Mechanical Properties of Fluids Class 11 Physics Notes

The pressure at any point in a fluid at rest equals the atmospheric pressure P₀ plus ρgh, where ρ is the fluid density, g the gravitational acceleration and h the depth below the free surface. Hence pressure increases uniformly with depth and is the same in all directions at a given depth.

• The pressure does not depend on the shape of the container — only on depth.
• Two vessels of different shapes connected at the bottom reach the same liquid level.
• Gauge pressure = P − P_atm = ρgh; the manometer and barometer work on this principle.

MCQ Questions

1 Mark Questions

1. 1
   Define pressure. Write its SI unit.
2. 2
   Write the expression for pressure at a depth h in a liquid.
3. 3
   Define gauge pressure.

2 Mark Questions

1. 1
   Find the pressure at a depth of 10 m below the surface of water. (ρ = 1000 kg m⁻³, g = 10 m/s², atmospheric pressure = 10⁵ Pa.)
2. 2
   Why is a dam wall thicker at the bottom than at the top?
3. 3
   Distinguish between absolute pressure and gauge pressure.

3 Mark Questions

1. 1
   Derive an expression for the pressure at a depth h inside a liquid at rest, taking atmospheric pressure into account.
2. 2
   A rectangular tank is filled with water up to a height of 2 m. Find the total force on a wall of dimensions 2 m × 3 m due to the water. (ρ = 1000 kg m⁻³, g = 10 m/s².)

The hydraulic lift, hydraulic press and hydraulic brakes all exploit Pascal's law. A small force F₁ applied on a small piston of area A₁ produces the same pressure throughout, giving a much larger force F₂ on a large piston of area A₂, since F₂ = F₁(A₂/A₁). The mechanical advantage is the ratio of the areas.

• Hydraulic brakes use Pascal's law to transmit the driver's pedal force to all four wheels.
• A small input force can lift a heavy car on a hydraulic lift.
• The principle works because liquids are nearly incompressible.

MCQ Questions

1 Mark Questions

1. 1
   State Pascal's law.
2. 2
   Name two machines that work on Pascal's law.
3. 3
   Write the formula for the force amplification in a hydraulic press.

2 Mark Questions

1. 1
   A hydraulic lift has pistons of areas 10 cm² and 200 cm². What force is needed on the small piston to lift a car of weight 2000 N?
2. 2
   Explain the working of hydraulic brakes using Pascal's law.
3. 3
   State Pascal's law and explain why it is the basis of the hydraulic press.

3 Mark Questions

1. 1
   State and explain Pascal's law, and derive the force amplification in a hydraulic lift.
2. 2
   The small piston of a hydraulic press has a diameter of 2 cm and the large piston a diameter of 20 cm. If a force of 10 N is applied to the small piston, find the force on the large piston.

For floatation, the weight of the floating body equals the weight of the fluid displaced. A body sinks if its density exceeds that of the fluid and floats if its density is lower. This is why a steel ship floats — its average density (including the air inside) is less than water.

• The fraction of a floating body submerged equals the ratio of its density to the fluid's density.
• A hydrometer measures the density of a liquid by the depth it sinks.
• An iceberg floats with about 90% of its volume submerged because ice is slightly less dense than water.

MCQ Questions

1 Mark Questions

1. 1
   State Archimedes' principle.
2. 2
   What is the condition for a body to float?
3. 3
   State the condition for a body to sink in a liquid.

2 Mark Questions

1. 1
   A body of volume 2 × 10⁻³ m³ is fully immersed in water. Find the buoyant force on it. (ρ = 1000 kg m⁻³, g = 10 m/s².)
2. 2
   Explain why a solid steel ball sinks in water but a steel ship floats.
3. 3
   A body floats with one quarter of its volume above water. Find the ratio of its density to that of water.

3 Mark Questions

1. 1
   State Archimedes' principle and use it to show the condition for floatation.
2. 2
   A cube of side 5 cm floats in water with 60% of its volume submerged. Find the density of the cube. (ρ_water = 1000 kg m⁻³.)

The most important consequence is that where the fluid speed is higher, the pressure is lower. This explains the lift on an aeroplane wing (faster air over the top, lower pressure), the working of a venturi meter, the atomiser and blow, and the curved flight of a spinning ball (Magnus effect).

MCQ Questions

1 Mark Questions

1. 1
   State the equation of continuity for an incompressible fluid.
2. 2
   Write Bernoulli's equation for a horizontal pipe.
3. 3
   State Bernoulli's principle.

2 Mark Questions

1. 1
   Water flows in a pipe of area 40 cm² at 2 m/s. Find the speed in a section of the pipe of area 10 cm².
2. 2
   State and explain Bernoulli's principle, and give one application.
3. 3
   Derive the equation of continuity for an incompressible fluid.

3 Mark Questions

1. 1
   Derive Bernoulli's equation for the steady flow of an ideal incompressible fluid and state the assumptions made.
2. 2
   Water flows through a horizontal pipe of cross-section 30 cm² where its speed is 2 m/s and pressure 2 × 10⁵ Pa. In a narrower section of area 10 cm² the speed rises; find the pressure there. (ρ = 1000 kg m⁻³.)

The SI unit of viscosity is the poiseuille (Pa s); the cgs unit is the poise. Viscosity of liquids decreases with temperature, while that of gases increases. Honey is far more viscous than water.

MCQ Questions

1 Mark Questions

1. 1
   Define viscosity.
2. 2
   State Stokes' law.
3. 3
   Define terminal velocity.

2 Mark Questions

1. 1
   A sphere of radius 1 mm falls through water of viscosity 0.001 Pa s at a terminal velocity of 2 cm/s. Find the viscous force on it.
2. 2
   Why does the viscosity of liquids decrease with temperature but that of gases increase?
3. 3
   Explain how a parachute achieves a constant terminal velocity.

3 Mark Questions

1. 1
   State Stokes' law and derive an expression for the terminal velocity of a sphere falling through a viscous fluid.
2. 2
   A rain drop of radius 0.1 mm falls through air with viscosity 1.8 × 10⁻⁵ Pa s. Estimate its terminal velocity. (ρ_water = 1000 kg m⁻³, ρ_air ≈ 0, g = 10 m/s².)

Surface tension arises because molecules at the surface have fewer neighbours and hence higher potential energy — the liquid pulls them inward. It explains why a needle floats on water, why drops are spherical, and the rise (or fall) of a liquid in a narrow capillary tube: h = 2S cosθ/(ρgr). Water rises in a glass tube because it wets it (θ < 90°).

• Small liquid drops and soap bubbles are spherical because a sphere has minimum surface area.
• Surface tension decreases with temperature.
• Capillary rise is greater in narrower tubes.

MCQ Questions

1 Mark Questions

1. 1
   Define surface tension. Write its SI unit.
2. 2
   Write the expression for the capillary rise of a liquid.
3. 3
   Why are small liquid drops spherical?

2 Mark Questions

1. 1
   Water rises to 4 cm in a capillary tube of radius 0.5 mm. Find the surface tension of water. (ρ = 1000 kg m⁻³, cosθ ≈ 1, g = 10 m/s².)
2. 2
   Explain why a needle can float on the surface of water although steel is denser than water.
3. 3
   State the factors on which the capillary rise of a liquid depends.

3 Mark Questions

1. 1
   Explain the phenomenon of surface tension and derive the ascent of a liquid in a capillary tube.
2. 2
   A liquid of surface tension 0.072 N m⁻¹ rises to 2 cm in a capillary tube of radius 0.36 mm. Find the density of the liquid. (cosθ = 1, g = 10 m/s².)

Exercises (NCERT Q9.1–Q9.20)

• 9.1) 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.
• 9.2) 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.) (c) Surface tension of a liquid is independent of the area of the surface (d) Water with detergent dissolved in it should have small angles of contact. (e) A drop of liquid under no external forces is always spherical in shape.
• 9.3) 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) (b) Viscosity of gases ... with temperature, whereas viscosity of liquids ... with temperature (increases / decreases) (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) (d) For a fluid in a steady flow, the increase in flow speed at a constriction follows (conservation of mass / Bernoulli's principle) (e) For the model of a plane in a wind tunnel, turbulence occurs at a ... speed for turbulence for an actual plane (greater / smaller)
• 9.4) 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 (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 (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.
• 9.5) 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?
• 9.6) 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.
• 9.7) 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.
• 9.8) 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 cm². What maximum pressure would the smaller piston have to bear?
• 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 cm of water in one arm and 12.5 cm of spirit in the other. What is the specific gravity of spirit?
• 9.10) 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)
• 9.11) Can Bernoulli's equation be used to describe the flow of water through a rapid in a river? Explain.
• 9.12) Does it matter if one uses gauge instead of absolute pressures in applying Bernoulli's equation? Explain.
• 9.13) 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 × 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].
• 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⁻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.
• 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?
• 9.16) 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?
• 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 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?
• 9.18) Figure 9.21 (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.
• 9.19) 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 × 10⁻¹ N m⁻¹. The atmospheric pressure is 1.01 × 10⁵ Pa. Also give the excess pressure inside the drop.
• 9.20) 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 × 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).

Answers at a glance

1. 1
   Q9.1 → (a) P = hρg and the feet lie below a taller blood column (b) air density halves by 6 km; the tenuous upper air adds little pressure (c) hydrostatic pressure acts in all directions, so it is a scalar.
2. 2
   Q9.2 → (a) weak adhesion of mercury versus strong adhesion of water to glass (b) cohesive mercury forms drops, adhesive water spreads (c) it is a per-unit-length force (d) small θ gives fast capillary rise (e) a sphere minimizes surface area.
3. 3
   Q9.3 → (a) decreases (b) increases; decreases (c) shear strain; rate of shear strain (d) conservation of mass (e) greater.
4. 4
   Q9.4 → (a) faster air above lowers pressure (b) continuity: smaller area, higher speed (c) the needle area, not the thumb, controls rate (d) reaction to the momentum of the jet (e) the Magnus/Bernoulli sideways force curves the flight.
5. 5
   Q9.5 → Pressure = 6.24 × 10⁶ N m⁻².
6. 6
   Q9.6 → The wine column would be 10.5 m high.
7. 7
   Q9.7 → Yes: ocean pressure at 3 km is only 2.94 × 10⁷ Pa, well below 10⁹ Pa.
8. 8
   Q9.8 → Maximum pressure on the smaller piston = 6.917 × 10⁵ Pa.
9. 9
   Q9.9 → Specific gravity of spirit = 0.8.
10. 10
    Q9.10 → The mercury levels differ by about 0.22 cm.
11. 11
    Q9.11 → No — a rapid is turbulent flow, and Bernoulli's equation requires streamline flow.
12. 12
    Q9.12 → No — the atmospheric term cancels; gauge and absolute pressures give the same difference.
13. 13
    Q9.13 → Pressure difference = 9.8 × 10² Pa; Reynolds number ≈ 0.3, so the flow is laminar.
14. 14
    Q9.14 → Lift ≈ 1.51 × 10³ N.
15. 15
    Q9.15 → Figure (a) is incorrect — pressure (and hence the liquid level) falls at the high-speed constriction.
16. 16
    Q9.16 → Speed of ejection ≈ 0.64 m s⁻¹.
17. 17
    Q9.17 → Surface tension = 2.5 × 10⁻² N m⁻¹.
18. 18
    Q9.18 → Figures (b) and (c) also support 4.5 × 10⁻² N, the same liquid and slider length giving the same force.
19. 19
    Q9.19 → Excess pressure = 310 Pa; total pressure inside ≈ 1.01 × 10⁵ Pa.
20. 20
    Q9.20 → Soap bubble excess = 20 Pa; air bubble at 40 cm depth: total pressure ≈ 1.06 × 10⁵ Pa.

MCQ Questions

1 Mark Questions

1. 1
   State Pascal's law for fluids at rest.
2. 2
   Define the buoyant force and state Archimedes' principle.
3. 3
   Define surface tension and give its SI unit.
4. 4
   State Bernoulli's theorem for a streamline flow of an ideal fluid.

2 Mark Questions

1. 1
   A submarine is at a depth of 20 m in water of density 1000 kg/m³. Find the pressure on it (g = 10 m/s², P₀ = 10⁵ Pa). (Ans: P = P₀ + ρgh = 3 × 10⁵ Pa)
2. 2
   Water flows in a horizontal pipe that narrows from 4 cm² to 1 cm². If the speed in the narrow part is 8 m/s, find the speed in the wide part. (Ans: v₁A₁ = v₂A₂ ⇒ v₁ = 2 m/s)
3. 3
   A drop of water of radius 1 mm falls with terminal velocity in air. If the viscosity of air is 1.8 × 10⁻⁵ Pa s and effective density is 1000 kg/m³, state the balance of forces. (Ans: 6πrv = (4/3)πr³ρg)

3 Mark Questions

1. 1
   Derive Bernoulli's equation P + ½ρv² + ρgh = constant for a streamline flow and state the assumptions made.
2. 2
   Derive the expression for the excess pressure inside a liquid drop and inside a soap bubble. (Ans: 2T/r; 4T/r)
3. 3
   Explain Stokes' law and derive the expression for the terminal velocity of a sphere falling in a viscous fluid. (Ans: vt = 2r²(ρ − σ)g/(9η))

5 Mark Questions

1. 1
   Describe the phenomenon of capillarity, derive the expression for the height of a liquid column in a capillary tube, and explain its dependence on radius and angle of contact. (Ans: h = 2T cos θ/(rρg))
2. 2
   Derive the equation of continuity and Bernoulli's equation, and explain with the help of a diagram how the venturi meter and the atomizer work on Bernoulli's principle.