Unit 8: Fluids

Pressure

• Definition: Pressure is force per unit area. $$P = \frac{F}{A}$$
  • SI Unit: Pascal (Pa), where $1 \text{ Pa} = 1 N/m^2$.
  • Pressure is a **scalar** (has no direction).
  • Acts equally in all directions in a fluid.
• Definition of Pressure with Perpendicular Force: $$P \equiv \frac{F_{\perp}}{A}$$
  • Only the **perpendicular component** of force contributes to pressure.
  • Any **parallel component** does not create pressure.

Fluid Pressure

• Pressure in a Fluid: $$P_{\text{fluid}} = \rho g h$$
  • **Depends on:**
  • $\rho$ - Density of the fluid.
  • $g$ - Gravitational field strength.
  • $h$ - Depth below the surface.
  • **Does NOT depend** on surface area.
• Total Pressure at Depth: $$P_{\text{total}} = P_{\text{atm}} + \rho g h$$
• Gauge and Absolute Pressure:
  • Gauge Pressure: $$P_{\text{gauge}} = \rho g h$$
  • Absolute Pressure: $$P_{\text{absolute}} = P_{\text{atm}} + P_{\text{gauge}}$$

Buoyancy and Archimedes’ Principle

• Buoyant Force: The **upward force** exerted by a fluid on a submerged object. $$F_B = \rho_f V_f g$$
  • Equal to the weight of the **fluid displaced** by the object.
  • Acts **opposite to gravity**.
• Archimedes’ Principle:
  • A floating object displaces **its own weight** in fluid.
  • A submerged object displaces **its own volume** in fluid.
• Floating and Sinking Conditions:
  • If **$F_B > mg$** → Object rises.
  • If **$F_B < mg$** → Object sinks.
  • If **$F_B = mg$** → Object remains at equilibrium.

Ideal Fluid Flow

• For a fluid to be considered **ideal**, it must satisfy:
  1. Nonviscous – No internal friction.
  2. Incompressible – Constant density.
  3. Steady Flow – Velocity at any point is constant over time.
  4. Irrotational – No turbulence.

Equation of Continuity (Mass Conservation)

$$A_1 v_1 = A_2 v_2$$

• $A$ is the cross-sectional area.
• $v$ is the velocity of the fluid.
• As the **area decreases**, the **velocity increases**.

Bernoulli’s Principle

• Bernoulli’s Equation: $$P + \frac{1}{2} \rho v^2 + \rho g y = \text{constant}$$
  • Describes the conservation of energy in a **steady, ideal** fluid.
  • As **velocity increases, pressure decreases** (inverse relationship).

Torricelli’s Theorem

• For fluid exiting a tank: $$v_2 = \sqrt{2 g h}$$
• Derived from Bernoulli’s equation assuming **no initial velocity** at the fluid surface.

Key Takeaways

• ✔ **Pressure increases with depth** but is independent of surface area.
• ✔ **Buoyant force is equal to the weight of displaced fluid** (Archimedes' Principle).
• ✔ **Fluids flow faster through narrower pipes** (Continuity Equation).
• ✔ **Faster fluid motion results in lower pressure** (Bernoulli’s Principle).
• ✔ **Fluid exits a hole at speed $\sqrt{2gh}$** (Torricelli’s Theorem).