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.

Answer

The frequency `gamma` of a vibrating string depends
(i) applied force (F) (ii) length (l) (iii) mass per unit length (m)

Explanation

`gamma prop F^(x) l^(y) m^(z) prop gamma = K F^(x) l^(y) m^(z)`
substitute the dimensional formulae of the above quantities
`[M^(0)L^(0)T^(-1)] = [MLT^(-2)]^(x) [L] [ML^(-1)]^(z)`
`[M^(0)L^(0)T^(-1)] = [M^(x + z)L^(x+y-z)T^(-2x)]`
Comparing the powers of M,L,T on both sides,
`x+z = 0, x+y-z = 0, -2x = -1`
Solving for x,y,z, we get
`x = 1/2` `y = -1` `z = -1/2`
Substitute x,y,z values in equ (1)
`gamma prop F^(1//2) l^(-1) m^(-1//2) :. gamma prop 1/l sqrt(F/m)`