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Ch 12: Fluid Mechanics
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 12: Fluid MechanicsProblem 12a
Chapter 12, Problem 12a

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/m3. (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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To find the gauge pressure at the oil-water interface, use the formula for pressure due to a fluid column: \( P = \rho g h \), where \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity (approximately 9.81 m/s²), and \( h \) is the height of the fluid column. Here, \( \rho = 600 \text{ kg/m}^3 \) and \( h = 0.120 \text{ m} \).
Calculate the pressure at the oil-water interface using the given values: \( P_{\text{interface}} = 600 \times 9.81 \times 0.120 \). This will give you the gauge pressure at the interface.
To find the gauge pressure at the bottom of the barrel, first calculate the pressure due to the oil layer as done in the previous step.
Next, calculate the pressure due to the water layer. Use the same formula \( P = \rho g h \), but this time \( \rho = 1000 \text{ kg/m}^3 \) (density of water) and \( h = 0.250 \text{ m} \).
Add the pressure due to the oil layer and the pressure due to the water layer to find the total gauge pressure at the bottom of the barrel: \( P_{\text{bottom}} = P_{\text{interface}} + (1000 \times 9.81 \times 0.250) \).

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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 the pressure measured in a system minus the atmospheric pressure. In fluid mechanics, gauge pressure is used to determine the pressure exerted by a fluid at a specific depth, excluding atmospheric pressure.
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Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth and is calculated using the formula P = ρgh, where 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 determining pressure at different depths in the barrel.
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Density

Density is a measure of mass per unit volume, expressed in kg/m^3. It is a fundamental property of materials that affects how they interact with forces like gravity. In this problem, the density of oil is used to calculate the hydrostatic pressure exerted by the oil layer on the water beneath it, influencing the gauge pressure at the oil-water interface.
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Related Practice
Textbook Question

Oceans on Mars. Scientists have found evidence that Mars may once have had an ocean 0.500 km deep. The acceleration due to gravity on Mars is 3.71 m/s2. (a) What would be the gauge pressure at the bottom of such an ocean, assuming it was freshwater? (b) To what depth would you need to go in the earth's ocean to experience the same gauge pressure?

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Textbook Question

In intravenous feeding, a needle is inserted in a vein in the patient's arm and a tube leads from the needle to a reservoir of fluid (density 1050 kg/m3) located at height h above the arm. The top of the reservoir is open to the air. If the gauge pressure inside the vein is 5980 Pa, what is the minimum value of h that allows fluid to enter the vein? Assume the needle diameter is large enough that you can ignore the viscosity of the liquid.

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Textbook Question

You are designing a diving bell to withstand the pressure of seawater at a depth of 250 m. (a) What is the gauge pressure at this depth? (You can ignore changes in the density of the water with depth.)

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Textbook Question

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Textbook Question

You are designing a diving bell to withstand the pressure of seawater at a depth of 250 m. What is the gauge pressure at this depth? (You can ignore changes in the density of the water with depth.) At this depth, what is the net force due to the water outside and the air inside the bell on a circular glass window 30.0 cm in diameter if the pressure inside the diving bell equals the pressure at the surface of the water? (Ignore the small variation of pressure over the surface of the window.)

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Textbook Question

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