Question

The speed $\left(v\right)$ of ripples on surface of water depends upon the surface tension $\left(\sigma \right)$ density $\rho$ and wave length $\left(\lambda \right)$ the speed v is proportional to

• A. ${\left(\frac{\sigma }{\lambda \rho }\right)}^{1/2}$
• B. ${\left(\frac{\rho }{\lambda \sigma }\right)}^{1/2}$
• C. ${\left(\frac{\lambda }{\sigma \rho }\right)}^{1/2}$
• D. ${\left(\rho g\sigma \right)}^{1/2}$

Answer 1

The correct option is C ${\left(\frac{\lambda }{\sigma \rho }\right)}^{1/2}$

Let, $v\propto {\sigma }^a{\rho }^b{\lambda }^c$

$\left[M^{0}L{T}^{-1}\right]\propto \left[M{T}^{-2}{]}^a\right[M{L}^{-3}{]}^b[L{]}^c$

$\propto \left[ML{]}^{a+b}\right[L{]}^{-3b+c}[T{]}^{-2a}$

equating the powers, we get,

$a+b=0$

$-3b+c=1$

$-2a=-1$

$\therefore a=b=c=\frac{1}{2}$

$\therefore v\propto {\sigma }^{\frac{1}{2}}{\rho }^{-\frac{1}{2}}{\lambda }^{-\frac{1}{2}}$

$v\propto \sqrt{\frac{\sigma }{\rho \lambda }}$