Bernoulli’s Equation

Bernoulli's equation is essentially the Conservation of Energy applied to fluids. It relates pressure, velocity, and height.

$$ P_1 + \frac{1}{2}\rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g y_2 $$

• $P$: Static Pressure (Energy density from pressure)
• $\frac{1}{2}\rho v^2$: Dynamic Pressure (Kinetic energy per unit volume)
• $\rho g y$: Hydrostatic Pressure (Gravitational potential energy per unit volume)

Key Relationship

• High Velocity $\rightarrow$ Low Pressure
• Low Velocity $\rightarrow$ High Pressure

💡 Common Applications

• Torricelli’s Law (The Leaking Bucket): Calculate the speed ($v_1$) of water leaking from a hole at a distance $h$ below the surface.

  • Top surface: $P_{atm}$, $v \approx 0$ (assumed large tank), height $y_2$.
  • Hole: $P_{atm}$, velocity $v_1$, height $y_1$.
  • Result: $v_1 = \sqrt{2g(y_2 - y_1)} = \sqrt{2gh}$.
  • This is the same speed as an object dropped from free fall height h.

• Airplane Wings (Lift): Wings are curved on top. Air travels a longer distance over the top, forcing it to move faster ($v_{high}$). By Bernoulli's principle, higher speed means lower pressure ($P_{top} < P_{bottom}$). The pressure difference creates an upward lift force.

• Bunsen Burners & Perfume Atomizers: A stream of fast-moving gas decreases the pressure in the tube, which "sucks" the fuel or perfume up from the reservoir to be mixed with the air stream.

⚠️ Key Confusing Point: The "Lift" Misconception: Students often assume high pressure pushes fluids faster. Bernoulli teaches the opposite for moving fluids: in a horizontal pipe, if fluid speeds up (constriction), the pressure drops. This principle explains why airplane wings generate lift. """