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What is Bernoulli's principle?

Class 11 PhysicsNCERT chapter: Mechanical Properties of Fluidsइसे हिंदी में पढ़ें

Exam-ready answer

Bernoulli's principle states that as we move along a streamline, the sum of the pressure (P), the kinetic energy per unit volume (½ρv²) and the potential energy per unit volume (ρgh) remains constant: P + ½ρv² + ρgh = constant. It holds for steady flow of an incompressible, non-viscous fluid. Where the speed is higher, the pressure is lower.

Explanation

Consider a fluid flowing steadily through a pipe of varying cross-section and height. Pressure from the fluid behind pushes a small volume forward and does work on it, while pressure from the fluid ahead does negative work. Applying the work-energy theorem, the net work done by these pressure forces equals the change in kinetic energy plus the change in gravitational potential energy of that volume of fluid.

Dividing by the volume gives P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, which is Bernoulli's equation. Here ρ is the density of the fluid. For a horizontal pipe, h is the same at both points, so the equation reduces to P + ½ρv² = constant. By the equation of continuity, the fluid speeds up where the pipe is narrow, so the pressure there is lower.

Dynamic lift: an aircraft wing (aerofoil) is shaped so that air flows faster over its upper surface than below it. Faster flow means lower pressure on top, so there is a net upward force on the wing. The same idea explains the swing of a spinning ball. Bernoulli's equation is an ideal model: it ignores viscosity, so real fluids lose some energy to internal friction.

Points to remember

  1. Statement: along a streamline, P + ½ρv² + ρgh = constant.
  2. Terms: pressure, kinetic energy per unit volume and potential energy per unit volume.
  3. Conditions: steady (streamline) flow, incompressible and non-viscous fluid.
  4. Derived by applying the work-energy theorem to a fluid element.
  5. Higher speed means lower pressure; this gives dynamic lift on an aerofoil.

Exam tip

Write the equation with each term named, list the three conditions (steady, incompressible, non-viscous), and draw a pipe of varying cross-section and height marking P₁, v₁, h₁ and P₂, v₂, h₂.

Questions students also ask

What is a streamline?
A streamline is the path followed by a fluid particle in steady flow. The tangent at any point on it gives the direction of the fluid velocity there. No two streamlines can cross, because a particle cannot have two velocities at one point.
What is Torricelli's law?
It gives the speed of efflux of a liquid from a small hole in an open tank. Applying Bernoulli's equation at the liquid surface and at the hole gives v = √(2gh), where h is the depth of the hole below the surface. This is the speed a body gains falling freely through height h.
Why are roofs sometimes blown off during a storm?
Strong wind blows fast over the roof, so the air pressure above it falls. The air inside the house is nearly still and stays at higher pressure. This pressure difference produces a large upward force that can lift a weak roof.

Chapter labels help you find the topic in your textbook. Check your own edition: chapter names and order can change. These are Know.it explanations, not official NCERT solutions.