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The speed of a wave produced in water is given by $v=\lambda^a g^b \rho^c$. Where $\lambda$, g and $\rho$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of $a , b$ and $c$ respectively, are
$\frac{1}{2}, \frac{1}{2}, 0$
$1,1,0$
$1,-1,0$
$\frac{1}{2}, 0, \frac{1}{2}$
Solution
$v=\lambda^a g^b \rho^c$
using dimension formula
$\Rightarrow\left[ M ^0 L ^1 T ^{-1}\right]=\left[ L ^1\right]^{ a }\left[ L ^1 T ^{-2}\right]^{ b }\left[ M ^1 L ^{-3}\right]^{ c }$
$\Rightarrow\left[ M ^0 L ^1 T ^{-1}\right]=\left[ M ^{ c } L ^{ a + b -3 c } T ^{-2 b}\right]$
$\therefore c =0, a + b -3 c =1,-2 b =-1 \Rightarrow b =\frac{1}{2}$
Now $a+b-3 c=1$
$\Rightarrow a+\frac{1}{2}-0=1$
$\Rightarrow a=\frac{1}{2}$
$\therefore a=\frac{1}{2}, b=\frac{1}{2}, c=0$
