Assuming that the frequency `gamma` of a vibrating string may depend upon (i) applied force (F) (ii) length (l) (iii) mass per unit lengt (m), prove that `gamma prop1/l sqrt(F/m)` using dimensional analysis.

Explanation

`gamma prop l^(a)F^(b)m^(c)`
`gamma=Kl^(a)F^(b)m^(c)`
K-dimensionless constnat of proportionality
a,b,c, powers of l,F,m
Dimensional Formula of `F=[MLT^(-2)]`
Dimensional Formula of linear density
`m=("mass")/("length")=([M])/([L])=[M^(-1)L^(-1)]`
Writing dimensiions of `gamma=Kl^(a)F^(b)m^(c)`
`=L^(a)M^(b)L^(a)T^(-2b)M^(c)L=[M]^(b+c)[L]^(a+b-c)[T]^(-2b)`
`M^(0)L^(0)T^(-1)=[M]^(b+c)[L]^(a+b-c)[T]^(-2b)`
Applying the principle of homogeneity of dimension.
`b+c=0`............1
`a+b-c=0`.............2
`-2b=-1` or `b=+1/2`
From 1 `c=-b=-1/2" "c=-1/2`
From 2 `a+1/2-(-1/2)=0" "a+1/2+1/2=0`
`:.a=-1`
Substituting the values a,b,c in `:'K=1`
`gamma=Kl^(-1)F^(1/2)m^(-1/2)`
`gamma=1/lsqrt(F/m)`