A longitudinal rectangular surface is hanged into the water such that its top and bottom points are at depth of 1.5 and 6.0 m respectively. The depth of center of pressure (m) from the top surface is ........
Try it first: pick an option, then read the explanation.
Asked in SSC JE 22.1.2018 (Morning Shift)
Detailed Explanation
When a vertical rectangular plate is submerged with its upper edge below the free surface, the pressure distribution is trapezoidal rather than triangular. The centre of pressure is calculated using the centroid depth and centroidal moment of inertia.
1. Determine plate dimensions and depths:
- Top edge depth \(h_1 = 1.5\text{ m}\)
- Bottom edge depth \(h_2 = 6.0\text{ m}\)
- Height of rectangular plate \(H = h_2 - h_1 = 6.0 - 1.5 = 4.5\text{ m}\)
- Width of plate \(= b\)
2. Calculate centroid depth (\(\bar{h}\)):
\[ \bar{h} = h_1 + \frac{H}{2} = 1.5 + \frac{4.5}{2} = 1.5 + 2.25 = 3.75\text{ m} \]
3. Calculate depth of centre of pressure (\(h^*\)):
\[ h^* = \bar{h} + \frac{I_G}{A \bar{h}} = \bar{h} + \frac{\frac{b H^3}{12}}{(b H) \bar{h}} = \bar{h} + \frac{H^2}{12 \bar{h}} \]
4. Substitute \(H = 4.5\text{ m}\) and \(\bar{h} = 3.75\text{ m}\):
\[ \frac{H^2}{12 \bar{h}} = \frac{(4.5)^2}{12 \times 3.75} = \frac{20.25}{45} = 0.45\text{ m} \]
\[ h^* = 3.75\text{ m} + 0.45\text{ m} = 4.20\text{ m} \]
Exam Note: In the official SSC JE answer key, option (a) 3.8 was mistakenly marked, which is simply the rounded centroid depth (3.75 ≈ 3.8). The exact mathematical and physical answer is option (b) 4.20 m.
Why “4.2” Is Correct
The centroid of the plate lies at \(3.75\text{ m}\) depth. Adding the eccentricity \(e = I_G / (A \bar{h}) = H^2 / (12 \bar{h}) = 4.5^2 / (12 \times 3.75) = 0.45\text{ m}\) gives an exact centre of pressure depth of \(4.20\text{ m}\).
Why the Other Options Are Wrong
- A. 3.8: 3.8 m results from erroneously taking the centroid depth (3.75 m rounded to 3.8 m) as the centre of pressure.
- C. 4.6: 4.6 m results from adding H/2 (2.25 m) to 1.5 m and adding an arbitrary 0.85 m.
- D. 4.8: 4.8 m results from an incorrect eccentricity calculation: 3.75 + 1.05 = 4.8 m.
Key Point
Centre of pressure for submerged rectangle: h* = h̄ + H² / (12·h̄) = 3.75 + 0.45 = 4.20 m.
Key Concepts for Students
- For submerged surfaces, h* > h̄ always; centre of pressure is never above or at the centroid.
- Eccentricity e = h* − h̄ = H² / (12·h̄) decreases as depth of submersion increases.
- Exam discrepancy: SSC JE key erroneously gave 3.8 (the centroid depth); mathematically 4.2 is exact.
Practice more: this is question 21 in Hydrostatic Pressure MCQs – Part 2. Try the whole set and check every answer instantly.