Answer to Question #104076 in Mechanics | Relativity for MOGBOLUWAGA

Question #104076
The speed of sound v in a medium depends on its wavelength λ, the young modulus E, and the density ρ, of the medium. Use the method of dimensional analysis to derive a formula for the speed of sound in a medium
1
Expert's answer
2020-03-02T09:07:48-0500

"v=\\lambda ^\\alpha E^\\beta \\rho ^\\gamma"


"[\\lambda]=[m]=L"


"[E]=[Pa]=ML^{-1}T^{-2}"


"[\\rho]=[kg\/m^3]=ML^{-3}"


"[v]=[m\/s]=LT^{-1}"


"LT^{-1}=L^\\alpha(ML^{-1}T^{-2})^\\beta(ML^{-3})^\\gamma"


We have


"\\alpha-\\beta-3\\gamma=1"

"\\beta+\\gamma=0"

"-2\\beta=-1"


The solution of the system gives


"\\alpha=0"

"\\beta=1\/2"

"\\gamma =-1\/2"


"v=\\lambda ^0 E^{1\/2} \\rho ^{-1\/2}=\\sqrt{\\frac{E}{\\rho}}" Answer.








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