`v,T, rho` and `lambda` denote ,surface tension, mass density and wavelength, respectively In an experiment v depends on `T,p` and `lambda` respectively .The value of v is proportional to

Options

A. `sqrt((T)/(lambda))
B. `sqrt((T)/(p lambda))`
C. `sqrt((lambda)/(p T))`
D. `sqrt((T)/( p lambda))

Answer

b

Explanation

`v = KT^(a)p rho^(b)lambda^(c )`
`LHS = [M^(0)LT^(-1)]`
`RHS = [M^(0)LT^(-1)]`
`= [MT^(-2)]^(a) [ML^(-3)]^(b) [L]^(c )`
`= [M^(a+b)L^(-3b+c) T^(-2a)]`
According to homogeneity principal
LHS - RHS
`[M^(0)LT^(-1)] = [M^(a+b) L^(-3b+c)T^(-2a)]`
`:. a+b = 0`
`:. a = -b`...(i)
`-3b+c = 1`...(ii)
`-2a= -1`...(iii)
`:. c = (-1)/(2)`
`b = -a = -(1)/(2)`
`-3b + c = 1`
`rArr -3 ((-1)/(2))`
`:' v = KT^(a) rho^(b) lambda^(c ) = KT^((1)/(2)) rho^(-(1)/(2)) lambda^(-(1)/(2)) = K sqrt((T)/(rho lambda))`.