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Q10.7:
A vertical off-shore structure is built to withstand a maximum stress of $10^9$ 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.
Solution :
Step 1: Identification of Physical Variables
To determine the suitability of the structure, we must calculate the hydrostatic pressure exerted by the ocean at the specified depth. Given parameters:
- Depth of the ocean, $h = 3 \text{ km} = 3000 \text{ m}$
- Maximum allowable stress of the structure, $\sigma_{max} = 10^9 \text{ Pa}$
- Density of seawater, $\rho \approx 1030 \text{ kg/m}^3$ [Standard empirical value for ocean water]
- Acceleration due to gravity, $g = 9.8 \text{ m/s}^2$
- Atmospheric pressure, $P_{atm} = 1.013 \times 10^5 \text{ Pa}$
Step 2: Calculating Hydrostatic Pressure
The pressure exerted at a depth $h$ below the surface of a fluid is given by the hydrostatic pressure formula [Per Pascal’s Law and Fluid Statics]:
$P = P_{atm} + \rho gh$
Substituting the given values into the equation:
$P = 1.013 \times 10^5 + (1030 \times 9.8 \times 3000)$
$P = 1.013 \times 10^5 + 30,282,000$
$P = 0.1013 \times 10^8 + 302.82 \times 10^5$
$P \approx 3.03 \times 10^7 \text{ Pa}$
Step 3: Comparative Analysis
We compare the calculated pressure ($P$) against the maximum stress capacity ($\sigma_{max}$) of the offshore structure:
| Variable | Magnitude (Pa) |
|---|---|
| Hydrostatic Pressure ($P$) | $3.03 \times 10^7 \text{ Pa}$ |
| Maximum Stress ($\sigma_{max}$) | $10 \times 10^8 \text{ Pa}$ |
Since $\sigma_{max} \gg P$ (specifically, the capacity is approximately 33 times greater than the pressure exerted), the structural integrity is maintained under these conditions.
Visual Representation of Force Vectors
Final Conclusion
Because the hydrostatic pressure exerted by the ocean at a depth of 3 km ($3.03 \times 10^7 \text{ Pa}$) is significantly lower than the structural limit of the platform ($10^9 \text{ Pa}$), the structure is well within the safety margin to withstand the pressure.
Final Solution: Yes, the structure is suitable for use on top of the oil well, as the hydrostatic pressure at 3 km depth is approximately $3.03 \times 10^7 \text{ Pa}$, which is well below the maximum stress limit of $10^9 \text{ Pa}$.
More Questions from Class 11 Physics Mechanical Properties of Fluids EXERCISES
- Q9.1(a): Explain why (a) The blood pressure in humans is greater at the feet than at the brain
- Q9.1(b): Explain why (b) Atmospheric pressure at a height of about (6\text{ km}) decreases to nearly half of its value at the sea level, though the height of the atmosphere is more than (100\text{ km})
- Q9.1(c): Explain why (c) Hydrostatic pressure is a scalar quantity even though pressure is force divided by area.
- Q9.10: If (15.0\text{ cm}) of water and spirit each are further poured into the respective arms of the tube, what is the difference in the levels of mercury in the two arms ? (Specific gravity of mercury = (13.6))
- Q9.11: Can Bernoulli’s equation be used to describe the flow of water through a rapid in a river ? Explain.
- Q9.12: Does it matter if one uses gauge instead of absolute pressures in applying Bernoulli’s equation ? Explain.
- Q9.13: Glycerine flows steadily through a horizontal tube of length (1.5\text{ m}) and radius (1.0\text{ cm}). If the amount of glycerine collected per second at one end is (4.0 \times 10^{-3}\text{ kg s}^{-1}), what is the pressure difference between the two ends of the tube ? (Density of glycerine = (1.3 \times 10^{3}\text{ kg m}^{-3}) and viscosity of glycerine = (0.83\text{ Pa s})). [You may also like to check if the assumption of laminar flow in the tube is correct].
- Q9.14: In a test experiment on a model aeroplane in a wind tunnel, the flow speeds on the upper and lower surfaces of the wing are (70\text{ m s}^{-1})and (63\text{ m s}^{-1}) respectively. What is the lift on the wing if its area is (2.5\text{ m}^{2}) ? Take the density of air to be (1.3\text{ kg m}^{-3}).
- Q9.15: Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?
- Q9.16: The cylindrical tube of a spray pump has a cross-section of (8.0\text{ cm}^{2}) one end of which has (40) fine holes each of diameter (1.0\text{ mm}). If the liquid flow inside the tube is (1.5\text{ m min}^{-1}), what is the speed of ejection of the liquid through the holes ?
- Q9.17: A U-shaped wire is dipped in a soap solution, and removed. The thin soap film formed between the wire and the light slider supports a weight of (1.5 \times 10^{-2}\text{ N}) (which includes the small weight of the slider). The length of the slider is (30\text{ cm}). What is the surface tension of the film ?
- Q9.18: Figure 9.21 (a) shows a thin liquid film supporting a small weight = (4.5 \times 10^{-2}\text{ N}). What is the weight supported by a film of the same liquid at the same temperature in Fig. (b) and (c) ? Explain your answer physically.
- Q9.19: What is the pressure inside the drop of mercury of radius (3.00\text{ mm}) at room temperature ? Surface tension of mercury at that temperature ((20^{\circ}\text{C})) is (4.65 \times 10^{-1}\text{ N m}^{-1}). The atmospheric pressure is (1.01 \times 10^{5}\text{ Pa}). Also give the excess pressure inside the drop.
- Q9.2(a): Explain why (a) The angle of contact of mercury with glass is obtuse, while that of water with glass is acute.
- Q9.2(b): Explain why (b) Water on a clean glass surface tends to spread out while mercury on the same surface tends to form drops. (Put differently, water wets glass while mercury does not.)
- Q9.2(c): Explain why (c) Surface tension of a liquid is independent of the area of the surface
- Q9.2(d): Explain why (d) Water with detergent disolved in it should have small angles of contact.
- Q9.2(e): Explain why (e) A drop of liquid under no external forces is always spherical in shape
- Q9.20: What is the excess pressure inside a bubble of soap solution of radius (5.00\text{ mm}), given that the surface tension of soap solution at the temperature ((20^{\circ}\text{C})) is (2.50 \times 10^{-2}\text{ N m}^{-1}) ? If an air bubble of the same dimension were formed at depth of (40.0\text{ cm}) inside a container containing the soap solution (of relative density (1.20)), what would be the pressure inside the bubble ? (1 atmospheric pressure is (1.01 \times 10^{5}\text{ Pa})).
- Q9.3(a): Fill in the blanks using the word(s) from the list appended with each statement: (a) Surface tension of liquids generally . . . with temperatures (increases / decreases)
- Q9.3(b): Fill in the blanks using the word(s) from the list appended with each statement: (b) Viscosity of gases . . . with temperature, whereas viscosity of liquids . . . with temperature (increases / decreases)
- Q9.3(c): Fill in the blanks using the word(s) from the list appended with each statement: (c) For solids with elastic modulus of rigidity, the shearing force is proportional to . . . , while for fluids it is proportional to . . . (shear strain / rate of shear strain)
- Q9.3(d): Fill in the blanks using the word(s) from the list appended with each statement: (d) For a fluid in a steady flow, the increase in flow speed at a constriction follows (conservation of mass / Bernoulli’s principle)
- Q9.3(e): Fill in the blanks using the word(s) from the list appended with each statement: (e) For the model of a plane in a wind tunnel, turbulence occurs at a ... speed for turbulence for an actual plane (greater / smaller)
- Q9.4(a): Explain why (a) To keep a piece of paper horizontal, you should blow over, not under, it
- Q9.4(b): Explain why (b) When we try to close a water tap with our fingers, fast jets of water gush through the openings between our fingers
- Q9.4(c): Explain why (c) The size of the needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor while administering an injection
- Q9.4(d): Explain why (d) A fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel
- Q9.4(e): Explain why (e) A spinning cricket ball in air does not follow a parabolic trajectory
- Q9.5: A (50\text{ kg}) girl wearing high heel shoes balances on a single heel. The heel is circular with a diameter (1.0\text{ cm}). What is the pressure exerted by the heel on the horizontal floor ?
- Q9.6: Toricelli’s barometer used mercury. Pascal duplicated it using French wine of density (984\text{ kg m}^{-3}). Determine the height of the wine column for normal atmospheric pressure.
- Q9.7: A vertical off-shore structure is built to withstand a maximum stress of (10^{9}\text{ 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\text{ km}), and ignore ocean currents.
- Q9.8: A hydraulic automobile lift is designed to lift cars with a maximum mass of (3000\text{ kg}). The area of cross-section of the piston carrying the load is (425\text{ cm}^{2}). What maximum pressure would the smaller piston have to bear ?
- Q9.9: A U-tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in level with (10.0\text{ cm}) of water in one arm and (12.5\text{ cm}) of spirit in the other. What is the specific gravity of spirit ?
- Q10.1(a): Explain why (a) The blood pressure in humans is greater at the feet than at the brain
- Q10.1(b): Explain why (b) Atmospheric pressure at a height of about 6 km decreases to nearly half of its value at the sea level, though the height of the atmosphere is more than 100 km
- Q10.1(c): Explain why (c) Hydrostatic pressure is a scalar quantity even though pressure is force divided by area.
- Q10.10: In the previous problem, if 15.0 cm of water and spirit each are further poured into the respective arms of the tube, what is the difference in the levels of mercury in the two arms ? (Specific gravity of mercury = 13.6)
- Q10.11: Can Bernoulli’s equation be used to describe the flow of water through a rapid in a river ? Explain.
- Q10.12: Does it matter if one uses gauge instead of absolute pressures in applying Bernoulli’s equation ? Explain.
- Q10.13: Glycerine flows steadily through a horizontal tube of length 1.5 m and radius 1.0 cm. If the amount of glycerine collected per second at one end is $4.0 \times 10^{-3} kg s^{-1}$, what is the pressure difference between the two ends of the tube ? (Density of glycerine = $1.3 \times 10^3 kg m^{-3}$ and viscosity of glycerine = $0.83 Pa s$). [You may also like to check if the assumption of laminar flow in the tube is correct].
- Q10.14: In a test experiment on a model aeroplane in a wind tunnel, the flow speeds on the upper and lower surfaces of the wing are $70 m s^{-1}$ and $63 m s^{-1}$ respectively. What is the lift on the wing if its area is $2.5 m^2$ ? Take the density of air to be $1.3 kg m^{-3}$.
- Q10.15: Figures 10.23(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?
- Q10.16: The cylindrical tube of a spray pump has a cross-section of $8.0 cm^2$ one end of which has 40 fine holes each of diameter 1.0 mm. If the liquid flow inside the tube is $1.5 m min^{-1}$, what is the speed of ejection of the liquid through the holes ?
- Q10.17: A U-shaped wire is dipped in a soap solution, and removed. The thin soap film formed between the wire and the light slider supports a weight of $1.5 \times 10^{-2} N$ (which includes the small weight of the slider). The length of the slider is 30 cm. What is the surface tension of the film ?
- Q10.18: Figure 10.24 (a) shows a thin liquid film supporting a small weight = $4.5 \times 10^{-2} N$. What is the weight supported by a film of the same liquid at the same temperature in Fig. (b) and (c) ? Explain your answer physically.
- Q10.19: What is the pressure inside the drop of mercury of radius 3.00 mm at room temperature ? Surface tension of mercury at that temperature (20 °C) is $4.65 \times 10^{-1} N m^{-1}$. The atmospheric pressure is $1.01 \times 10^5 Pa$. Also give the excess pressure inside the drop.
- Q10.2(a): Explain why (a) The angle of contact of mercury with glass is obtuse, while that of water with glass is acute.
- Q10.2(b): Explain why (b) Water on a clean glass surface tends to spread out while mercury on the same surface tends to form drops. (Put differently, water wets glass while mercury does not.)
- Q10.2(c): Explain why (c) Surface tension of a liquid is independent of the area of the surface
- Q10.2(d): Explain why (d) Water with detergent disolved in it should have small angles of contact.
- Q10.2(e): Explain why (e) A drop of liquid under no external forces is always spherical in shape
- Q10.20: What is the excess pressure inside a bubble of soap solution of radius 5.00 mm, given that the surface tension of soap solution at the temperature (20 °C) is $2.50 \times 10^{-2} N m^{-1}$ ? If an air bubble of the same dimension were formed at depth of 40.0 cm inside a container containing the soap solution (of relative density 1.20), what would be the pressure inside the bubble ? (1 atmospheric pressure is $1.01 \times 10^5 Pa$).
- Q10.3(a): Fill in the blanks using the word(s) from the list appended with each statement: (a) Surface tension of liquids generally . . . with temperatures (increases / decreases)
- Q10.3(b): Fill in the blanks using the word(s) from the list appended with each statement: (b) Viscosity of gases . . . with temperature, whereas viscosity of liquids . . . with temperature (increases / decreases)
- Q10.3(c): Fill in the blanks using the word(s) from the list appended with each statement: (c) For solids with elastic modulus of rigidity, the shearing force is proportional to . . . , while for fluids it is proportional to . . . (shear strain / rate of shear strain)
- Q10.3(d): Fill in the blanks using the word(s) from the list appended with each statement: (d) For a fluid in a steady flow, the increase in flow speed at a constriction follows (conservation of mass / Bernoulli’s principle)
- Q10.3(e): Fill in the blanks using the word(s) from the list appended with each statement: (e) For the model of a plane in a wind tunnel, turbulence occurs at a ... speed for turbulence for an actual plane (greater / smaller)
- Q10.4(a): Explain why (a) To keep a piece of paper horizontal, you should blow over, not under, it
- Q10.4(b): Explain why (b) When we try to close a water tap with our fingers, fast jets of water gush through the openings between our fingers
- Q10.4(c): Explain why (c) The size of the needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor while administering an injection
- Q10.4(d): Explain why (d) A fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel
- Q10.4(e): Explain why (e) A spinning cricket ball in air does not follow a parabolic trajectory
- Q10.5: A 50 kg girl wearing high heel shoes balances on a single heel. The heel is circular with a diameter 1.0 cm. What is the pressure exerted by the heel on the horizontal floor ?
- Q10.6: Toricelli’s barometer used mercury. Pascal duplicated it using French wine of density 984 $kg m^{-3}$. Determine the height of the wine column for normal atmospheric pressure.
- Q10.8: A hydraulic automobile lift is designed to lift cars with a maximum mass of 3000 kg. The area of cross-section of the piston carrying the load is $425 cm^2$. What maximum pressure would the smaller piston have to bear ?
- Q10.9: A U-tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in level with 10.0 cm of water in one arm and 12.5 cm of spirit in the other. What is the specific gravity of spirit ?
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