The frequency (f) of a stretched string depends upon the tension F (dimensions of force), length l of the string and the mass per unit length μ of string. Derive the formula for frequency.

Suppose, that the frequency f depends on the tension raised to the power a, length raised to power b  and mass per unit length raised to power c. Then,f∝Falbμc or, f=kFalbμcHere, k is a dimensionless constant. Thus, [f]=[F]a[η]b[μ]c  or, M0 L0 T-1=MLT-2a[ L]bML-1c or, M0 L0 T-1=Ma+c La+b-c T-2aFor dimensional balance, the dimensions on both sides should be same.Thus, a+c=0… (ii) a+b-c=0...(iii) and -2a=-1… (iv) Solving these three equations, we geta=12,  c=-12 and b=-1Substituting these values in Eq. (i), we getf=k(F)1/2(l)-1(μ)-1/2  or, f=klFμExperimentally, the value of k is found to be 12Hence,f=12lFμ