Speed of Sound in Gases: Dimensional Analysis

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Questions and Answers

What is the value of the exponent x in the formula for the speed of sound?

1
2
0
1/2 (correct)

Which equation represents the relationship between the exponents x, y, and z for the speed of sound?

x - 3y + z = -1 (correct)
x + 3y + z = -1
x - 2y + z = -1
x + 2y + z = -1

What is the value of the exponent y?

0
1/2
-1/2 (correct)
-1

When applying dimensional analysis to the speed of sound, what is the dimension of pressure?

<p>$M L^{-2} T^{-2}$ (A)</p> Signup and view all the answers

In the dimensional equation for speed, the term representing volume contributes what value to the overall dimensional analysis?

<p>$L^{3}$ (D)</p> Signup and view all the answers

Flashcards

Dimensional Analysis Variables

Using units to determine the relationship between physical quantities in an equation.

Speed of Sound Formula

V = p^x ρ^y V^z where V is speed, p is pressure, ρ is density and V is volume, and x, y, z are exponents.

Units of Pressure

Pressure is measured in kg/(m*s^2) or ML^-1 T^-2

Units of Density

Density is measured in kg/m^3 or ML^-3

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Dimensional Analysis Exponents

Finding exponents (x, y, z) that make the units on both sides of an equation match.

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Study Notes

Speed of Sound in a Gas

Sound speed (V) in a gas, pressure (p), density (ρ), and volume (V) are related.
Dimensional analysis is used to find the exponents x, y, and z in the equation V = p^xρ^yV^z.

Dimensional Analysis

Pressure (p) = mass (m) × acceleration (a) / area (A)
p = kg⋅m^-1⋅s^-2 = M L^-1 T^-2
Density (ρ) = mass (m) / volume (V)
ρ = kg⋅m^-3 = M L^-3
Volume (V) = length³ = L³
Speed (V) = length / time = L T^-1
Substituting the units into the equation V = p^xρ^yV^z gives M^x L^-x T^-2x × M^y L^-3y × L^z = M L T^-1
Equating the exponents produces a system of equations to solve for x, y, and z.

Solving for Exponents

Simplifying the equation gives:
- For mass (M): x + y = 1
- For length (L): -x - 3y + z = 1
- For time (T): -2x = -1
Solving the system of equations:
- x = 1/2
- y = 1/2
- z = 0
The equation for the speed of sound in a gas is V = √(γp/ρ).

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Description

Explore the relationship between sound speed, pressure, density, and volume in gases through dimensional analysis. This quiz will guide you in deriving the exponents x, y, and z in the equation relating these physical properties. Test your understanding of pressure, density, and how they influence sound speed.