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1.Units, Dimensions and Measurement
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The velocity of water waves $v$ may depend upon their wavelength $\lambda $, the density of water $\rho $ and the acceleration due to gravity $g$. The method of dimensions gives the relation between these quantities as
A
${v^2} \propto \lambda {g^{ - 1}}{\rho ^{ - 1}}$
B
${v^2} \propto g\lambda \rho $
C
${v^2} \propto g\lambda $
D
${v^2} \propto {g^{ - 1}}{\lambda ^{ - 3}}$
Solution
Dimension of Velocity is $LT ^{-1}$
Dimension of wavelength $\lambda$ is $L$
Dimension of acceleration due to gravity $g$ is $LT ^{-2}$
Dimension of Density of water $p$ is $ML ^{-3}$
Let $v \propto \lambda^{ a } g ^{ b } \rho^{ c }$
Using Dimensional method
$a+b-3 c=1$
$c =0$
$-2 b =-1$
$b =0.5$
Hence, $a =0.5$
$\therefore v \propto \sqrt{ g \lambda}$
$v ^{2} \propto g \lambda$
Standard 11
Physics
