Question

The speed $(v)$ of ripples on the surface of water depends on surface tension $(\sigma)$, density $(\rho)$ and wavelength $(\lambda)$. The square of speed $(v)$ is proportional to

• A. $\frac{\sigma}{\rho \lambda}$
• B. $\frac{\rho}{\sigma \lambda}$
• C. $\frac{\lambda}{\sigma \rho}$
• D. $\rho\lambda\sigma$

Answer 1

Answer: (A)

Let $v \propto \sigma^{a} \rho^{b} \lambda^{c}$
Equating dimensions on both sides,
$\left[ M ^{0} L ^{1} T ^{-1}\right] \propto\left[ MT ^{-2}\right]^{a}\left[ ML ^{-3}\right]^{b}[ L ]^{c}$
$\propto[ M ]^{a+b}[ L ]^{-3 b+c}[ T ]^{-2 a}$
Equating the powers of M, L, T on both sides, we get
$a +b=0$
$-3 b+ c=1$
$-2 a=-1$
Solving, we get
$a=\frac{1}{2}, b=-\frac{1}{2}, c=-\frac{1}{2}$
$\therefore v \propto \sigma^{1 / 2} \rho^{-1 / 2} \lambda^{-1 / 2}$
$\because v^{2} \propto \frac{\sigma}{\rho \lambda}$