Question 7 of 18intermediate🔧 ApplyNumerical3 marks

A vertical off-shore structure is built to withstand a maximum stress of 10⁹ Pa. Is the structure suitable for putting up on top of an oil well in the ocean? Take the depth of the ocean to be roughly 3 km, and ignore ocean currents.

Correct Answer

Yes, the structure is suitable for putting up on top of an oil well in the ocean. The pressure at a depth of 3 km is approximately 3.1×103.1 \times 10⁷ Pa, which is much less than the maximum stress of 10⁹ Pa that the structure can withstand.

Exercise: EXERCISES | Q: 9.7 | (Page 21)
For More Understanding

Explanation

This question requires calculating the pressure exerted by sea water at a depth of 3 km and comparing it with the maximum stress the structure can withstand. Using the formula P = Pₐ + ρgh from Example 9.4 in the textbook, we calculate the absolute pressure at 3 km depth. The density of sea water (1.03×101.03 \times 10³ kg/m³) and g = 10 m/s² are taken from Example 9.4. The atmospheric pressure Pₐ = 1.01×101.01 \times 10⁵ Pa is also from the same example.

Solution Steps

  1. Given: Depth h = 3 km = 3000 m, Density of sea water ρ = 1.03×101.03 \times 10³ kg/m³, g = 10 m/s², Atmospheric pressure Pₐ = 1.01×101.01 \times 10⁵ Pa

  2. Formula: From Eq. (9.7), Pressure at depth h is P = Pₐ + ρgh

  3. Calculation: P = 1.01×101.01 \times 10⁵ Pa + (1.03×101.03 \times 10³ kg/m³) × (10 m/s²) × (3000 m)

  4. Solving: P = 1.01×101.01 \times 10⁵ Pa + 3.09×103.09 \times 10⁷ Pa = 3.1001×103.1001 \times 10⁷ Pa ≈\approx 3.1×103.1 \times 10⁷ Pa

  5. Comparison: Pressure at 3 km depth (≈\approx 3.1×103.1 \times 10⁷ Pa) is much less than maximum stress (10⁹ Pa)

  6. Conclusion: Since 3.1×103.1 \times 10⁷ Pa << 10⁹ Pa, the structure can easily withstand the ocean pressure and is suitable for the oil well.

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