Mechanical Properties of Fluids - Class 11 Physics

Fluids

Overview: Study of liquids and gases at rest (Hydrostatics) and in motion (Hydrodynamics).

1. Pressure & Pascal's Law

Pressure (P)

Normal force per unit area. Scalar quantity. Unit: Pascal (Pa).

P = F / A

Density (ρ): Mass per unit volume. ρ = M/V.

Pascal's Law

Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the vessel.

P₁ = P₂ ⇒ F₁/A₁ = F₂/A₂ (Hydraulic Lift)

Pressure with Depth

P = P_atm + ρgh

2. Archimedes' Principle

A body immersed in fluid experiences an upward Buoyant Force equal to the weight of the fluid displaced.

F_B = V_displaced × ρ_fluid × g

Law of Floatation: A body floats if Weight of body = Buoyant Force.

3. Fluid Dynamics

Continuity Equation

For conservation of mass in steady flow:

A₁v₁ = A₂v₂

Bernoulli's Principle

For ideal fluid (incompressible, non-viscous), sum of pressure, kinetic and potential energy density is constant.

P + ½ρv² + ρgh = constant

4. Viscosity & Surface Tension

Viscosity

Internal resistance to flow. Drag force given by Stokes' Law:

F = 6πηrv

Where η is coefficient of viscosity.

Terminal Velocity

Constant velocity achieved by falling body in viscous fluid.

v_t = 2r²(ρ - σ)g / 9η

Surface Tension (S)

Property to minimize surface area. Force per unit length.

S = F / L

Excess Pressure in Drop: P = 2S/R.

Excess Pressure in Bubble: P = 4S/R.

Numericals - Mechanical Properties of Fluids

Numericals

Pressure Depth

Q1. Calculate pressure at depth 10m in water. (g=10, P_atm=10⁵ Pa)

P = P_atm + ρgh

ρ = 1000 kg/m³, h=10

P = 10⁵ + 1000 × 10 × 10

P = 10⁵ + 10⁵

P = 2 × 10⁵ Pa = 2 atm

Hydraulic Lift

Q2. Pistons of hydraulic press have areas 10 cm² and 100 cm². Force of 20N on small piston. Force on large?

F1/A1 = F2/A2

20 / 10 = F2 / 100

2 = F2 / 100

F2 = 200 N

Continuity Equation

Q3. Water flows through pipe. At A, area=4cm², v=5m/s. At B, area=2cm². Find velocity at B.

A1 v1 = A2 v2

4 × 5 = 2 × v2

20 = 2 v2

v2 = 10 m/s

Velocity increases as area decreases.

Bernoulli's Theorem

Q4. Horizontal pipe. At point 1, P1=5000Pa, v1=2m/s. At point 2, v2=8m/s. Find P2. (ρ=1000)

P1 + ½ρv1² = P2 + ½ρv2² (h is constant)

5000 + 0.5×1000×4 = P2 + 0.5×1000×64

5000 + 2000 = P2 + 32000

7000 = P2 + 32000

P2 = -25000 Pa (Ideally P1 should be higher to avoid negative pressure).

Terminal Velocity

Q5. A rain drop radius 1mm falls. η_air = 1.8x10^-5. Ignore air density. Find v_t.

v_t = 2r²ρg / 9η

r=10^-3, ρ=1000

v_t = 2(10^-6)(1000)(10) / 9(1.8×10^-5)

v_t = 0.02 / 16.2×10^-5

v_t = 2000 / 16.2 ≈ 123 m/s

Surface Tension Energy

Q6. Work done to blow a soap bubble from radius 2cm to 3cm. (S = 0.03 N/m).

Soap bubble has 2 surfaces. Area = 2 × 4πR² = 8πR²

ΔArea = 8π (R2² - R1²)

ΔA = 8π (9 - 4) × 10^-4 = 40π × 10^-4

W = S × ΔA = 0.03 × 40π × 10^-4

W = 3.76 × 10^-4 J

Capillary Rise

Q7. Water rises to height h in capillary. If radius is halved, new height?

h = 2S cosθ / rρg

h ∝ 1/r

If r becomes r/2, h becomes 2h.

Height doubles.

Buoyancy

Q8. Block of wood (density 600) floats in water. Fraction submerged?

Weight of block = Buoyant force

V ρ_body g = V_sub ρ_water g

V_sub/V = ρ_body / ρ_water

Fraction = 600 / 1000 = 0.6

Torricelli's Law

Q9. Tank has hole at depth 5m. Velocity of efflux?

v = √(2gh)

v = √(2 × 10 × 5)

v = √100 = 10 m/s

Excess Pressure

Q10. Calculate excess pressure inside water drop of radius 1mm. S=0.07 N/m.

P = 2S/r

P = 2 × 0.07 / 10^-3

P = 0.14 × 1000

P = 140 Pa

Formulas & Facts - Fluids

Equations & Formulas

Concept	Formula
Pressure	P = F / A
Gauge Pressure	P_g = ρgh
Archimedes	F_B = V ρ_liq g
Continuity Eq	A₁v₁ = A₂v₂
Bernoulli's Eq	P + ½ρv² + ρgh = k
Torricelli (Efflux)	v = √(2gh)
Viscous Force	F = - η A (dv/dx)
Stokes' Law	F = 6πηrv
Terminal Velocity	v_t = 2r²(ρ-σ)g / 9η
Capillary Rise	h = 2S cosθ / rρg

50 NEET Facts

Key points for Mechanical Properties of Fluids.

1. Fluid definition Substance that can flow (Liquids and Gases). No Shear modulus.

2. Gauge Pressure Pressure excess over atmospheric pressure. P - P_a = hρg.

3. Hydrostatic Paradox Pressure depends only on depth h, not on the shape of the vessel.

4. Pascal's Law Pressure change is transmitted undiminished. Basis of Hydraulic brakes and lifts.

5. Archimedes' Principle Loss of weight of body = Weight of fluid displaced.

6. Floatation Condition A body floats if its density is less than or equal to liquid density.

7. Center of Buoyancy Point through which Buoyant force acts. It is the center of gravity of displaced fluid.

8. Streamline Flow Velocity at every point is constant in time. No turbulence.

9. Turbulent Flow Disordered flow. Occurs when Reynold's number > 3000.

10. Critical Velocity Velocity above which flow becomes turbulent.

11. Equation of Continuity Conservation of Mass. Av = constant. Narrower pipe -> Faster flow.

12. Bernoulli's Principle Conservation of Energy. Where speed is high, pressure is low (Aerofoil lift).

13. Dynamic Lift Upward force on airplane wings due to pressure difference (High speed on top, low pressure).

14. Magnu's Effect Curving of path of a spinning ball due to pressure difference created by spin.

15. Venturimeter Device to measure rate of flow of liquid using Bernoulli's principle.

16. Viscosity Internal friction between fluid layers. Liquids: decreases with temp. Gases: increases with temp.

17. Ideal Fluid Incompressible and Non-viscous (zero viscosity).

18. Terminal Velocity Constant velocity when Drag + Buoyancy = Weight.

19. Reynold's Number (Re) Re = ρvd/η. Re < 1000 Laminar. Re> 2000 Turbulent.

20. Surface Tension Force per unit length. Arises due to cohesive forces. Decreases with temperature.

21. Sphere Shape Raindrops are spherical to minimize surface area (Surface Tension).

22. Surface Energy Work done to increase surface area. E = S × ΔArea.

23. Cohesive Force Force between molecules of same substance (Water-Water).

24. Adhesive Force Force between molecules of different substances (Water-Glass).

25. Angle of Contact (θ) Angle between tangent to liquid surface and solid surface.

26. Wetting Liquid wets solid if θ < 90° (Adhesion> Cohesion).

27. Non-Wetting Mercury does not wet glass (θ > 90°). Cohesion > Adhesion.

28. Capillarity Rise or fall of liquid in narrow tube. Due to surface tension.

29. Ascent Formula h ∝ 1/r. Thinner tube, higher rise.

30. Excess Pressure Bubble P = 4S/R because bubble has 2 free surfaces (inside and outside).

31. Excess Pressure Drop P = 2S/R (One free surface).

32. Detergents Reduce surface tension of water, helping it penetrate pores and clean better.

33. Blood Pressure Measured using Sphygmomanometer (Gauge pressure). 120/80 mmHg.

34. Atmospheric Pressure Weight of air column. 1 atm = 1.013 × 10⁵ Pa = 760 torr.

35. Torricelli's Law Velocity of efflux is same as free fall from height h. v = sqrt(2gh).

36. Bunsen Burner Works on Bernoulli's principle. High speed gas creates low pressure, sucking in air.

37. Heart Attack Artery constriction speeds up blood (Continuity), lowering pressure (Bernoulli), causing artery to collapse.

38. Deep Sea Diver Experiences huge pressure. Nitrogen dissolves in blood (bends). Needs special suit.

39. Ice Floating Ice is less dense than water (anomalous expansion). 90% is submerged.

40. Ship Stability Ship is stable if Metacenter is above Center of Gravity.

41. Poiseuille's Eq Volume flow rate Q = πPr^4 / 8ηL. Very sensitive to radius (r^4).

42. Siphon Tube used to transfer liquid from high to low level. Works on atm pressure.

43. Barometer Measures atmospheric pressure. Height of mercury column (76 cm at sea level).

44. Specific Gravity Relative density. Density of substance / Density of water at 4°C. No unit.

45. Ball Pen Works on gravity and surface tension (capillary action for ink flow).

46. Insects on Water Can walk on water due to surface tension film.

47. Oil on Water Spreads if S_water > S_oil + S_interface. Typically oil spreads (calms waves).

48. Atomizer Spray gun. Uses Bernoulli's principle to suck liquid and spray it.

49. Blowing ROOF During storms, high wind speed over roof creates low pressure. Roof is lifted up.

50. Free Fall of Liquid In a falling vessel, effective g=0. Pressure does not increase with depth.

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