) / 12.
I_xx = (b * (4.5)³) / 12 = (b * 91.125) / 12
4. Calculate the Center of Pressure (h_cp):
h_cp = h_c + (I_xx / (A * h_c))
h_cp = 3.75 + (((b * 91.125) / 12) / ((b * 4.5) * 3.75))
h_cp = 3.75 + ((b * 91.125) / 12) / (b * 16.875)
h_cp = 3.75 + (91.125 / 12) / 16.875 (b cancels out)
h_cp = 3.75 + (7.59375 / 16.875)
h_cp = 3.75 + 0.45
h_cp = 4.2 m
Therefore, the correct answer is Option C: 4.2 m.