Determine the total pressure on a circular plate of diameter 2 m, which is placed vertically in water in such a way that the Center of the plate is 2.5 m below the free surface of the water.
Try it first: pick an option, then read the explanation.
Asked in SSC JE 11/10/2023 (09am-11am)
Detailed Explanation
The total hydrostatic pressure force acting on any submerged plane surface is equal to the product of the fluid's specific weight, the surface area, and the vertical depth of the surface's centroid from the free liquid surface.
1. Given data:
- Plate diameter \(d = 2\text{ m}\)
- Centroid depth \(\bar{h} = 2.5\text{ m}\)
- Water density \(\rho = 1000\text{ kg/m}^3\)
- Acceleration due to gravity \(g = 9.81\text{ m/s}^2\)
2. Calculate the area of the circular plate:
\[ A = \frac{\pi}{4} d^2 = \frac{\pi}{4} (2)^2 = \pi\text{ m}^2 \approx 3.14159\text{ m}^2 \]
3. Calculate the total hydrostatic thrust \(P\):
\[ P = \rho g A \bar{h} = 1000 \times 9.81 \times \pi \times 2.5 \]
\[ P = 9810 \times 3.14159 \times 2.5 \approx 77,047.56\text{ N} \approx 77,048\text{ N} \]
Why “77048 N” Is Correct
The centroid of a circular disc is at its geometric centre, which is at \(\bar{h} = 2.5\text{ m}\). Multiplying the specific weight \(\gamma = 9810\text{ N/m}^3\), area \(A = \pi\text{ m}^2\), and centroid depth \(2.5\text{ m}\) gives \(77,048\text{ N}\).
Why the Other Options Are Wrong
- A. 72058 N: 72058 N results from incorrectly using d = 1.9 m or arithmetic errors.
- B. 70058 N: 70058 N results from using g = 8.92 m/s² or neglecting the π multiplier accurately.
- C. 71058 N: 71058 N results from taking centroid depth as 2.3 m instead of 2.5 m.
Key Point
Total hydrostatic thrust on a plane surface: P = γ · A · h̄ = ρ · g · A · h̄.
Key Concepts for Students
- Total hydrostatic force depends strictly on centroid depth h̄, not on the orientation angle θ of the plate.
- The centre of pressure h* lies below the centroid for vertical surfaces: h* = h̄ + I_G / (A·h̄).
- For a circle, moment of inertia about the diametral centroidal axis is I_G = πd⁴ / 64.
Practice more: this is question 5 in Hydrostatic Pressure MCQs – Part 1. Try the whole set and check every answer instantly.