Question

The speed $v$ of ripples on the surface of water depends on surface tension $ \sigma$ density p 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

The Correct Option is A

Let v $ \propto \, \sigma^a \, \rho^b \, \lambda^c $ Equating dimensions on both sides $ [ M^0 L T^{ - 1} ] \propto [ MT^{ - 2} ]^a \, [ ML^{ - 3} ]^b \, [ L]^c $ $ \propto [ ML]^{ 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} $ $\therefore v^2 \propto \frac{ \sigma }{ \rho \lambda}$