The velocity of water waves v may depend upon their wavelength λ , the density of water ρ and the acceleration due to gravity g . The method of dimensions gives the relation between these quantities as

Question

The velocity of water waves v may depend upon their wavelength λ , the density of water ρ and the acceleration due to gravity g . The method of dimensions gives the relation between these quantities as (A) v2 propto ρ g (B) v2 propto g λ ρ  (C) v2 propto g λ  (D) v2 propto g- 1 λ - 3

Tardigrade Admin · Accepted Answer

Let vx = k gy λ z ρ δ . Now by substituting the dimensions of each quantities and equating the powers of M, L and T we get δ = 0 and x = 2 , y = 1 , z = 1 .

Q. The velocity of water waves $v$ may depend upon their wavelength $λ$ , the density of water $ρ$ and the acceleration due to gravity $g$ . The method of dimensions gives the relation between these quantities as

$v^{2} \propto ρ g$

$v^{2} \propto g λ ρ$

$v^{2} \propto g λ$

$v^{2} \propto g^{- 1} λ^{- 3}$

Let $v^{x} = k g^{y} λ^{z} ρ^{δ} .$ Now by substituting the dimensions of each quantities and equating the powers of M, L and T we get $δ = 0$ and $x = 2, y = 1, z = 1$ .

Q. The velocity of water waves v may depend upon their wavelength λ , the density of water ρ and the acceleration due to gravity g . The method of dimensions gives the relation between these quantities as

v2∝ρg

v2∝gλρ

v2∝gλ

v2∝g−1λ−3