Units & Dimensions

JEE Main level, single correct, medium

81 of 149

Question 81 of 149

The frequency (ν)\left(\nu \right) of an oscillating liquid drop may depend upon radius (r)\left(r\right) of the drop, density (ρ)\left(\rho \right) of liquid and the surface tension (s)\left(s\right) of the liquid as: ν=raρbsc\nu ={r}^{a}{\rho }^{b}{s}^{c}. The values of a,ba,b and cc respectively are

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Answer: (A) (−32,−12,12)\left(-\dfrac{3}{2},-\dfrac{1}{2},\dfrac{1}{2}\right)

Here, we have ν=raρbsc\nu ={r}^{a}{\rho }^{b}{s}^{c}
Equating the dimensions of each quantity on both sides
[T−1]=[L1]a[M1L−3]b[MLT−2L]c⇒T−1=Mb+cLa−3bT−2c\left[{T}^{-1}\right]={\left[{L}^{1}\right]}^{a}{\left[{M}^{1}{L}^{-3}\right]}^{b}{\left[\dfrac{ML{T}^{-2}}{L}\right]}^{c} \Rightarrow {T}^{-1}={M}^{b+c}{L}^{a-3b}{T}^{-2c}
Comparing the dimensions, we get
c=12,b=−12,a−3b=0c=\dfrac{1}{2},b=-\dfrac{1}{2},a-3b=0
⇒a+32=0⇒a=−32\Rightarrow a+\dfrac{3}{2}=0\Rightarrow a=-\dfrac{3}{2}
Thus, the value of aa, bb and cc are (−32,−12,12)\left(-\dfrac{3}{2},-\dfrac{1}{2},\dfrac{1}{2}\right).

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