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