Bernoulli’s Principle – Friendly Notes

1. Flow Basics

A fluid in smooth, steady motion forms streamlines. Two streamlines never cross, so a particle always follows one clear path. If you slice the flow with imaginary cross-sections, the same amount of fluid crosses each slice every second. For an incompressible fluid, the product of area and speed stays fixed:

\(A v = \text{constant}\)

This means narrow parts of a pipe force the fluid to speed up, while wide parts let it slow down.:contentReference[oaicite:0]{index=0}

2. Bernoulli’s Equation

A moving fluid juggles three kinds of energy per unit volume: pressure \(P\), kinetic energy \(\frac{1}{2}\rho v^{2}\), and gravitational potential energy \(\rho g h\). Along one streamline their sum never changes:

\(P + \dfrac{1}{2}\rho v^{2} + \rho g h = \text{constant}\)

If you compare two points (1 and 2) you can also write

\(P_{1} + \dfrac{1}{2}\rho v_{1}^{2} + \rho g h_{1} = P_{2} + \dfrac{1}{2}\rho v_{2}^{2} + \rho g h_{2}\)

The relation comes straight from energy conservation: pressure work changes the fluid’s speed and height.:contentReference[oaicite:1]{index=1}

Needed conditions

• Incompressible fluid (density \(\rho\) stays the same).
• Negligible internal friction (zero viscosity).
• Steady, non-turbulent flow.

When the fluid rests (\(v=0\)), Bernoulli’s equation reduces to the familiar hydrostatic result \(P_{1}-P_{2}= \rho g(h_{2}-h_{1})\).:contentReference[oaicite:2]{index=2}

3. Speed of Efflux (Torricelli’s Law)

Let a tank have a small hole a depth \(h\) below the free surface. If the tank is open to the atmosphere, the liquid shoots out with speed

\(v = \sqrt{2 g h}\)

This is the same speed an object would gain in free fall through height \(h\). If the tank is pressurized, replace \(2 g h\) by \((P-P_{a})/\rho\).:contentReference[oaicite:3]{index=3}

4. Dynamic Lift

4.1 Non-spinning ball

Air flows symmetrically around a smooth ball that does not spin, so the pressures above and below match and the net vertical force is zero.:contentReference[oaicite:4]{index=4}

4.2 Spinning ball (Magnus effect)

A spinning ball drags nearby air, speeding the flow on one side and slowing it on the other. Faster flow means lower pressure. The resulting pressure difference pushes the ball toward the low-speed side, creating the sideways or upward “swerve” players love.:contentReference[oaicite:5]{index=5}

4.3 Aircraft wing

A wing (aerofoil) tilts so that air travels farther and faster over the top surface than underneath. The upper pressure drops, the lower pressure rises, and the difference produces lift that can balance the plane’s weight. In one worked example, a fully loaded airplane of mass \(3.3\times10^{5}\,\text{kg}\) with \(500\,\text{m}^{2}\) of wing area needs a pressure difference of about \(6.5\times10^{3}\,\text{N\,m}^{-2}\). A mere 8 % increase in air speed over the top surface achieves this.:contentReference[oaicite:6]{index=6}

5. Quick Practice Question

Try this: Water exits a hole \(0.5\text{ m}\) below the surface. What speed does it have at the hole? (Answer: \(v \approx 3.1\,\text{m/s}\)).

6. High-Yield Ideas for NEET

• Continuity equation \(A v = \text{constant}\) and how area–speed changes affect pressure.
• Bernoulli’s equation and its energy interpretation.
• Torricelli’s law for liquid jets: \(v = \sqrt{2 g h}\).
• Magnus effect: lift on a spinning ball.
• Dynamic lift on wings and the basic pressure-speed argument for airplane flight.

Stay curious and keep practicing!