Section 6.8: Review Examples

This section provides a set of review problems to consolidate your understanding of fluid dynamics, pressure, Bernoulli’s principle, and related concepts.

Example 1: Pressure in a Fluid

A liquid exerts a pressure of 5000 Pa at a depth of 2 m. Find the density of the liquid. Take \( g = 9.8 \, \text{m/s²} \).

\( P = \rho g h \Rightarrow \rho = \frac{P}{g h} = \frac{5000}{9.8 \times 2} \approx 255.1 \, \text{kg/m³} \)

Example 2: Flow Rate with Continuity

Water flows through a pipe of area 0.03 m² at 4 m/s. If the pipe narrows to 0.01 m², find the velocity in the narrower section.

Using continuity: \( A_1 v_1 = A_2 v_2 \Rightarrow v_2 = \frac{A_1 v_1}{A_2} = \frac{0.03 \times 4}{0.01} = 12 \, \text{m/s} \)

Example 3: Bernoulli’s Principle

Water flows through a horizontal pipe. At a wide section, velocity is 2 m/s and pressure is 1.5 × 10⁵ Pa. At a narrow section, velocity is 6 m/s. Find the pressure in the narrow section. Density of water \( \rho = 1000 \, \text{kg/m³} \).

Bernoulli: \( P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \)
\( P_2 = P_1 + \frac{1}{2}\rho (v_1^2 - v_2^2) = 1.5\times10^5 + 0.5 \cdot 1000 (4 - 36) \)
\( P_2 = 1.5\times10^5 - 16{,}000 = 1.34 \times 10^5 \, \text{Pa} \)

Practice Problems

  1. A fluid with density 850 kg/m³ is 3 m deep. Find the pressure at the bottom.
  2. Water flows at 1.5 m/s in a pipe of radius 0.05 m. The pipe narrows to 0.02 m. Find the velocity in the narrow part.
  3. In a horizontal pipe, the pressure at one point is 2 × 10⁵ Pa and the velocity is 3 m/s. If velocity increases to 6 m/s at another point, find the new pressure (ρ = 1000 kg/m³).
  4. Explain qualitatively how viscosity affects fluid motion in a pipe.
  5. Describe the effect of narrowing a pipe on velocity and pressure according to Bernoulli’s principle.
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