Section 1.7: Mixed Practice (Fluids)
This section provides a range of problems covering pressure, density, buoyancy, and fluid dynamics. Use these to reinforce concepts from Sections 1.1–1.6.
Problem 1
A scuba diver descends to a depth of 25 m in seawater (\( \rho = 1025 \,\text{kg/m}^3 \)). What is the absolute pressure at this depth?
\( P = P_0 + \rho g h \)
\( = 1.01\times 10^5 + 1025 \cdot 9.8 \cdot 25 \approx 3.52 \times 10^5 \,\text{Pa} \)
Problem 2
A cylinder of radius 0.05 m and height 0.20 m floats upright in oil of density \( 850 \,\text{kg/m}^3 \). If its mass is 1.8 kg, what fraction of its height is submerged?
Weight = \( mg = 1.8 \cdot 9.8 = 17.64 \,\text{N} \)
Cross-sectional area = \( \pi r^2 = \pi (0.05)^2 = 0.00785 \,\text{m}^2 \)
Submerged height = \( h = \frac{W}{\rho g A} = \frac{17.64}{850 \cdot 9.8 \cdot 0.00785} \approx 0.27 \,\text{m} \)
Since the total height is 0.20 m, the cylinder sinks completely and would not float upright. (Answer: it sinks.)
Problem 3
Water flows through a pipe that narrows from diameter 0.20 m to 0.10 m. If the speed in the wide section is 1.5 m/s, what is the speed in the narrow section?
Continuity: \( A_1 v_1 = A_2 v_2 \)
\( v_2 = \frac{A_1}{A_2} v_1 = \left(\frac{0.20^2}{0.10^2}\right) \cdot 1.5 = 4 \cdot 1.5 = 6.0 \,\text{m/s} \)
Problem 4
Air (density \( 1.2 \,\text{kg/m}^3 \)) flows through a horizontal pipe. At one point, the pipe narrows and the velocity increases from 4.0 m/s to 12.0 m/s. Find the pressure difference between the two points.
Bernoulli: \( \Delta P = \tfrac{1}{2}\rho (v_2^2 - v_1^2) \)
\( = 0.5 \cdot 1.2 \cdot (144 - 16) = 0.6 \cdot 128 = 76.8 \,\text{Pa} \)