Units & Dimensions

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68 of 149

Question 68 of 149

The speed of a wave produced in water is given by ν=λagbρc\nu ={\lambda }^{a}{g}^{b}{\rho }^{c}. Where λ,g\lambda ,g and ρ\rho are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a,ba,b and cc respectively, are

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Answer: (D) 12,12,0\dfrac{1}{2},\dfrac{1}{2},0

The dimensional form of velocity can be written as
[v]=[λagbρc]\left[v\right]=\left[{\lambda }^{a}{g}^{b}{\rho }^{c}\right]
⇒[M0LT−1]=[L]a[LT−2]b[ML−3]c\Rightarrow \left[{M}^{0}L{T}^{-1}\right]={\left[L\right]}^{a}{\left[L{T}^{-2}\right]}^{b}{\left[M{L}^{-3}\right]}^{c}
Comparing the exponents for M,L,TM,L,T,
ForM,Mc=M0ForL,La+b−3c=L1ForT,T−2b=T−1ForM, {M}^{c}={M}^{0} ForL, {L}^{a+b-3c}={L}^{1} ForT, {T}^{-2b}={T}^{-1}
So,
c=0...(i)a+b−3c=1...(ii)−2b=−1...(iii)c=0...(i) a+b-3c=1...(ii) -2b=-1...(iii)
From equations (i),(ii),(iii),
a=12,b=12,c=0a=\dfrac{1}{2},b=\dfrac{1}{2},c=0

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