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

Given : The frequency `gamma` of a vibrating segment is `gamma prop F^(a)(l)^(b)(m)^(c )`
Rewriting the above equation with dimensions,
`[T]^(-1)=[MLT^(-2)]^(a)[L]^(b)[ML^(-1)]^(c )`
`M^(0)L^(0)T^(-1)=[M]^(a+c)[L]^(a+b-c)[T]^(-2a)`
Comparing the powers on both sides,
`a+c=0" ...(1)"`
`a+b-c=0" ...(2)"`
`-2a=-1" ....(3)"`
From (3) we get,
`-2a=-1`
`a=(1)/(2)" ...(4)"`
(4) in (1) we get
`(1)/(2)+c=0 rArr c=(-1)/(2)" ...(5)"`
(4) and (5) in (2) we get,
`(1)/(2)+c=0 rArr c=(-1)/(2)" ...(5)"`
(4) and (5) in (2) we get,
`(1)/(2)+b-(-(1)/(2))=0`
`b+1 = 0 rArr b=-1" ...(6)"`
From the above, `gamma prop F^(1//2)(l)^(-1)(m)^(-1//2)`
`rArr " "gamma prop (1)/(l) sqrt(F//m)`