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Chapter 1: Problem 8

If \(p\) is a pressure, \(V\) a velocity, and \(\rho\) a fluid density, what are the dimensions (in the \(M L T\) system) of (a) \(p / \rho\) (b) \(p V \rho,\) and (c) \(p / \rho V^{2} ?\)

Short Answer

Expert verified
The dimensions in M L T system are: (a) \(L^{2} T^{-2}\), (b) \(M^{2} L^{-4} T^{-3}\), and (c) \(M^{0} L^{0} T^{0}\) (which is dimensionless)

Step by step solution

01

Identify the Dimensions of the given Parameters

Let's start by identifying the dimensions in the \(M L T\) system: - Pressure (\(p\)), has units of force per area, which is \(Mass \cdot Acceleration / Area\). Thus, \(p\) has dimensions \(M L^{-1} T^{-2}\). - Velocity (\(V\)), is distance over time, which gives us dimensions of \(L T^{-1}\)- Density (\(\rho\)), is mass over volume, so its dimensions are \(M L^{-3}\)
02

Evaluate the Dimensions for p/蟻

For part (a), we want the dimensions of \(p / \rho = (M L^{-1} T^{-2}) / (M L^{-3})\). Canceling terms, we find that the units of \(p / \rho\) are \(L^{2} T^{-2}\)
03

Evaluate the Dimensions for pV蟻

For part (b), we want the dimensions of \(p V \rho = (M L^{-1} T^{-2}) \cdot (L T^{-1}) \cdot (M L^{-3})\). Multiplying these together, we find that the units of \(p V \rho\) are \(M^{2} L^{-4} T^{-3}\)
04

Evaluate the Dimensions for p/蟻V虏

For part (c), we want the dimensions of \(p / (\rho V^{2}) = (M L^{-1} T^{-2}) / ((M L^{-3}) \cdot (L T^{-1})^{2})\). Dividing these terms, we find that the units of \(p / (\rho V^{2})\) are \(M^{0} L^{0} T^{0}\), or dimensionless

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Key Concepts

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

Pressure Units
Pressure is a fundamental concept in physics and engineering, and it is defined as the amount of force exerted per unit area. Understanding pressure units requires us to break it down into its dimensional form in terms of mass (M), length (L), and time (T). In the MLT system, pressure (\(p\)) has the dimensions:
  • Mass (\(M\)): This represents the amount of matter in an object. In the context of pressure, it relates to the mass required to exert force over an area.
  • Length (\(L\)): Pressure deals with area, which is a length squared. Hence, we use negative dimensions for length, \(L^{-1}\), as it refers to area being inversely related to pressure.
  • Time (\(T\)): The force is mass times acceleration, where acceleration is velocity over time squared, thus resulting in the dimension of time as \(T^{-2}\).
Therefore, the dimensional units of pressure are \(ML^{-1}T^{-2}\). Understanding these dimensions helps in performing operations like converting one unit of pressure to another or evaluating the effect of pressure in various equations.
Velocity Dimensions
Velocity is another important concept, especially in kinematics and fluid dynamics. Velocity is simply the speed of an object with a given direction, and it measures the change of position with respect to time.In dimensional terms within the MLT system, velocity (\(V\)) is expressed as:
  • Length (\(L\)): This refers to the displacement or distance traveled, contributing as a direct factor in velocity dimensions.
  • Time (\(T\)): Time is crucial here as velocity is associated with the rate of change, leading to the dimension \(T^{-1}\).
Hence, the dimensional representation of velocity becomes \(LT^{-1}\). This dimensional analysis is helpful when solving problems related to motion or comparing different velocities, as it provides insight into their relationships in equations and mathematical models.
Density in Fluid Mechanics
Density is an essential property in fluid mechanics that describes how much mass is contained in a given unit volume. It is crucial for understanding the behavior of fluids under different conditions.The dimensions of density (\(\rho\)) in the MLT system are:
  • Mass (\(M\)): This reflects the amount of substance present, making it a direct component of density.
  • Volume: As volume is a measure of space occupied (a cubic length), this leads to a dimension for length of \(L^{-3}\), where it inversely affects density.
Thus, density is expressed dimensionally as \(ML^{-3}\). Understanding the dimensionality of density is particularly important in fluid mechanics for calculating buoyancy, pressure differences, and various fluid flow properties.

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Most popular questions from this chapter

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