Question #89830
Use dimensional analysis to express the wave velocity in terms of the tension in the string and mass per unit length of the string.
1
Expert's answer
2019-05-20T13:47:44-0400

Let the wave velocity is dependent on the tension and mass per unit length. Then


v=Fαμβv=F^{\alpha}\mu^{\beta}

Since dimensions of velocity, force and mass per unit length are as follows,


[v]=LT1[v]=LT^{-1}

[F]=MLT2[F]=MLT^{-2}

[μ]=ML1[\mu]=ML^{-1}

we obtain

LT1=(MLT2)α(ML1)βLT^{-1}=\left(MLT^{-2}\right)^{\alpha}(ML^{-1})^{\beta}

LT1=Mα+βLαβT2αLT^{-1}=M^{\alpha+\beta}L^{\alpha-\beta}T^{-2\alpha}

So


{α+β=0αβ=12α=1\left\{\begin{matrix} \alpha+\beta &=& 0 \\ \alpha-\beta &=& 1\\ -2\alpha &=& -1 \end{matrix}\right.

Solution

{α=1/2β=1/2\left\{\begin{matrix} \alpha &=& 1/2\\ \beta &=& -1/2 \end{matrix}\right.

Finally

v=F1/2μ1/2=Fμv=F^{1/2}\mu^{-1/2}=\sqrt{\frac{F}{\mu}}


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