The speed of a wave produced in water is given by ν = λ a g b ρ c . Where λ , g and ρ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a , b and c r

Question

The speed of a wave produced in water is given by ν = λ a g b ρ c . Where λ , g and ρ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a , b and c respectively, are

TestHub · Accepted Answer

The dimensional form of velocity can be written as
$[v] = [λ^{a} g^{b} ρ^{c}]$
$\Rightarrow [M^{0} L T^{- 1}] = {[L]}^{a} {[L T^{- 2}]}^{b} {[M L^{- 3}]}^{c}$
Comparing the exponents for $M, L, T$ ,
$For M, M^{c} = M^{0} For L, L^{a + b - 3 c} = L^{1} For T, T^{- 2 b} = T^{- 1}$
So,
$c = 0 . . . (i) a + b - 3 c = 1 . . . (i i) - 2 b = - 1 . . . (i i i)$
From equations (i),(ii),(iii),
$a = \frac{1}{2}, b = \frac{1}{2}, c = 0$

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