Centre of pressure on an inclined plane is .........
Try it first: pick an option, then read the explanation.
Asked in SSC JE 1 MARCH 2017 Evening Shift
Detailed Explanation
For any plane surface submerged in a static liquid at an angle \(\\theta\) to the horizontal, hydrostatic pressure increases linearly with vertical depth. This non-uniform loading displaces the resultant force line of action below the geometric centroid.
1. Formula for depth of centre of pressure on an inclined plane submerged at angle \(\theta\) to the free surface:
\[ h^* = \bar{h} + \frac{I_G \sin^2\theta}{A \bar{h}} \]
2. Analyze the components of the second term:
- Centroidal moment of inertia \(I_G > 0\) for any real area.
- For an inclined plane (\(0^\circ < \theta \le 90^\circ\)), \(\sin^2\theta > 0\).
- Surface area \(A > 0\) and centroid depth \(\bar{h} > 0\).
3. Therefore, the eccentricity is strictly positive:
\[ h^* - \bar{h} = \frac{I_G \sin^2\theta}{A \bar{h}} > 0 \;\implies\; h^* > \bar{h} \]
Thus, the centre of pressure is always located below the centroid.
Why “Below the centroid” Is Correct
Because liquid pressure increases with depth, the lower half of an inclined surface experiences greater pressure than the upper half. Taking moments of these distributed forces shifts the resultant force point of application below the centroid.
Why the Other Options Are Wrong
- A. At the centroid: Centre of pressure coincides with the centroid only for a horizontal plane (θ = 0°), where pressure is uniform.
- B. Above the centroid: Centre of pressure can never lie above the centroid in a uniform gravitational field.
- D. At metacentre: Metacentre is a stability parameter for floating or submerged bodies, not the centre of pressure on a plane surface.
Key Point
For any inclined or vertical plane surface, centre of pressure always lies below the centroid (h* > h̄).
Key Concepts for Students
- Horizontal surface: θ = 0° ⇒ sin θ = 0 ⇒ h* = h̄ (centre of pressure coincides with centroid).
- Vertical surface: θ = 90° ⇒ sin θ = 1 ⇒ h* = h̄ + I_G / (A·h̄) (maximum eccentricity).
- As depth h̄ → ∞, eccentricity approaches zero and centre of pressure asymptotically approaches centroid.
Practice more: this is question 23 in Hydrostatic Pressure MCQs – Part 2. Try the whole set and check every answer instantly.