Question

Where will be the location of centre of pressure if a beaker contains water up to a height of 'h'

• A. \(h/3\)
• B. \(h/2\) from the surface
• C. \(2h/3\) from the surface
• D. \(h/6\) from the surface

Hint:

For any vertical rectangular plate with its top edge aligned with the liquid surface, the pressure distribution is triangular. The centroid of a triangle is located at \(\frac{2}{3}\) of its height from the apex (surface), which yields \(2h/3\) instantly.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The center of pressure is the point on a submerged surface where the total sum of the hydrostatic pressure field acts.
For any submerged surface, the center of pressure always lies below its centroid due to the non-uniform, triangular distribution of hydrostatic pressure with depth.
Key Formula or Approach:
For a vertically submerged plane surface, the depth of the center of pressure (\(h_{cp}\)) is given by:
\[ h_{cp} = \bar{h} + \frac{I_G}{A \cdot \bar{h}} \]
Where:
- \(\bar{h}\) is the depth of the centroid of the area from the free surface.
- \(I_G\) is the area moment of inertia about its centroidal axis.
- \(A\) is the area of the submerged surface.

Step 2: Detailed Explanation:
Let us consider a vertical rectangular section of the beaker wall of width \(b\) and height \(h\), extending from the free water surface:
- The area is:
\[ A = b \cdot h \]
- The depth to the centroid of this rectangle is:
\[ \bar{h} = \frac{h}{2} \]
- The area moment of inertia about the horizontal centroidal axis is:
\[ I_G = \frac{b \cdot h^3}{12} \]
Substitute these values into the center of pressure equation:
\[ h_{cp} = \frac{h}{2} + \frac{\frac{b \cdot h^3}{12}}{(b \cdot h) \cdot \frac{h}{2}} \]
Simplify the fractional term:
\[ \frac{\frac{b \cdot h^3}{12}}{\frac{b \cdot h^2}{2}} = \frac{2 \cdot b \cdot h^3}{12 \cdot b \cdot h^2} = \frac{h}{6} \]
Now, add the terms:
\[ h_{cp} = \frac{h}{2} + \frac{h}{6} \]
Find a common denominator:
\[ h_{cp} = \frac{3h + h}{6} = \frac{4h}{6} = \frac{2}{3}h \]
Therefore, the center of pressure is located at a depth of \(\frac{2}{3}h\) (or \(2h/3\)) measured from the free surface of the water (which is also equivalent to \(h/3\) from the bottom).

Step 3: Final Answer:
The location of the center of pressure is \(2h/3\) from the surface.