Question

The velocity of water waves may depend on their wavelength $\lambda ,$ the density of water p and the acceleration due to gravity g. The method of dimensions gives the relation between these quantities as

• A. $v^2\propto {\lambda }^{-1}{g}^{-1}{\rho }^{-1}$
• B. $v^2\propto g\lambda$
• C. $v^2\propto g\lambda \rho$
• D. $v^2\propto {\lambda }^3{g}^{-1}{\rho }^{-1}$

Answer 1

The correct option is B

$v^2\propto g\lambda$

$v\propto {\lambda }^a{\rho }^a{g}^c\Rightarrow L{T}^{-1}=\left[L{]}^a\right[M{L}^{-3}{]}^b[L{T}^{-2}{]}^{c}b=0.......(1)a-3b+c=1......(2)and-2c=-1.......(3\left)Solve\right(1),(2\left)and\right(3)foranswer.$