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.

Assuming that the frequency gamma of a vibrating string may depend upo

1.	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» Solution :`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)`