Estimate the distance to the centre of pressure below the free surface for a vertical triangular plane area submerged in water (as in the given figure) with one side on the free surface, vertex downward and altitude h.

Asked in SSC JE 25-09-2019 (Morning)

Detailed Explanation

For a vertical triangular area with its horizontal base resting on the water surface and vertex pointing downwards, the centre of pressure depth is evaluated using the parallel axis theorem for area moment of inertia.

1. Geometric properties of the triangle:

• Base width along water surface \(= b\)
• Vertical altitude \(= h\)
• Area \(A = \frac{1}{2} b h\)
• Depth to centroid \(\bar{h} = \frac{h}{3}\) (measured from base)
• Moment of inertia about centroidal axis \(I_G = \frac{b h^3}{36}\)

2. Formula for depth of centre of pressure (\(h^*\)):

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

3. Substitute the geometric expressions:

\[ h^* = \frac{h}{3} + \frac{\frac{b h^3}{36}}{\left(\frac{1}{2} b h\right) \left(\frac{h}{3}\right)} = \frac{h}{3} + \frac{\frac{b h^3}{36}}{\frac{b h^2}{6}} = \frac{h}{3} + \frac{h}{6} = \frac{3h}{6} = \frac{h}{2} \]

Why “h/2” Is Correct

Because the base is at the surface, more area is concentrated near the top, placing the centroid at \(h/3\). The linearly increasing pressure distribution shifts the resultant force downward to exactly \(h/2\).

Why the Other Options Are Wrong

• A. 3h/4: 3h/4 is the centre of pressure when the vertex is at the free surface and the base is at depth h.
• B. h/3: h/3 is the centroid depth (h̄), not the centre of pressure.
• C. 2h/3: 2h/3 is the centre of pressure for a vertical rectangular lamina with top edge at the surface.

Key Concepts for Students

• Base at surface: h̄ = h/3, h* = h/2.
• Vertex at surface: h̄ = 2h/3, h* = 3h/4.
• Centre of pressure is always located deeper than the centroid (h* > h̄).

Practice more: this is question 17 in Hydrostatic Pressure MCQs – Part 2. Try the whole set and check every answer instantly.