JEE/NEET Physics · Mechanical Properties of Fluids series · Part 4 of 7 · All parts →
- Continuity: A₁v₁ = A₂v₂ — narrow pipe, faster flow (mass booked-keeping)
- Bernoulli: P + ½ρv² + ρgh = constant along a streamline (energy conservation for fluids)
- Faster flow ↔ lower pressure — the counter-intuitive core
- Applications: aeroplane wings, atomisers, venturi meters, swing bowling
- Torricelli: efflux speed from a hole = √(2gh) — free fall from the surface
Blow between two balloons and they move TOWARD each other. Spray perfume and fluid climbs the tube. Airplanes climb into the sky. All the same rule: where fluid speeds up, its pressure drops. Part 4 of the Mechanical Properties of Fluids series.
- Continuity: the booking rule
- Bernoulli: the energy budget
- Fast means thin (in pressure)
- Torricelli’s hole
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
Continuity: The Booking Rule
Fluid isn’t created or destroyed in a pipe: what enters must leave. Per second, that’s volume rate = Av, so A₁v₁ = A₂v₂. Put your thumb over the hose and the jet speeds up — smaller area, larger velocity, same litres per second.
Bernoulli: The Energy Budget
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| P | static pressure at the point | Pa |
| ½ρv² | dynamic pressure — motion’s share | Pa |
| ρgh | height’s share | Pa |
Fast Means Thin (in Pressure)
At the same height, if v rises, P must fall — the fluid pays for speed out of its pressure account. This single trade explains lift, swing, atomisers and roof-stripping storms.
Torricelli’s Hole
Water leaving a small hole at depth h exits at v = √(2gh) — exactly the speed of a stone dropped from the surface: potential energy converting to kinetic, unchanged by being liquid.
Solved Examples
v₂ = A₁v₁/A₂ = 4×2/1 = 8 m/s.
✔
Answer: 8 m/s
v = √(2×10×5) = 10 m/s.
✔
Answer: 10 m/s
ΔP = ½ρ(v₁² − v₂²) = ½×1000×(4 − 36) = −16,000 Pa.
The fluid’s pressure account paid for the speed-up — 0.16 atm lighter where it’s faster. ✔
Answer: Drops by 16 kPa
- ‘Fast fluid pushes harder.’ Opposite: in level flow, faster means LOWER pressure — the exam’s favourite reversal.
- Using Bernoulli between different streamlines or turbulent flows. It holds along a smooth streamline, steady flow, ideal fluid.
- Continuity with density changes. A₁v₁ = A₂v₂ assumes incompressible flow (liquids; gases at low speed).
- Applying Torricelli with the tank draining fast. √(2gh) assumes the surface level is steady (large tank, small hole).
This Physics in Your Daily Life
- Aeroplane wings — curved tops force air to travel faster above: lower pressure above, lift below. Bernoulli carrying 400 tonnes across oceans (with Newton’s third law helping — the full story).
- Shower curtains sucking inward — fast water/air inside, still air outside: the pressure difference pushes the curtain at you.
- Perfume atomisers and insecticide sprays — fast air over a tube lowers pressure there; liquid climbs up and shatters into spray.
- Swing bowling and banana kicks — the spinning ball drags air on one side: different speeds, different pressures, curved flight.
- Roofs lifting off in storms — fast wind OVER the roof lowers its pressure; the slower attic air underneath pushes the roof off.
A river narrows and the current quickens — nothing pushed it; it MUST quicken to get the same water through. But speeding up costs energy, and in a level riverbed the only account holding spendable energy is pressure. Speed is bought with pressure, always: the fluid pays its own toll.
Water doubling speed 2→4 m/s: Δ(½ρv²) = ½×1000×(16−4) = 6000 Pa of pressure spent. Over a 10 cm² wing section that’s 6 N of lift per patch — scale to wing-size and aeroplanes fly. The books balance per cubic metre, everywhere.
Picture three stacked bars at each point of a pipe — pressure, speed, height shares — of equal total height. At a narrowing, the speed bar grows and the pressure bar visibly shrinks, the total unchanged. The bar chart IS Bernoulli.
Practice set (answers hidden — try first)
(NEET-level) Area halves: velocity
(JEE Main-level) Water speeds 1→3 m/s (level): ΔP =
(NEET-level) Hole 1.25 m below the surface (g=10): exit speed =
(Concept) Two balloons, blow between them:
(JEE Main-level) 4 cm² at 3 m/s → 12 m²/s flow rate; at 2 cm²: v =
- A₁v₁ = A₂v₂ — flow rate booked
- P + ½ρv² + ρgh = constant
- faster ↔ lower pressure (level flow)
- Torricelli: v = √(2gh)
- wing lift = pressure difference
- 🔁 continuity equation
- 🔁 Bernoulli terms and meaning
- 🔁 speed-pressure trade
- 🧠 Chant: ‘narrow-fast-cheap(in pressure)’.
- 🧠 Reverse intuition: ‘speed is bought with pressure’.
- 🏠 Daily: shower curtain attack = Bernoulli.
- 🏠 Daily: atomisers lift liquid with fast air.
Quick revision
- Continuity: A₁v₁ = A₂v₂ — narrow pipe, faster flow (mass booked-keeping)
- Bernoulli: P + ½ρv² + ρgh = constant along a streamline (energy conservation for fluids)
- Faster flow ↔ lower pressure — the counter-intuitive core
- Applications: aeroplane wings, atomisers, venturi meters, swing bowling
- Torricelli: efflux speed from a hole = √(2gh) — free fall from the surface
- Continuity: the booking rule
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