Answer Checking (Speed of Sound formula, dimensional analysis)

AI Thread Summary
The discussion focuses on the formula for measuring the speed of sound in air, specifically the units of the constant k in the equation v = √(kTg/m). Participants clarify that k could represent either the adiabatic index or the Boltzmann Constant, but emphasize that the main task is to determine its units based on dimensional analysis. The second part of the discussion evaluates the dimensional consistency of a proposed equation for wave speed, concluding that it is not dimensionally proportionate. Overall, the conversation highlights the importance of understanding units and dimensions in physics equations. Clear guidance is provided on how to approach these types of problems effectively.
Phonetic
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Homework Statement


One formula for measuring the speed of sound in air is:
<br /> v=\sqrt\frac{kTg}{m}<br />

v=velocity
T=temperature
g=dimensionless constant
m=average mass of an air molecule

If all units are SI, what are the units of k?


The Attempt at a Solution



I've done a lot of Googling to first try and find formulas for the speed of sound. The most similar I've seen on Wikipedia was
cc8e66050f6b9537574f750498ec6eb1.png

with the last part of the formula being the most relevant
Cideal = ideal speed
gamma = adiabatic index
k = Boltzmann Constant
T = temperature in Kelvin
m = mass of a single molecule in kg

That makes me think that k is a placeholder for the adiabatic index or the Boltzmann Constant, but I'm not sure which (if either) it is.

Homework Statement



In this question, the length of a wave, λ, has dimensions L, wave speed (v) has dimensons L/T, and gravitational acceleration (g) has units of L/T². Could the following be an equation for wave speed:

Homework Equations



<br /> v=\sqrt\frac{g}{λ}<br />

The Attempt at a Solution


λ=L
v=D/T
g=L/T²

<br /> D/T = \sqrt\frac{L/T²}{L}<br />

<br /> D/T=\sqrt{T²}<br />

<br /> D/T=T<br />

No it could not be an equation for wave speed because it is not dimensionally proportionate.

If I didn't get these right, where did I go wrong? I feel like I got the second one right, but I'm completely clueless on the first one. Thanks in advance!
 
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Phonetic said:

Homework Statement


One formula for measuring the speed of sound in air is:
<br /> v=\sqrt\frac{kTg}{m}<br />

v=velocity
T=temperature
g=dimensionless constant
m=average mass of an air molecule

If all units are SI, what are the units of k?


The Attempt at a Solution



I've done a lot of Googling to first try and find formulas for the speed of sound. The most similar I've seen on Wikipedia was
cc8e66050f6b9537574f750498ec6eb1.png

with the last part of the formula being the most relevant
Cideal = ideal speed
gamma = adiabatic index
k = Boltzmann Constant
T = temperature in Kelvin
m = mass of a single molecule in kg

That makes me think that k is a placeholder for the adiabatic index or the Boltzmann Constant, but I'm not sure which (if either) it is.

No need to look for similar equations. They just wanted you to get the units of k.

you have velocity on the left side in m/s. Write down the units of the rest of units of the dimensions. Then put them into the formula and re-arrange to find k.

Phonetic said:

Homework Statement



In this question, the length of a wave, λ, has dimensions L, wave speed (v) has dimensons L/T, and gravitational acceleration (g) has units of L/T². Could the following be an equation for wave speed:

Homework Equations



<br /> v=\sqrt\frac{g}{λ}<br />

The Attempt at a Solution


λ=L
v=D/T
g=L/T²

<br /> D/T = \sqrt\frac{L/T²}{L}<br />

<br /> D/T=\sqrt{T²}<br />

<br /> D/T=T<br />

No it could not be an equation for wave speed because it is not dimensionally proportionate.

If I didn't get these right, where did I go wrong? I feel like I got the second one right, but I'm completely clueless on the first one. Thanks in advance!

This one is correct.
 
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