The speed `(v)` of ripples on the surface of waterdepends on surface tension `(sigma)`, density `(rho)` and wavelength `(lambda)`. The square of speed `(v)` is proportional to

Options

A. `(rho)/(sigma lambda)`
B. `(sigma)/(rho lambda)`
C. `(lambda)/(sigma rho)`
D. `rh lambda sigma `

Answer

b

Explanation

Let `v prop sigma^(a) rho^(b) lambda^(c )`
Equating dimension of both sides,
`[M^(0) L^(1)T^(-1)] prop [MT^(-2)]^(a) [ML^(-3)]^(b)[L]^(c )`
`prop [M]^(a +b) [L]^(- 3b +c )[T]^(-2a)`
Equating the powers of `M,L,T ` on both sides , we get
`a + b= 0`
`- 3b + c = 1`
`- 2a = -1`
Solving we get
`a = (1)/(2) , b = -(1)/(2), c = -(1)/(2)`
`:. v prop sigma^(1//2) rho^(-1//2) lambda^(-1//2)`
`:. v^(2) prop (sigma)/(rho lambda)`