The frequency 'n' of a stretched string depends on length 'l' of a string, mass per unit length or linear density (m) and tension (T) or (F) acting on the string. Derive an equation connecting these quantities using Dimensional Analysis.

Explanation

Let ` n prop l^a m ^b T^ c `
where ` [n] = [M^0 L^0T^( -1) ] `
` [m] =[ ( M)/(L)] = [ML^( -1) ] `
`[T] = ["Tension"]= [M^ 1 L ^1 T^( -2) ] `
and (length ) = [L]
i.e., ` n = kl^am^b T^c `
` [M^0L^0T^(-1)] = [L]^a [ML^( -1) ]^ b [M^1L^1T^(-2)]^c `
`[M^0L^0T^( -1) ] = [ M^ ( b + c ) L ^( a + b + c ) T ^( -2c ) ] `
From the principle of homogeneity, equating the dimensions of M, L & T we write
` b + c = 0 " " (1) `
` a - b + c = 0 " " ` (2)
` - 2c = -1 " " (3) `
From (3) , ` c = 1//2 ` using 'c' in (1) we get
` b + 1//2 = 0 " " or b = -1//2 `
using b & c in (2) we get
` a - (-1//2) + 1//2 = 0 " " a + 1//2 + 1//2 = 0 " " therefore a = - 1 `
Hence ` n = kl ^ ( -1) m ^( -1//2) T^( 1//2 ) `
i.e., ` n = (K)/(l) sqrt((T) /(m)) `