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Question

If the velocity of water wave v depends upon the wavelength λ, the density of water ρ and the acceleration due to gravity g, then which of the following options depicts correct relation dimensionally?

A
v2λg1ρ1
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B
v2gλρ
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C
v2gλ
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D
v2g1λ3
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Solution

The correct option is C v2gλ
Dimensional formula of velocity, v
[v]=[M0L1T1]
Wavelength, λ
[λ]=[M0L1T0]
Density, ρ
[ρ]=[M1L3T0]
Acceleration due to gravity, g
[g]=[M0L1T2]
Let us suppose the relation is
v2λagbρc
[v2]=[λagbρc]
[(M0L1T1)2]=[M0L1T0]a×[M0L1T2]b×[M1L3T0]c
[M0L2T2]=[M(c)L(a+b3c)T(2b)]
On comparing, we get,
c=0(1)
a+b3c=2(2)
2b=2(3)
On solving these equations, we get,
a=1,b=1 and c=0
Thus,
v2λagbρcv2gλ

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