Mechanical Properties of Fluids

### Two narrow bores of diameters 3.0 mm and 6.0 mm are joined together to form a U-tube open at both ends. If the U-tube contains water, what is the difference in its levels in the two limbs of the tube? Surface tension of water at the temperature of the experiment is 7.3 × 10–2 N m–1. Take the angle of contact to be zero and density of water to be 1.0 × 103 kg m–3 (g = 9.8 m s–2).

Answer : According to the question, the diameter of the first bore is d1 = 3.0 mm = 3 × 10-3 m The radius of the first bore is r1 = 3/2 = 1.5 x 10-3 m. The diameter of the second bore is d2 =6mm The...

### Mercury has an angle of contact equal to 140° with soda lime glass. A narrow tube of radius 1.00 mm made of this glass is dipped in a trough containing mercury. By what amount does the mercury dip down in the tube relative to the liquid surface outside ? Surface tension of mercury at the temperature of the experiment is 0.465 N m–1. Density of mercury = 13.6 × 103 kg m–3

Answer : According to the question, the density of mercury is ρ =13.6 × 103 kg/m3 Acceleration due to gravity, g = 9.8 m/s2 The angle of the contact between mercury and soda-lime glass is θ = 140°...

### In Millikan’s oil drop experiment, what is the terminal speed of an uncharged drop of radius 2.0 × 10–5 m and density 1.2 × 103 kg m–3. Take the viscosity of air at the temperature of the experiment to be 1.8 × 10–5 Pa s. How much is the viscous force on the drop at that speed? Neglect buoyancy of the drop due to air.

Answer According to the question, the acceleration due to gravity is g = 9.8 m/s2 The radius of the uncharged drop is r = 2.0 × 10-5 m The density of the uncharged drop is ρ = 1.2 × 103 kg m-3 The...

### A plane is in level flight at constant speed and each of its wings has an area of 25 m2. If the speed of the air is 180 km/h over the lower wing and 234 km/h over the upper wing surface, determine the plane’s mass. (Take air density to be 1 kg/m3), g = 9.8 m/s2

Answer : Area of the wings of the plane, A=2×25=50 m2 Speed of air over the lower wing, V1 ​=180km/h= 180 x (5/18) = 50 m/s Speed of air over the upper wing, V2 =234km/h= 234 x (5/18) = 65 m/s...

### (a) What is the largest average velocity of blood flow in an artery of radius 2×10–3m if the flow must remain lanimar? (b) What is the corresponding flow rate ? (Take viscosity of blood to be 2.084 × 10–3 Pa s).

Answer : According to the question, the radius of the vein is r = 2 × 10-3 m And the diameter of the vein is d = 2 × 1 × 10-3 m = 2 × 10-3 m The viscosity of blood is given by η = 2.08 x 10-3 m The...

### During a blood transfusion, the needle is inserted in a vein where the gauge pressure is 2000 Pa. At what height must the blood container be placed so that blood may just enter the vein? [Use the density of whole blood from Table 10.1]

Answer - According to the question, the density of whole blood is ρ = 1.06 × 103 kg m-3 Gauge pressure is P = 2000 Pa Acceleration due to gravity is g = 9.8 m/s2 =P/ρg =200/(1.06 x 103 x 9.8)...

### Two vessels have the same base area but different shapes. The first vessel takes twice the volume of water that the second vessel requires to fill upto a particular common height. Is the force exerted by the water on the base of the vessel the same in the two cases ? If so, why do the vessels filled with water to that same height give different readings on a weighing scale ?

Answer : The pressure and hence the force applied on the two vessels will be the same because the base area is the same. When the walls of the vessel are not perpendicular to the base, force is also...

### A manometer reads the pressure of a gas in an enclosure as shown in Fig. 10.25 (a) When a pump removes some of the gas, the manometer reads as in Fig. 10.25 (b) The liquid used in the manometers is mercury and the atmospheric pressure is 76 cm of mercury. (a) Give the absolute and gauge pressure of the gas in the enclosure for cases (a) and (b), in units of cm of mercury. (b) How would the levels change in case (b) if 13.6 cm of water (immiscible with mercury) is poured into the right limb of the manometer? (Ignore the small change in the volume of the gas).

Answer ; (a) For diagram (a): According to the question, atmospheric pressure, P0 = 76 cm of Hg Gauge pressure is the difference in mercury levels between the two arms. The gauge pressure is 20 cm...

### A tank with a square base of area 1.0 m2 is divided by a vertical partition in the middle. The bottom of the partition has a small-hinged door of area 20 cm2. The tank is filled with water in one compartment, and an acid (of relative density 1.7) in the other, both to a height of 4.0 m. compute the force necessary to keep the door close.

Answer : According to the given, the area of the hinged door is a = 20 cm2= 20 × 10-4 m The base area of given tank A = 2 m2 and the density of water is ρ1 = 103 kg/m3and the density of acid is ρ2 =...

### 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 × 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 × 105 Pa)

Answer : According to the question, the surface tension of the soap solution is S = 2.50 × 10-2 N/m r = 5.00 mm = 5 × 10-3 m We know that the density of the soap solution is ρ = 1.2 × 103 kg/m3 and...

### 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 × 10–1 N m–1. The atmospheric pressure is 1.01 × 105 Pa. Also give the excess pressure inside the drop.

Ans: According to the question, the Surface tension of mercury is S = 4.65 × 10-1 N m-1 The radius of the drop of mercury is r = 3.00 mm = 3 × 10-3 m And the atmospheric pressure, P0 = 1.01 × 105 Pa...

### Figure (a) shows a thin liquid film supporting a small weight = 4.5 × 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.

Answer : (a) According to the question, the length of the film which is supporting the weight is = 40 cm = 0.4 m. and the total weight supported = 4.5 x 10-2 N. We know that the film has two free...

### 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 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 ?

Answer : The maximum weight the film can support, W = 1.5 × 10-2 N Length of the slider, l = 30 cm = 0.3 m Total length of liquid film, l = 2 x 30 cm = 60 cm = 0.6 m because the liquid film has two...

### The cylindrical tube of a spray pump has a cross-section of 8.0 cm2 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?

Answer : According to the question, Number of holes, n = 40 Spray pump's cross-sectional area, A1 = 8 cm -2= 8 × 10-4 m-2 Radius of each hole, r = 0.5 × 10-3 m Cross-sectional area the said hole, a...

### Figures (a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?

Answer : Figure (a) is incorrect. The reason for this is that the velocity of liquid flow near the kink is high, hence applying Bernoulli's theorem, the pressure is low. As a result, when there is a...

### 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 m2 ? Take the density of air to be 1.3 kg m -3

Answer : According to the question, the Speed of wind on the upper side of the wing is V1 = 70 m/s And the speed of wind on the lower side of the wing is V2 = 63 m/s We are given that the area of...

### 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 × 10–3 kg s–1, what is the pressure difference between the two ends of the tube? (Density of glycerine = 1.3 × 103 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].

Answer : According to the question, the Length of the horizontal tube, l = 1.5 m and the radius of the tube is r = 1 cm = 0.01 m Therefore diameter of the tube becomes- d = 2r = 0.02 m It is given...

### Does it matter if one uses gauge instead of absolute pressures in applying Bernoulli’s equation? Explain

Answer : No, it doesn't matter if Bernoulli's equation is applied using gauge pressures rather than absolute pressures, as long as the atmospheric pressures at the two sites where Bernoulli's...

### Can Bernoulli’s equation be used to describe the flow of water through a rapid in a river? Explain.

Answer : Bernoulli's equation cannot be used to describe the flow of water in a river since it only applies to perfect liquids in a streamlined flow, whereas the water in a stream is turbulent.

### In the previous problem, if 15.0 cm of water and spirit each is 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)

Answer ; According to the question, h1 = 10.0 cm and ρ1 = 1 g cm-3 and for the spirit column in another arm of the U-tube, h2 = 12.5 cm We have to determine the value of p2 Let h be the difference...

### 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?

Answer : According to the question, the height of the spirit column is h1 = 12.5 cm = 0.125 m And the height of the water column is h2 = 10 cm = 0.1 m Let A and B represent the points of contact...

### 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

Ans: Given: According to the question, the maximum stress that a structure can handle is P = 109 Pa And the depth of the sea is d = 3 km = 3 × 103 m We also have the density of water as ρ...

### 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

Ans: According to the question, the density of mercury is ρ1 = 13.6 × 103 kg/m3 And the height of the mercury column is h1 = 0.76 m Also, the density of French wine is : ρ2 = 984 kg/m3 We have to...

### A 50 kg girl wearing high heel shoes balances on a single heel. The heel is circular with a diameter of 1.0 cm. What is the pressure exerted by the heel on the horizontal floor?

Answer : According to the question, the radius of the heel, r = d/2​ = 0.005 m And the mass of the lady (m) = 50 kg We know that the expression for the Area of the heel can be written as - A = πr2 =...

### Explain why (d) A fluid flowing out of a small hole in a vessel results in a backward thrust on the vessel (e) A spinning cricket ball in air does not follow a parabolic trajectory

Answer : (d) This is due to the conservation of momentum principle. The vessel gains backward motion while the flowing fluid gains forward momentum. (e) If there had been no air, a spinning cricket...

### 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

Answer : (c) The needle size determines the flow velocity, while thumb pressure determines the pressure. Bernoulli's theorem states that P + 1/2pv2 = constant. The pressure P has a single power in...

### Explain why (a) To keep a piece of paper horizontal, you should blow over, not under, it (b) When we try to close a water tap with our fingers, fast jets of water gush through the openings between our fingers

Answer : (a) When we blow over a piece of paper, the velocity of the air above it becomes greater than the velocity of the air below it. According to Bernoulli's theorem, as the K.E. of air above...

### Fill in the blanks using the word(s) from the list appended with each statement:

(c) For solids with an elastic modulus of rigidity, the shearing force is proportional to … , while for fluids, it is proportional to … (shear strain/rate of shear strain) (d) For a fluid in a...

### Fill in the blanks using the word(s) from the list appended with each statement: (a) The surface tension of liquids generally … with temperatures (increases/decreases) (b) The viscosity of gases … with temperature, whereas the viscosity of liquids … with temperature (increases/decreases)

Answer : (a) Decreases (b) increases; decreases

### Explain why (c) Surface tension of a liquid is independent of the area of the surface (d) Water with detergent disolved in it should have small angles of contact. (e) A drop of liquid under no external forces is always spherical in shape

Answer : (c) Because of surface tension, a liquid will always try to obtain the smallest possible surface area. A liquid drop will always take the shape of a sphere under zero external pressures...

### Explain why (a) The angle of contact of mercury with glass is obtuse, while that of water with glass is acute. (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.)

Answer : (a) Water molecules exhibit weak intermolecular interactions but are attracted to solids by a strong force. As a result, they flow out. Mercury molecules, on the other hand, have a stronger...