Section 1.1: Fluids

This section introduces fluid mechanics, including pressure, density, buoyancy, and fluid flow. We examine how fluids behave under different forces and conditions.

Key Equations & Concepts:
  • Density: \( \rho = \frac{m}{V} \)
  • Pressure: \( P = \frac{F}{A} \)
  • Hydrostatic Pressure: \( P = P_0 + \rho g h \)
  • Buoyant Force: \( F_b = \rho_{fluid} V_{displaced} g \)
  • Continuity Equation: \( A_1 v_1 = A_2 v_2 \)
  • Bernoulli's Equation: \( P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant} \)

Example 1

A 2 kg block is submerged in water. Find the buoyant force acting on it if the displaced volume is 0.002 m³.

\( F_b = \rho_{water} V g = 1000 \cdot 0.002 \cdot 9.8 = 19.6 \text{ N} \)

Example 2

Water flows through a pipe with cross-section 0.05 m² at 2 m/s. The pipe narrows to 0.02 m². Find the flow speed in the narrow section.

Using continuity equation: \( A_1 v_1 = A_2 v_2 \Rightarrow 0.05 \cdot 2 = 0.02 \cdot v_2 \Rightarrow v_2 = 5 \text{ m/s} \)

Practice Problems

  1. Calculate the pressure at 3 m depth in freshwater.
  2. A cube of side 0.1 m is submerged in water. Find the buoyant force.
  3. Water flows through a pipe: area 0.1 m², speed 1 m/s. The pipe narrows to 0.05 m². Find new speed.
  4. Air exerts 101 kPa at sea level. Find force on 0.2 m² surface.
  5. A metal block weighs 50 N in air and 30 N in water. Find buoyant force.
  6. Compute hydrostatic pressure at bottom of a 5 m oil tank (\( \rho = 850 \text{ kg/m³} \)).
  7. Explain why ships float even if made of steel.
  8. Velocity of fluid doubles. How does pressure change (Bernoulli)?
  9. A cylinder of radius 0.1 m, area 0.0314 m², has 200 N force applied. Find pressure.
  10. Water flows from wide section (0.03 m²) at 3 m/s to narrow (0.01 m²). Find speed.
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