Question

The velocity of waves on the surface of water is proportional to ${\lambda ^\alpha }{\rho ^\beta }{g^\gamma }$ where $\lambda $ is wavelength ,$\rho $ is density and $g$ is acceleration due to gravity, which of the following relation is correct
A. $\alpha = \beta = \gamma $
B. $\beta = \gamma \ne \alpha $
C. $\gamma = \alpha \ne \beta $
D. $\alpha \ne \beta \ne \gamma $

Answer 1

Hint: A unit of measurement is a definite magnitude of a quantity that is used as a norm for measuring the same kind of quantity. It is described and accepted by convention or by law. A length, for example, is a physical quantity. The metre is a unit of measurement that defines a specific length.

Complete step by step answer:
The forces to which the fundamental quantities are lifted to represent other physical quantities are known as dimensions. Dimensional formula is a mathematical expression that expresses the dimensions of a physical quantity in terms of fundamental quantities. The term "wave velocity" is often used to refer to speed, but it actually refers to both speed and direction. The sum of a wave's wavelength and frequency (number of vibrations per second) is its velocity, which is independent of its intensity.

Dimension of : velocity of waves= $\left[ {L{T^{ - 1}}} \right]$
Wavelength= $\left[ L \right]$
Density= $M{L^{ - 3}}$
$g = \left[ {L{T^{ - 2}}} \right]$
$\Rightarrow V = \left[ {{\lambda ^\alpha }{\rho ^\beta }{g^\gamma }} \right]$
$\Rightarrow \left[ {\left( {L{T^{ - 1}}} \right)} \right] = \left[ {{{\left( L \right)}^\alpha }{{\left( {M{L^{ - 3}}} \right)}^\beta }{{\left( {L{T^{ - 2}}} \right)}^\gamma }} \right] \\
\Rightarrow \left[ {L{T^{ - 1}}} \right] = \left[ {{L^{\alpha - 3\beta + \gamma }}{M^\beta }{T^{ - 2\gamma }}} \right] \\ $
On equating the power we get,
$\therefore \beta = 0 \\
\therefore \gamma = 21 \\
\therefore \alpha = 21 $
Hence, $\gamma = \alpha \ne \beta $

So the correct option is C.

Note: The wavelength, frequency, medium, and temperature all influence the speed of a wave. The wavelength is multiplied by the frequency to get the wave speed \[\left( {speed{\text{ }} = {\text{ }}l{\text{ }} \times {\text{ }}f} \right)\] . The following equations are simple if certain conditions are met. In a given medium, speed is constant.