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Chapter 14: Problem 12

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

Short Answer

Expert verified
(a) 706.32 Pa; (b) 3158.82 Pa.

Step by step solution

01

Understand Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. It can be calculated using the formula:\[ P = \rho gh \]where \( P \) is the pressure, \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity (approximately 9.81 \( \text{m/s}^2 \)), and \( h \) is the height of the fluid column.
02

Calculate Gauge Pressure at Oil-Water Interface

First, calculate the pressure at the oil-water interface due to the oil layer:- Density of oil, \( \rho_{\text{oil}} = 600 \, \text{kg/m}^3 \)- Height of oil, \( h_{\text{oil}} = 0.120 \, \text{m} \)Substitute these into the formula:\[ P_{\text{interface}} = \rho_{\text{oil}} \times g \times h_{\text{oil}} \]\[ P_{\text{interface}} = 600 \, \text{kg/m}^{3} \times 9.81 \, \text{m/s}^2 \times 0.120 \, \text{m} \]\[ P_{\text{interface}} = 706.32 \, ext{Pa} \]
03

Calculate Gauge Pressure at the Bottom of the Barrel

Now, calculate the pressure at the bottom due to both the oil and water layers:- Density of water, \( \rho_{\text{water}} = 1000 \, \text{kg/m}^3 \)- Height of water, \( h_{\text{water}} = 0.250 \, \text{m} \)First, calculate the pressure due to the water:\[ P_{\text{water}} = \rho_{\text{water}} \times g \times h_{\text{water}} \]\[ P_{\text{water}} = 1000 \, \text{kg/m}^{3} \times 9.81 \, \text{m/s}^2 \times 0.250 \, \text{m} \]\[ P_{\text{water}} = 2452.5 \, ext{Pa} \]Next, add the pressures from both the oil and water:\[ P_{\text{bottom}} = P_{\text{interface}} + P_{\text{water}} \]\[ P_{\text{bottom}} = 706.32 \, ext{Pa} + 2452.5 \, ext{Pa} \]\[ P_{\text{bottom}} = 3158.82 \, ext{Pa} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gauge Pressure
Gauge pressure refers to the pressure of a fluid relative to the ambient atmospheric pressure. When you measure gauge pressure, you are essentially measuring the pressure above the local atmospheric level.
Unlike absolute pressure, which considers the pressure of the vacuum (zero pressure), gauge pressure simplifies analysis by removing atmospheric conditions from calculations.The formula for gauge pressure in a fluid is:
  • \( P = \rho gh \)
  • \( P \) is the gauge pressure
  • \( \rho \) is the density of the fluid
  • \( g \) is the acceleration due to gravity (approximately 9.81 \( \text{m/s}^2 \))
  • \( h \) is the height or depth of fluid column
Using this formula helps derive how pressure builds up depending on the fluid's characteristics and depth within a container or any structure. Gauge pressure is commonly used in applications where atmospheric pressure acts as the baseline.
Density
Density is a fundamental property of fluids, defined as the mass per unit volume. It indicates how much mass exists in a given space within the fluid, often expressed in kilograms per cubic meter (\( \text{kg/m}^3 \)).
In the context of fluid mechanics, density plays a crucial role in determining how substances interact when they come into contact with each other.

Why Density Matters in Fluid Mechanics:

  • Different layers of fluid with varying densities will naturally separate due to buoyancy effects, with denser fluids sinking beneath less dense ones.
  • The calculation of pressure, including gauge pressure, relies heavily on knowing the fluid's density:
    • A higher density will result in greater pressures at the same depth or height.
  • Density influences the force exerted by or on a fluid, impacting how objects within the fluid rise, float, or sink.
In practical applications, understanding density ensures accurate pressure calculations and fluid behavior predictions in various engineering and scientific fields.
Hydrostatic Pressure
Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases linearly with the depth of the fluid and is a function of the fluid's density, gravitational acceleration, and the depth of the fluid column.
This pressure is essential in determining how forces distribute within a fluid column and is critical for understanding and designing systems that hold or contain fluids.Here's why hydrostatic pressure is important:
  • Calculation of Forces: It assists in calculating the forces exerted by the fluid on the walls of containers, dams, and other structures.
  • Layered Fluids: When multiple fluid layers are present, hydrostatic pressure calculations help determine the contribution of each layer to the total pressure exerted at the base or interface.
  • Depth-Dependent: This property means that as the depth of a fluid increases, so too does the pressure, directly affecting submerged objects.
Using the formula \( P = \rho gh \), hydrostatic pressure thus influences gauge pressure, which omits atmospheric pressure but focuses solely on the pressure originating from the fluid column alone.
Oil-Water Interface
The oil-water interface is a crucial point where two immiscible liquids, oil and water, meet. This interface has unique characteristics pertinent to fluid mechanics due to the differing properties of the two fluids.
Understanding this interface is vital when calculating pressures and designing systems involving multiple fluids.

Important Characteristics of the Oil-Water Interface:

  • Density Difference: Oil typically has a lower density than water, which means it floats above the water layer. This density difference is what leads to a stratified layer formation.
  • Pressure Calculation: Measuring pressure at the oil-water interface exclusive of influences from above layers requires individual pressure considerations from each fluid separately:
    • At the interface, gauge pressure is calculated primarily from the oil layer sitting above.
  • Interfaces in Engineering: Recognizing changes in pressure and flow characteristics at such interfaces is crucial for engineering applications, such as in the design of fluid containers or systems where separation is necessary.
The stability and properties of an oil-water interface must be factored in for accurate pressure calculations and effective fluid system designs.

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Most popular questions from this chapter

A piece of wood is 0.600 \(\mathrm{m}\) long, 0.250 \(\mathrm{m}\) wide, and 0.080 \(\mathrm{m}\) thick. Its density is 600 \(\mathrm{kg} / \mathrm{m}^{3}\) . What volume of lead must be fastened underneath it to sink the wood in calm water so that its top is just even with the water level? What is the mass of this volume of lead?

SHM of a Floating Object. An object with height \(h\) , mass \(M,\) and a uniform cross-sectional area \(A\) floats upright in a liquid with density \(\rho\) . (a) Calculate the vertical distance from the surface of the liquid to the bottom of the floating object at equilibrium. (b) A downward force with magnitude \(F\) is applied to the top of the object. At the new equilibrium position, how much farther below the surface of the liquid is the bottom of the object than it was in part (a)? (Assume that some of the object remains above the surface of the liquid.) (c) Your result in part (b) shows that if the force is suddenly removed, the object will oscillate up and down in SHM. Calculate the period of this motion in terms of the density \(\rho\) of the liquid, the mass \(M,\) and cross-sectional area \(A\) of the object. You can ignore the damping due to fluid friction (see Section \(13.7 ) .\)

A hydrometer consists of a spherical bulb and a cylindrical stem with a cross- sectional area of 0.400 \(\mathrm{cm}^{2}\) (see Fig. 14.13a). The total volume of bulb and stem is \(13.2 \mathrm{cm}^{3} .\) When immersed in water, the hydrometer floats with 8.00 \(\mathrm{cm}\) of the stem above the water surface. When the hydrometer is immersed in an organic fluid, 3.20 \(\mathrm{cm}\) of the stem is above the surface. Find the density of the organic fluid. (Nore: This illustrates the precision of such a hydrometer. Relatively small density differences give rise to relatively large differences in hydrometer readings.)

You drill a small hole in the side of a vertical cylindrical water tank that is standing on the ground with its top open to the air. (a) If the water level has a height \(H,\) at what height above the base should you drill the hole for the water to reach its greatest distance from the base of the cylinder when it hits the ground? (b) What is the greatest distance the water will reach?

A uniform lead sphere and a uniform aluminum sphere have the same mass. What is the ratio of the radius of the aluminum sphere to the radius of the lead sphere?
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