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Magazine Chapter 1: Problem 120
A 7 -ft tall steel cylinder has a cross-sectional area of \(15 \mathrm{ft}^{2}\). At the bottom, with a height of \(2 \mathrm{ft}\), is liquid water, on top of which is a 4 -ft-high layer of gasoline. The gasoline surface is exposed to atmospheric air at 14.7 psia. What is the highest pressure in the water?
Short Answer
Expert verified
The highest pressure in the water is 68.68 psi.
Step by step solution
01
Understand the Problem
We are given a steel cylinder with a layer of water at the bottom and a layer of gasoline on top. The problem asks for the highest pressure in the water, where the pressure is due to the weight of the gasoline and water, plus atmospheric pressure at the gasoline surface.
02
Identify Layers and Pressures
The cylinder is divided into two layers: a 2-ft layer of water at the bottom and a 4-ft layer of gasoline on top. The surface of the gasoline is exposed to atmospheric pressure. We need to calculate the pressure at the bottom of the water layer.
03
Calculate Pressure due to Gasoline
Pressure due to gasoline is calculated using the formula: \( P =
ho g h \), where \( \rho \) is the gasoline density, \( g \) is the acceleration due to gravity (32.2 ft/s虏), and \( h \) is the height of the gasoline. Assume gasoline density is approximately 42 lb/ft鲁. Thus, \( P_{gasoline} = 42 \times 32.2 \times 4 \).
04
Calculate Pressure due to Water
Pressure due to water also uses the formula \( P =
ho g h \). The density of water is 62.4 lb/ft鲁. Thus, \( P_{water} = 62.4 \times 32.2 \times 2 \).
05
Add Atmospheric Pressure
The total pressure at the bottom of the water is the atmospheric pressure plus the pressures due to gasoline and water. Atmospheric pressure is 14.7 psi, so add this to the sum of the pressures calculated from the gasoline and water.
06
Calculate Total Pressure at Bottom
Calculate the total pressure at the bottom: \[ P_{total} = 14.7 + \frac{(42 \times 32.2 \times 4)}{144} + \frac{(62.4 \times 32.2 \times 2)}{144} \]Convert all pressures to psi for consistency (by dividing by 144 to change lb/ft虏 to psi).
07
Solve the Equation
Solving the above expression gives:Pressure from gasoline: \((42 \times 32.2 \times 4) \text{ becomes } 3765.6 \text{ lb/ft}虏\); converting to psi gives \(26.13 \text{ psi}\).Pressure from water: \((62.4 \times 32.2 \times 2) \text{ becomes } 4010.88 \text{ lb/ft}虏\); converting to psi gives \(27.85 \text{ psi}\).Adding these to the atmospheric pressure: \ \(14.7 + 26.13 + 27.85 = 68.68 \text{ psi}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Pressure in Liquids
Pressure in liquids is a fundamental concept that describes how the force exerted by a liquid acts perpendicularly to a surface. When you immerse an object in a fluid, the pressure exerted by the liquid is felt equally in all directions. This pressure is determined by several factors:
Water and gasoline, although both are liquids, have different densities, affecting how they individually contribute to pressure in a scenario like the one described in the exercise.
- The depth of the liquid
- The density of the liquid
- The gravitational force
Water and gasoline, although both are liquids, have different densities, affecting how they individually contribute to pressure in a scenario like the one described in the exercise.
Hydrostatic Pressure
Hydrostatic pressure is the pressure exerted by a fluid at any given point, due to the force of gravity. It depends on the depth of the fluid and is crucial in fluid statics, where fluids are at rest. The related mathematical formula is:
\[ P = \rho g h \]
Where:
\[ P = \rho g h \]
Where:
- \( P \) is the hydrostatic pressure
- \( \rho \) is density of the fluid
- \( g \) is the gravitational acceleration
- \( h \) is the height or depth of the fluid
Pressure Calculation
Pressure calculation within the setup involves breaking down the contributions to total pressure by each fluid layer along with atmospheric pressure. Each layer contributes to the pressure at the bottom of the cylinder:
- Calculate pressure from gasoline using its density, the gravitational constant, and height of 4 ft.
- Calculate pressure from water with its density, gravitational constant, and height of 2 ft.
Fluid Mechanics
Fluid mechanics is the study of fluids (liquids and gases) and the forces acting upon them. It is a branch of physics that deals with the behavior of fluid substances and is divided into fluid statics and fluid dynamics. In this context, we are focusing on fluid statics, where the fluids are at rest.
Some key principles of fluid mechanics involved in this exercise include:
Some key principles of fluid mechanics involved in this exercise include:
- Understanding fluid properties like density and how they affect pressure.
- Analyzing how depth influences pressure in a static fluid column.
- Recognizing the additive nature of atmospheric pressure to fluid pressure.
