Speed of a wave produced in water is given by v=λ^a g^b ρ^c . Where λ , g and ρ are wavelength o

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1.Units, Dimensions and Measurement

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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

A

$\frac{1}{2}, \frac{1}{2}, 0$

B

$1,1,0$

C

$1,-1,0$

D

$\frac{1}{2}, 0, \frac{1}{2}$

(JEE MAIN-2023)

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 $a+b-3 c=1$

$\Rightarrow a+\frac{1}{2}-0=1$

$\Rightarrow a=\frac{1}{2}$

$\therefore a=\frac{1}{2}, b=\frac{1}{2}, c=0$

Standard 11

Physics

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