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QJEEM23Phca97: JEE Main 2023 Physics - Class 11 JEEM23Phca97 Solution: The frequency ( ) of an oscillating liquid drop may depend upon …

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

Multiple Choice Options

  • A (32,12,−12)\left( {{3 \over 2},{1 \over 2}, - {1 \over 2}} \right)
  • B (−32,−12,12)\left( { - {3 \over 2}, - {1 \over 2},{1 \over 2}} \right)
  • C (32,−12,12)\left( {{3 \over 2}, - {1 \over 2},{1 \over 2}} \right)
  • D (−32,12,12)\left( { - {3 \over 2},{1 \over 2},{1 \over 2}} \right)

Step-by-Step Verified Solution

[v]=[T−1][v]=\left[\mathrm{T}^{-1}\right]

amp;[r]=L[s]=[MLT−2 L]amp;[ρ]=[ML3]=[ML−3]amp;⇒v=raρb scamp;⇒T−1=LaMb L−3bMc T−2camp;⇒T−1=M(b+c)L(a−3b)T−2camp;−2c=−1⇒c=12amp;b+c=0amp;⇒b=−12amp;a−3b=0⇒3b=a⇒a=−32amp;(a,b,c)=(−32,−12,12)\begin{aligned}& {[r]=\mathrm{L} \quad[s]=\left[\frac{\mathrm{MLT}^{-2}}{\mathrm{~L}}\right]} \\\\& {[\rho]=\left[\frac{\mathrm{M}}{\mathrm{L}^{3}}\right]=\left[\mathrm{ML}^{-3}\right]} \\\\& \Rightarrow v=r^{a} \rho^{b} \mathrm{~s}^{c} \\\\& \Rightarrow \mathrm{T}^{-1}=\mathrm{L}^{a} \mathrm{M}^{b} \mathrm{~L}^{-3 b} \mathrm{M}^{c} \mathrm{~T}^{-2 c} \\\\& \Rightarrow \mathrm{T}^{-1}=\mathrm{M}^{(b+c)} \mathrm{L}^{(a-3 b)} \mathrm{T}^{-2 c} \\\\& -2 c=-1 \Rightarrow c=\frac{1}{2} \\\\& b+c=0 \\\\& \Rightarrow b=-\frac{1}{2} \\\\& a-3 b=0 \Rightarrow 3 b=a \Rightarrow a=-\frac{3}{2} \\\\& (a, b, c)=\left(-\frac{3}{2},-\frac{1}{2}, \frac{1}{2}\right)\end{aligned}

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