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Physics · Textbook solutions

Mechanical Properties of Fluids

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 44 questions

Worked Examples

10 q

Solved Examples

Worked · 10
  1. Eg 9.1
    The two thigh bones (femurs), each of cross-sectional area 10 cm210\,\text{cm}^2 support the upper part of a human body of mass 40 kg40\,\text{kg}. Estimate the average pressure sustained by the femurs.
  2. Eg 9.2
    What is the pressure on a swimmer 10 m10\,\text{m} below the surface of a lake?
  3. Eg 9.3
    The density of the atmosphere at sea level is 1.29 kg/m31.29\,\text{kg/m}^3. Assume that it does not change with altitude. Then how high would the atmosphere extend?
  4. Eg 9.4
    At a depth of 1000 m1000\,\text{m} in an ocean (a) what is the absolute pressure? (b) What is the gauge pressure? (c) Find the force acting on the window of area 20 cm×20 cm20\,\text{cm} \times 20\,\text{cm} of a submarine at this depth, the interior of which is maintained at sea-level atmospheric pressure. (The density of sea water is 1.03×103 kg m−31.03 \times 10^{3}\,\text{kg m}^{-3}, g=10 m s−2g = 10\,\text{m s}^{-2}.)
  5. Eg 9.5
    Two syringes of different cross-sections (without needles) filled with water are connected with a tightly fitted rubber tube filled with water. Diameters of the smaller piston and larger piston are 1.0 cm1.0\,\text{cm} and 3.0 cm3.0\,\text{cm} respectively. (a) Find the force exerted on the larger piston when a force of 10 N10\,\text{N} is applied to the smaller piston. (b) If the smaller piston is pushed in through 6.0 cm6.0\,\text{cm}, how much does the larger piston move out?
  6. Eg 9.6
    In a car lift compressed air exerts a force F1F_1 on a small piston having a radius of 5.0 cm5.0\,\text{cm}. This pressure is transmitted to a second piston of radius 15 cm15\,\text{cm} (Fig 9.7). If the mass of the car to be lifted is 1350 kg1350\,\text{kg}, calculate F1F_1. What is the pressure necessary to accomplish this task? (g=9.8 ms−2g = 9.8\,\text{ms}^{-2}).
  7. Eg 9.7
    A fully loaded Boeing aircraft has a mass of 3.3×105 kg3.3 \times 10^{5}\,\text{kg}. Its total wing area is 500 m2500\,\text{m}^2. It is in level flight with a speed of 960 km/h960\,\text{km/h}. (a) Estimate the pressure difference between the lower and upper surfaces of the wings (b) Estimate the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface. [The density of air is ρ=1.2 kg m−3\rho = 1.2\,\text{kg m}^{-3}]
  8. Eg 9.8
    A metal block of area 0.10 m20.10\,\text{m}^2 is connected to a 0.010 kg0.010\,\text{kg} mass via a string that passes over an ideal pulley (considered massless and frictionless), as in Fig. 9.13. A liquid with a film thickness of 0.30 mm0.30\,\text{mm} is placed between the block and the table. When released the block moves to the right with a constant speed of 0.085 m s−10.085\,\text{m s}^{-1}. Find the coefficient of viscosity of the liquid.
  9. Eg 9.9
    The terminal velocity of a copper ball of radius 2.0 mm2.0\,\text{mm} falling through a tank of oil at 20∘C20^\circ\text{C} is 6.5 cm s−16.5\,\text{cm s}^{-1}. Compute the viscosity of the oil at 20∘C20^\circ\text{C}. Density of oil is 1.5×103 kg m−31.5 \times 10^{3}\,\text{kg m}^{-3}, density of copper is 8.9×103 kg m−38.9 \times 10^{3}\,\text{kg m}^{-3}.
  10. Eg 9.10
    The lower end of a capillary tube of diameter 2.00 mm2.00\,\text{mm} is dipped 8.00 cm8.00\,\text{cm} below the surface of water in a beaker. What is the pressure required in the tube in order to blow a hemispherical bubble at its end in water? The surface tension of water at temperature of the experiments is 7.30×10−2 Nm−17.30 \times 10^{-2}\,\text{Nm}^{-1}. 1 atmospheric pressure =1.01×105 Pa= 1.01 \times 10^{5}\,\text{Pa}, density of water =1000 kg/m3= 1000\,\text{kg/m}^3, g=9.80 m s−2g = 9.80\,\text{m s}^{-2}. Also calculate the excess pressure.

Exercises

34 q
  1. Ex 9.1(a)
    Explain why (a) The blood pressure in humans is greater at the feet than at the brain
  2. Ex 9.1(b)
    Explain why (b) Atmospheric pressure at a height of about 6 km6\,\text{km} decreases to nearly half of its value at the sea level, though the height of the atmosphere is more than 100 km100\,\text{km}
  3. Ex 9.1(c)
    Explain why (c) Hydrostatic pressure is a scalar quantity even though pressure is force divided by area.
  4. Ex 9.2(a)
    Explain why (a) The angle of contact of mercury with glass is obtuse, while that of water with glass is acute.
  5. Ex 9.2(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.)
  6. Ex 9.2(c)
    Explain why (c) Surface tension of a liquid is independent of the area of the surface
  7. Ex 9.2(d)
    Explain why (d) Water with detergent disolved in it should have small angles of contact.
  8. Ex 9.2(e)
    Explain why (e) A drop of liquid under no external forces is always spherical in shape
  9. Ex 9.3(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)
  10. Ex 9.3(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)
  11. Ex 9.3(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)
  12. Ex 9.3(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)
  13. Ex 9.3(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)
  14. Ex 9.4(a)
    Explain why (a) To keep a piece of paper horizontal, you should blow over, not under, it
  15. Ex 9.4(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
  16. Ex 9.4(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
  17. Ex 9.4(d)
    Explain why (d) A fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel
  18. Ex 9.4(e)
    Explain why (e) A spinning cricket ball in air does not follow a parabolic trajectory
  19. Ex 9.5
    A 50 kg50\,\text{kg} girl wearing high heel shoes balances on a single heel. The heel is circular with a diameter 1.0 cm1.0\,\text{cm}. What is the pressure exerted by the heel on the horizontal floor?
  20. Ex 9.6
    Toricelli's barometer used mercury. Pascal duplicated it using French wine of density 984 kg m−3984\,\text{kg m}^{-3}. Determine the height of the wine column for normal atmospheric pressure.
  21. Ex 9.7
    A vertical off-shore structure is built to withstand a maximum stress of 109 Pa10^{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 km3\,\text{km}, and ignore ocean currents.
  22. Ex 9.8
    A hydraulic automobile lift is designed to lift cars with a maximum mass of 3000 kg3000\,\text{kg}. The area of cross-section of the piston carrying the load is 425 cm2425\,\text{cm}^2. What maximum pressure would the smaller piston have to bear?
  23. Ex 9.9
    A U-tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in level with 10.0 cm10.0\,\text{cm} of water in one arm and 12.5 cm12.5\,\text{cm} of spirit in the other. What is the specific gravity of spirit?
  24. Setup carried over from the preceding question (Exercise 9.9), which this question refers back to as "the previous problem": a U-tube contains water and methylated spirit separated by mercury, and the mercury columns in the two arms are in level with 10.0 cm10.0\,\text{cm} of water in one arm and 12.5 cm12.5\,\text{cm} of spirit in the other. The book prints no restatement; this preamble is editorial, so that the question stands on its own.
    Ex 9.10
    In the previous problem, if 15.0 cm15.0\,\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= 13.6)
  25. Ex 9.11
    Can Bernoulli's equation be used to describe the flow of water through a rapid in a river? Explain.
  26. Ex 9.12
    Does it matter if one uses gauge instead of absolute pressures in applying Bernoulli's equation? Explain.
  27. Ex 9.13
    Glycerine flows steadily through a horizontal tube of length 1.5 m1.5\,\text{m} and radius 1.0 cm1.0\,\text{cm}. If the amount of glycerine collected per second at one end is 4.0×10−3 kg s−14.0 \times 10^{-3}\,\text{kg s}^{-1}, what is the pressure difference between the two ends of the tube? (Density of glycerine =1.3×103 kg m−3= 1.3 \times 10^{3}\,\text{kg m}^{-3} and viscosity of glycerine =0.83 Pa s= 0.83\,\text{Pa s}). [You may also like to check if the assumption of laminar flow in the tube is correct].
  28. Ex 9.14
    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−170\,\text{m s}^{-1} and 63 m s−163\,\text{m s}^{-1} respectively. What is the lift on the wing if its area is 2.5 m22.5\,\text{m}^2? Take the density of air to be 1.3 kg m−31.3\,\text{kg m}^{-3}.
  29. Ex 9.15
    Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect? Why?
  30. Ex 9.16
    The cylindrical tube of a spray pump has a cross-section of 8.0 cm28.0\,\text{cm}^2 one end of which has 40 fine holes each of diameter 1.0 mm1.0\,\text{mm}. If the liquid flow inside the tube is 1.5 m min−11.5\,\text{m min}^{-1}, what is the speed of ejection of the liquid through the holes?
  31. Ex 9.17
    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 1.5×10−2 N1.5 \times 10^{-2}\,\text{N} (which includes the small weight of the slider). The length of the slider is 30 cm30\,\text{cm}. What is the surface tension of the film?
  32. Ex 9.18
    Figure 9.21 (a) shows a thin liquid film supporting a small weight =4.5×10−2 N= 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.
  33. Ex 9.19
    What is the pressure inside the drop of mercury of radius 3.00 mm3.00\,\text{mm} at room temperature? Surface tension of mercury at that temperature (20 ∘C)(20\,^\circ\text{C}) is 4.65×10−1 N m−14.65 \times 10^{-1}\,\text{N m}^{-1}. The atmospheric pressure is 1.01×105 Pa1.01 \times 10^{5}\,\text{Pa}. Also give the excess pressure inside the drop.
  34. Ex 9.20
    What is the excess pressure inside a bubble of soap solution of radius 5.00 mm5.00\,\text{mm}, given that the surface tension of soap solution at the temperature (20 ∘C)(20\,^\circ\text{C}) is 2.50×10−2 N m−12.50 \times 10^{-2}\,\text{N m}^{-1}? If an air bubble of the same dimension were formed at depth of 40.0 cm40.0\,\text{cm} inside a container containing the soap solution (of relative density 1.201.20), what would be the pressure inside the bubble? (1 atmospheric pressure is 1.01×105 Pa1.01 \times 10^{5}\,\text{Pa}).