Chapter 10 - MECHANICAL PROPERTIES OF FLUIDS

A hot air balloon is a sphere of radius 8 m. The air inside is at a
temperature of 60°C. How large a mass can the balloon lift when
the outside temperature is 20°C?

(a) Pressure decreases as one ascends the atmosphere. If the
density of air is ρ, what is the change in pressure dp over a
differential height dh?
(b) Considering the pressure p to be proportional to the density,
find the pressure p at a height h if the pressure on the surface
of the earth is p0.
(c) If p0 = 1.03×105 N m–2, ρ0 = 1.29 kg m–3 and g = 9.8 m s–2, at
what height will the pressure drop to (1/10) the value at the
surface of the earth?
(d) This model of the atmosphere works for relatively small distances.
Identify the underlying assumption that limits the model.

The sufrace tension and vapour pressure of water at 20°C is
7.28×10–2 Nm–1 and 2.33×103 Pa, respectively. What is the radius
of the smallest spherical water droplet which can form without
evaporating at 20°C?

If a drop of liquid breaks into smaller droplets, it results in lowering
of temperature of the droplets. Let a drop of radius R, break into N
small droplets each of radius r. Estimate the drop in temperature.

Two mercury droplets of radii 0.1 cm. and 0.2 cm. collapse into
one single drop. What amount of energy is released? The surface
tension of mercury T= 435.5 × 10–3 N m–1.

The free surface of oil in a tanker, at rest, is horizontal. If the tanker
starts accelerating the free surface will be titled by an angle θ. If
the acceleration is a m s–2, what will be the slope of the free surface?.

A cubical block of density ρ is floating on the surface of water.
Out of its height L, fraction x is submerged in water. The vessel is
in an elevator accelerating upward with acceleration a . What is
the fraction immersed?

A vessel filled with water is kept on a weighing pan and the
scale adjusted to zero. A block of mass M and density ρ is
suspended by a massless spring of spring constant k. This block
is submerged inside into the water in the vessel. What is the
reading of the scale?

Is surface tension a vector?

Is viscosity a vector?

What is Bernoulli’s theorem? Show that sum of pressure, potential and kinetic
energy in the streamline flow is constant?

What is equation of continuity? Water plows through a horizontal pipe of
radius, 1cm at a speed of 2m/s. What should be the diameter of nozzle if water
is to come out at a speed of 10m/s?

What is terminal velocity and derive an expression for it?

Water flows through a horizontal pipe of which the cross – section is not
constant. The pressure is 1cm of mercury where the velocity is 0.35m/s. Find
the pressure at a point where the velocity is 0.65m/s.

What is stoke’s law and what are the factors on which viscous drag depends?

Water flows faster than honey. Why?

Give two areas where Bernoulli’s theorem is applied?

What is the significance of Reynolds’s Number?

Define viscosity?

State the principle on which Hydraulic lift work and explain its working?

A liquid drop of diameter 4 mm breaks into 1000 droplets of equal size.
Calculate the resultant change in the surface energy. Surface tension of the
liquid is 0.07 N/m?

Calculate the radius of new bubble formed when two bubbles of radius r₁ and

r₂ coalesce?

Find the work done in blowing a soap bubble of surface tension 0.06 N/m
from 2cm radius to 5cm radius?

Hydrostatic pressure is a scalar quantity even though pressure is force
divided by area, and force is a vector. Explain?

State the angle of contact and on what values do the angle of contact depends?

Define surface tension?

The blood pressure of humans is greater at the feet than at the brain?

State the law of floatation?

A metallic sphere of radius 1 × 10^-3 m and density 1 × 10⁴ kg | m3 enters a tank

of water after a free fall through a high ‘h’ in earth’s gravitational field. If its

velocity remains unchanged after entering water, determine the value of h.

Given :-

Co-efficient of viscosity of water = 1 × 10^-3 Ns | m2; g = 10 m | s²; density of

water = 1 × 10³ kg | m3?

Two capillary tubes of length 15 cm and 5 cm and radii 0.06 cm and 0.02 cm
respectively are connected in series. If the pressure difference a cross the end
faces is equal to the pressure of 15 cm high water column, then find the
pressure difference across the : 
1) first tube
2) Second tube.

A small ball of mass ‘m’ and density ‘d’ dropped in a viscous liquid of density ‘d’.
After some time, the ball falls with a constant velocity. What is the viscous force
on the ball?

A metal plate 5 cm × 5 cm rests on a layer of castor oil 1 mm thick whose coefficient

of viscosity is 1.55 Nsm^-2. What is the horizontal force required to more

the plate with a speed of 2 cm | s?

Why does the cloud seem floating in the sky?

If eight rain drops each of radius 1 mm are falling through air at a terminal
velocity of 5 cm | s. If they coalesce to form a bigger drop, what is the terminal
velocity of bigger drop?

What is the cause of viscosity in a fluid? How does the flow of fluid depend on
viscosity?

Find out the dimensions of co-efficient of viscosity?

The diameter of ball A is half that of ball B. What will be their ratio of their
terminal velocities in water?

What is terminal velocity? What is the terminal velocity of a body in a freely
falling system?

Why should detergents have small angles of contact?

What is the work done in splitting a drop of water of 1 mm radius into 64

droplets? Given the surface tension of water is 72 × 10^-3 N/m²?

What is the energy stored in a soap babble of diameter 4 cm, given the surface
tension = 0.07 N/m?

Find the work done in breaking a water drop of radius 1 mm into 1000 drops.

Given the surface tension of water is 72 × 10^-3 N/m?

If the radius of a soap bubble is r and surface tension of the soap solution is T.
Keeping the temperature constant, what is the extra energy needed to double
the radius of soap bubble?

The antiseptics used for cuts and wounds in human flesh have low surface
tension. Why?

Why is a soap solution a better cleansing agent than ordinary water?

What should be the pressure inside a small air bubble of 0.1mm radius situated

just below the water surface? Surface tension of water = 7.2 × 10^-2 N/m and

atmospheric pressure = 1.013 × 10⁵ N/m²?

Water rises in a capillary tube but mercury falls in the same tube. Why?

A drop of oil placed on the surface of water spreads out, but a drop of water
placed on oil contracts. Why?

A cubical block of iron 5cm on each side is floating on mercury in a vessel:-

1) What is the height of the block above mercury level?

2) Water is poured into vessel so that it just covers the iron block. What is the

height of the water column?

Given Density of mercury = 13.6g|cm3 and Density of iron = 7.2g|cm³

An object floats on water with 20% of its volume above the water time. What is

the density of object? Given Density of water = 1000kg|m³.

Two ballons that have same weight and volume contains equal amounts of
helium. One is rigid and other is free to expand as outside pressure decreases.
When released, which balloon will rise higher?

A liquid stands at the same level in the U – lube when at rest. If A is the area of
cross section of tube and g is the acceleration due to gravity, what will be the
difference in height of the liquid in the two limbs when the system is given
acceleration ‘a’?

Why is it painful to walk barefooted on a road covered with pebbles having sharp
edges?

At what depth in an ocean will a tube of air have one – fourth volume it will have
on reaching the surface? Given Atmospheric Pressure = 76 cm of Hg and density
of Hg = 13.6g/cc?

A boulder is thrown into a deep lake. As it sinks deeper and deeper into water,
does the buoyant force changes?

If a big drop of radius R is formed by 1000 small droplets of water, then find the
radius of small drop?

Why does not the pressure of atmosphere break windows?

Does Archimedes principle hold in a vessel in a free fall? [

Where ρ₀ = 1.25 kg m^-3
is the density at sea level, and y₀ is a constant. This density variation is called the law of atmospheres. Obtain this law assuming that the temperature of atmosphere remains a constant (isothermal conditions). Also assume that the value of gremains constant.
(b) A large He balloon of volume 1425 m³ is used to lift a payload of 400 kg. Assume that the balloon maintains constant radius as it rises. How high does it rise?
[Take y₀= 8000 m and ρ_He = 0.18 kg m^-3 ]

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).

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.

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.

A plane is in level flight at constant speed and each of its two 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 m^–3).

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

 In deriving Bernoulli’s equation, we equated the work done on the fluid in the tube to its change in the potential and kinetic energy. (a) What is the largest average velocity of blood flow in an artery of diameter 2 × 10^–3 m if the flow must remain laminar? (b) Do the dissipative forces become more important as the fluid velocity increases? Discuss qualitatively.

During 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? 

wo 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?

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) are poured into the right limb of the manometer? (Ignore the small change in the volume of the gas).

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.

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).

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.

Figure 10.24 (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.

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?

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?

Figures 10.23 (a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect? Why?

 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^–1and 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.

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].

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

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

 In problem 10.9, 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).

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?

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 cm². What maximum pressure would the smaller piston have to bear?

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.

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.

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?

Explain why

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

Explain why
(a) The blood pressure in humans is greater at the feet than at the brain
(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
(c) Hydrostatic pressure is a scalar quantity even though pressure is force divided by area.