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Ch 12: Fluid Mechanics
All textbooksYoung & Freedman Calc 14th EditionCh 12: Fluid MechanicsProblem 12
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Chapter 12, Problem 12

A barrel contains a 0.120-m layer of oil floating on water that is 0.250 m deep. The density of the oil is 600 kg/m^3. (a) What is the gauge pressure at the oil–water interface? (b) What is the gauge pressure at the bottom of the barrel?

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1
Identify the densities and depths of the fluids. For oil, the density (\(\rho_{oil}\)) is 600 kg/m^3 and the depth (\(h_{oil}\)) is 0.120 m. For water, the density (\(\rho_{water}\)) is typically 1000 kg/m^3 and the depth (\(h_{water}\)) is 0.250 m.
Calculate the gauge pressure at the oil-water interface using the formula for pressure due to a fluid column, \(P = \rho \cdot g \cdot h\), where \(\rho\) is the density of the fluid, \(g\) is the acceleration due to gravity (approximately 9.81 m/s^2), and \(h\) is the height of the fluid column. Here, use the density and depth of the oil.
To find the gauge pressure at the bottom of the barrel, consider the pressure contributions from both the oil and the water. First, calculate the pressure at the bottom of the oil layer as done in step 2.
Add the pressure due to the water layer to the pressure at the bottom of the oil layer. Use the same pressure formula, but now with the density and depth of the water. The total pressure at the bottom of the barrel is the sum of the pressures due to both the oil and the water.
Remember that gauge pressure is the pressure relative to atmospheric pressure. Since atmospheric pressure acts on the surface of the oil, it does not affect the gauge pressure calculations in this scenario.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. It is calculated by subtracting atmospheric pressure from the absolute pressure. In fluid mechanics, gauge pressure is important for understanding how pressure varies within a fluid column due to the weight of the fluid above a given point.
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Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the weight of the fluid above it. It is given by the formula P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. This concept is crucial for calculating pressures at different depths in a fluid.
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Density and Buoyancy

Density is the mass per unit volume of a substance, and it plays a key role in buoyancy, which is the ability of an object to float in a fluid. The density of the oil and water determines how they interact at the interface, affecting the pressure calculations at different levels within the barrel. Understanding these properties is essential for solving problems involving layered fluids.
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Textbook Question
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