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

Explanation

Suppose, the frequency f depends on the tension raised to the power a, length raised to the power b and mass per unit lengthraised to the power c.
Then, `fpropto(F)^(a)(l)^(b)(mu)^(c)`
or `f=k(F)^(a)(l)^(b)(mu)^(c)` .....(i)
Here k is a dimensionless constant.
Thus `[f]=[F]^(a)[l]^(b)[mu]^(c)`
or `[M^(0)L^(0)T^(-1)]=[MLT^(-2)]^(a)[L]^(b)[ML^(-1)]^(c)`
or `[M^(0)L^(0)T^(-1)]=[M^(a+c)L^(a+b-c)T^(-2)]`
For dimensional balance, the dimensions on sides should be same.
Thus `a+c=0` .....(ii)
`a+b-c=0` .....(iii)
and `-2a=-1` ....(iv)
Solving these three equatitions,we get
`a=(1)/(2),c=-(1)/(2) and b=-1`
Substituting these values in Eq. (i) we get
`f=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)/(2l)sqrt((F)/(mu))`