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Question: The velocity of waves on the surface of water is proportional to \({\lambda ^\alpha }{\rho ^\beta }{...

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

Explanation

Solution

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= [LT−1]\left[ {L{T^{ - 1}}} \right]
Wavelength= [L]\left[ L \right]
Density= ML−3M{L^{ - 3}}
g=[LT−2]g = \left[ {L{T^{ - 2}}} \right]
⇒V=[λαρβgγ]\Rightarrow V = \left[ {{\lambda ^\alpha }{\rho ^\beta }{g^\gamma }} \right]
⇒[(LT−1)]=[(L)α(ML−3)β(LT−2)γ] ⇒[LT−1]=[Lα−3β+γMβT−2γ] \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,
∴β=0 ∴γ=21 ∴α=21\therefore \beta = 0 \\\ \therefore \gamma = 21 \\\ \therefore \alpha = 21
Hence, γ=α≠β\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 (speed = l × f)\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.