Dimensional Analysis: Finding Relation Between v, p and p

AI Thread Summary
The discussion focuses on using dimensional analysis to establish a relationship between the velocity of sound in a gas (v), its density (ρ), and pressure (p). The key equations provided express the dimensions of these variables: [v] = LT-1, [ρ] = ML-3, and [p] = ML-1T-2. The approach involves assuming a functional form v = ρ^α * p^β and equating the dimensions on both sides to solve for the powers α and β. The importance of ensuring that both sides of the equation have the same dimensions is emphasized as a fundamental principle of dimensional analysis. This method ultimately aims to derive a valid relationship between the variables based on their dimensional properties.
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Homework Statement


The velocity,v of the sound in a gas depends on the density, p and the pressure, p of the gas. By using dimensional analysis, find a possible relation between v, p and p.


Homework Equations


[v] = LT-1
[p] = ML-3
[p] = ML-1T-2


The Attempt at a Solution


well i have no idea how to do it, so far i can only express the variables in dimension ways
 
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Here, have a ρ to express the density.

The whole concept of dimensional analysis is based on the fact that you can only equate two quantities if they have the same dimensions and units.

You are told that the velocity of sound in a gas, v depends on two factors, that is to say, that it is a function of these two factors.

v=f(\rho , p)

Naively, we say that it is a product of these two quantities, raised to some powers \alpha, \beta:

v=\rho^{\alpha}\cdot p^{\beta}

Now use dimensional analysis to find \alpha, \beta

Do this by constraining the system so that the dimensions on the RHS are the same as the dimensions on the LHS.
 
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