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The frequency (f) of a stretched string ...

The frequency (f) of a stretched string depends upon the tension F (dimensions of form ) of the string and the mass per unit length `mu` of string .Derive the formula for frequency

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Suppose, that the frequency f depends on the tension raised to the power a, length raised to the power b and mass per unit length raised to the power c. Then.
`f oo [F[F]^(a)[l]^(b)[mu]^(c)`
`"or "f=k[F]^(a)[l]^(b)[mu]^(c)`
Here, k is a dimensionless constant. Thus,
`[f]=[F]^(0)[l]^(b)[mu]^(c)`
For dimensional balance, the dimension on both sides should be same.
Thus,`" a+c=0" ...(ii)"`
`" "a+b-c=0" ...(iii)"`
`"and "-2a=-1" ...(iv)"`
Solving these three equation, we get
`a=(1)/(2), c=-(1)/(2) and b=-1`
Substituting these values in Eq. (i). we get
`k(F)^(1//2)(l)^(-1)(mu)^(-1//2)" or "f=(k)/(l)sqrt((F)/(mu))`
Experimentally, the value of k is found to be `(1)/(2)`
Hence,`" "f=(1)/(2)sqrt(F)/(mu).`
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