class 11 notes

Class 11 Physics Notes Chapter 10 (Chapter 10) – Examplar Problems (English) Book

Philoid

Philoid

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Examplar Problems (English)
Detailed Notes with MCQs of Chapter 10: Mechanical Properties of Fluids from your NCERT Exemplar. This chapter is crucial not just for your Class 11 understanding but also forms a significant part of the syllabus for various government exams. The Exemplar problems often test the depth of your understanding, so pay close attention.

Chapter 10: Mechanical Properties of Fluids - Detailed Notes

1. Introduction to Fluids:

  • Fluids: Substances that can flow. This includes liquids and gases. They do not possess a definite shape and take the shape of the container.
  • Key Difference: Liquids are largely incompressible and have a definite volume, while gases are highly compressible and expand to fill the entire volume of the container.
  • Thrust: The total normal force exerted by a fluid at rest on a given surface in contact with it. Unit: Newton (N).
  • Pressure (P): Thrust per unit area. It's a scalar quantity.
    • Formula: P = F/A (where F is the normal force, A is the area)
    • SI Unit: Pascal (Pa) or N/m².
    • Other Units: atm (atmosphere), bar, torr (1 atm ≈ 1.013 x 10⁵ Pa, 1 bar = 10⁵ Pa, 1 torr ≈ 133.3 Pa).
  • Density (ρ): Mass per unit volume.
    • Formula: ρ = M/V
    • SI Unit: kg/m³.
    • Relative Density (Specific Gravity): Ratio of the density of a substance to the density of water at 4°C. It's a dimensionless quantity.
      • Relative Density = Density of substance / Density of water (1000 kg/m³)

2. Pressure in a Fluid:

  • Pressure Variation with Depth: In a fluid at rest, pressure increases with depth.
    • Formula: P = P₀ + hρg
      • P = Pressure at depth h
      • P₀ = Pressure at the surface (often atmospheric pressure)
      • h = Depth
      • ρ = Density of the fluid
      • g = Acceleration due to gravity
    • Gauge Pressure: The difference between the absolute pressure and atmospheric pressure (P - P₀ = hρg).
  • Atmospheric Pressure: Pressure exerted by the Earth's atmosphere. Measured using a barometer. Standard atmospheric pressure at sea level is approximately 1.013 x 10⁵ Pa.
  • Hydrostatic Paradox: The pressure at a certain horizontal level in a fluid at rest is the same in all directions and is independent of the shape or base area of the container, depending only on the depth (h), density (ρ), and g.

3. Pascal's Law:

  • Statement: Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.
  • Principle: Based on the fact that liquids are largely incompressible.
  • Applications:
    • Hydraulic Lift: Uses Pascal's law to lift heavy objects. F₂/A₂ = F₁/A₁. A small force F₁ applied to a small piston (area A₁) generates a large force F₂ on a large piston (area A₂). Mechanical Advantage = A₂/A₁.
    • Hydraulic Brakes
    • Hydraulic Press

4. Archimedes' Principle:

  • Statement: When a body is immersed wholly or partially in a fluid at rest, it experiences an upward thrust (buoyant force) equal to the weight of the fluid displaced by it.
  • Buoyant Force (F<0xE2><0x82><0x92>): F<0xE2><0x82><0x92> = V<0xE1><0xB5><0xA2><0xE1><0xB5><0xA3><0xE1><0xB5><0xA1> ρ<0xE1><0xB5><0x9B> g
    * V<0xE1><0xB5><0xA2><0xE1><0xB5><0xA3><0xE1><0xB5><0xA1> = Volume of the fluid displaced (equal to the volume of the submerged part of the object)
    * ρ<0xE1><0xB5><0x9B> = Density of the fluid
    * g = Acceleration due to gravity
  • Apparent Weight: Weight of the object in fluid = Actual Weight (in air) - Buoyant Force.
  • Law of Floatation: A body floats in a fluid if the buoyant force acting on it is equal to its weight. For a floating body: Weight of body = Weight of fluid displaced.
    • If ρ<0xE1><0xB5><0xA5><0xE1><0xB4><0x84><0xE1><0xB4><0x8B><0xE1><0xB5><0x89><0xE1><0xB4><0x84><0xE1><0xB5><0x97> < ρ<0xE1><0xB5><0x9B>, the body floats with a fraction submerged.
    • If ρ<0xE1><0xB5><0xA5><0xE1><0xB4><0x84><0xE1><0xB4><0x8B><0xE1><0xB5><0x89><0xE1><0xB4><0x84><0xE1><0xB5><0x97> = ρ<0xE1><0xB5><0x9B>, the body floats fully submerged.
    • If ρ<0xE1><0xB5><0xA5><0xE1><0xB4><0x84><0xE1><0xB4><0x8B><0xE1><0xB5><0x89><0xE1><0xB4><0x84><0xE1><0xB5><0x97> > ρ<0xE1><0xB5><0x9B>, the body sinks.

5. Fluid Dynamics (Fluids in Motion):

  • Streamline Flow (Laminar Flow): Each particle of the fluid follows the same path as the preceding particle, and the velocity of the particle at any point remains constant over time (though it can vary from point to point). Streamlines do not cross each other. Occurs at low speeds.
  • Turbulent Flow: Irregular, chaotic flow with eddies and vortices. Occurs at high speeds or when there are obstacles. Velocity changes erratically.
  • Critical Velocity (v<0xE1><0xB5><0x84>): The velocity above which streamline flow becomes turbulent. Depends on the fluid's viscosity, density, and the dimensions of the pipe/channel.
  • Equation of Continuity: Based on the conservation of mass for an incompressible fluid in streamline flow.
    • Formula: A₁v₁ = A₂v₂ or Av = constant
      • A = Cross-sectional area of the pipe
      • v = Speed of the fluid flow
    • Meaning: Where the pipe is narrower, the fluid speed is higher, and vice versa.

6. Bernoulli's Principle:

  • Statement: For the streamline flow of an ideal (incompressible, non-viscous) fluid, the sum of pressure energy per unit volume, kinetic energy per unit volume, and potential energy per unit volume remains constant along a streamline.
    • Formula: P + ½ρv² + ρgh = constant
      • P = Pressure energy per unit volume (Pressure)
      • ½ρv² = Kinetic energy per unit volume
      • ρgh = Potential energy per unit volume
  • Assumptions: Fluid is ideal (non-viscous, incompressible), flow is steady (streamline), flow is irrotational.
  • Applications:
    • Venturi-meter: Measures the flow speed of a fluid.
    • Atomizer/Sprayer: High-speed air creates low pressure, drawing liquid up.
    • Aerofoil Lift (Airplane Wing): Air travels faster over the curved top surface, creating lower pressure above the wing compared to below, resulting in an upward lift force.
    • Spinning Ball (Magnus Effect): A spinning ball drags air, creating different relative speeds and pressures on opposite sides, leading to a curved path.
    • Blowing off of roofs during storms.

7. Viscosity:

  • Definition: The property of a fluid by virtue of which it opposes the relative motion between its adjacent layers. It's like internal friction in fluids.
  • Cause: Cohesive forces between liquid molecules.
  • Coefficient of Viscosity (η): Measure of viscosity.
    • Formula (Newton's Law of Viscosity): F = -ηA (dv/dx)
      • F = Viscous force
      • η = Coefficient of viscosity
      • A = Area of the layers in contact
      • dv/dx = Velocity gradient (change in velocity with distance perpendicular to flow)
    • SI Unit: Pa·s (Pascal-second) or N·s/m². CGS unit: Poise (1 Pa·s = 10 Poise).
  • Effect of Temperature: Viscosity of liquids decreases with increasing temperature; viscosity of gases increases with increasing temperature.
  • Stokes' Law: The viscous drag force (F<0xE1><0xB5><0x96>) acting on a small sphere moving with velocity (v) through a viscous fluid.
    • Formula: F<0xE1><0xB5><0x96> = 6πηrv
      • η = Coefficient of viscosity of the fluid
      • r = Radius of the sphere
      • v = Velocity of the sphere relative to the fluid
  • Terminal Velocity (v<0xE1><0xB5><0x97>): The constant maximum velocity acquired by a body falling through a viscous fluid when the net force (Weight - Buoyant Force - Viscous Drag) becomes zero.
    • Formula (for a sphere): v<0xE1><0xB5><0x97> = [2r²(ρ - σ)g] / 9η
      • ρ = Density of the sphere
      • σ = Density of the fluid
      • r = Radius of the sphere
      • η = Coefficient of viscosity of the fluid
      • g = Acceleration due to gravity

8. Surface Tension:

  • Definition: The property of a liquid surface film to contract and behave like a stretched elastic membrane. It arises due to cohesive forces between liquid molecules.
  • Surface Tension (S or T): The tangential force acting per unit length on an imaginary line drawn on the free surface of the liquid.
    • Formula: S = F/L
    • SI Unit: N/m or J/m².
  • Surface Energy: The potential energy per unit area of the surface film. Numerically equal to surface tension.
    • Work Done (W) to increase surface area by ΔA: W = S × ΔA
  • Angle of Contact (θ): The angle between the tangent to the liquid surface at the point of contact and the solid surface inside the liquid.
    • If θ < 90° (e.g., water and glass), the liquid wets the surface, meniscus is concave. Adhesive forces > Cohesive forces.
    • If θ > 90° (e.g., mercury and glass), the liquid does not wet the surface, meniscus is convex. Cohesive forces > Adhesive forces.
    • If θ = 90°, the liquid surface is horizontal.
  • Effect of Temperature: Surface tension generally decreases with increasing temperature (becomes zero at critical temperature).
  • Effect of Impurities: Soluble impurities can increase or decrease surface tension. Insoluble impurities (like grease) generally decrease surface tension. Detergents decrease surface tension.
  • Excess Pressure:
    • Inside a liquid drop: ΔP = 2S/R
    • Inside a soap bubble (two surfaces): ΔP = 4S/R
    • Inside an air bubble in a liquid: ΔP = 2S/R (R is the radius)
  • Capillarity (Capillary Action): The phenomenon of rise or fall of a liquid in a narrow tube (capillary tube) compared to the surrounding level.
    • Cause: Due to surface tension and the balance between adhesive and cohesive forces.
    • Ascent Formula (for liquids that wet the glass, θ < 90°): h = (2S cosθ) / (rρg)
      • h = Height of liquid column
      • S = Surface tension
      • θ = Angle of contact
      • r = Radius of the capillary tube
      • ρ = Density of the liquid
      • g = Acceleration due to gravity
    • If θ > 90°, cosθ is negative, h is negative, indicating a capillary depression (e.g., mercury in glass).

Key Takeaways for Exams:

  • Understand the difference between pressure and thrust.
  • Be comfortable with pressure calculations at depth (hρg).
  • Know Pascal's law and its application in hydraulic systems (F₁/A₁ = F₂/A₂).
  • Master Archimedes' principle: Buoyant force = weight of displaced fluid. Understand conditions for floating and sinking.
  • Equation of Continuity (Av = constant) is fundamental for flow rate problems.
  • Bernoulli's principle (P + ½ρv² + ρgh = constant) is crucial – understand its terms and applications (lift, Venturi, etc.). Remember the assumptions (ideal fluid, streamline flow).
  • Viscosity: Know Stokes' Law (6πηrv) and the formula for terminal velocity. Understand how viscosity changes with temperature for liquids and gases.
  • Surface Tension: Understand its origin, definition (F/L), surface energy (S x ΔA), excess pressure formulas (2S/R, 4S/R), and capillarity (h = 2S cosθ / rρg). Know the role of the angle of contact.

Multiple Choice Questions (MCQs)

  1. A hydraulic 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, and that of the smaller piston is 10 cm². What is the minimum pressure the smaller piston has to bear? (Assume g ≈ 9.8 m/s²)
    a) 6.92 x 10⁵ Pa
    b) 6.92 x 10⁶ Pa
    c) 7.50 x 10⁵ Pa
    d) 1.63 x 10⁷ Pa

  2. A large open tank has two holes in the wall. One is a square hole of side L at a depth y from the top and the other is a circular hole of radius R at a depth 4y from the top. When the tank is completely filled with water, the quantities of water flowing out per second from both holes are the same. Then, R is equal to:
    a) L / √(2π)
    b) 2πL
    c) L
    d) L / (2π)

  3. Bernoulli's principle is based on the conservation of:
    a) Mass
    b) Momentum
    c) Energy
    d) Angular Momentum

  4. A spherical ball of radius 'r' falls through a viscous liquid with a terminal velocity 'v'. If another spherical ball of radius '2r' but same density falls through the same liquid, its terminal velocity will be:
    a) v
    b) 2v
    c) 4v
    d) 8v

  5. Water rises to a height 'h' in a capillary tube. If the length of the capillary tube above the surface of water is made less than 'h', then:
    a) Water will rise up to the top of the capillary tube and stay there.

    • b) Water will rise up to the top of the capillary tube and overflow like a fountain.
      c) Water will rise up to a height slightly less than the length of the tube and the angle of contact will increase.
      d) Water will not rise at all.

  6. The excess pressure inside a soap bubble of radius R is proportional to:
    a) R
    b) 1/R
    c) R²
    d) 1/R²

  7. Two liquids of densities ρ₁ and ρ₂ (ρ₂ = 2ρ₁) are filled up behind a square wall of side 'l' as shown in the figure (ρ₁ up to height l/2, ρ₂ from l/2 to l). The ratio of the force due to ρ₁ to the force due to ρ₂ acting on the wall is: (Assume atmospheric pressure is ignored)
    a) 1/2
    b) 1/4
    c) 1/7
    d) 1/8

  8. Which of the following phenomena is NOT primarily explained by surface tension?
    a) Floating of a needle on water
    b) Spherical shape of raindrops
    c) Capillary rise
    d) Working of a hydraulic lift

  9. When the temperature of a liquid increases, its viscosity:
    a) Increases
    b) Decreases
    c) Remains unchanged
    d) First increases then decreases

  10. According to the equation of continuity (A₁v₁ = A₂v₂), when water flowing through a pipe enters a narrower section, its speed increases. What happens to its pressure according to Bernoulli's principle (assuming horizontal pipe)?
    a) Pressure increases
    b) Pressure decreases
    c) Pressure remains constant
    d) Pressure becomes zero

Let me know if you need clarification on any specific point or want to discuss some Exemplar problems related to these concepts. Good luck with your preparation!

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