Fluid Mechanics

JEE Main level, single correct, medium

175 of 192

Question 175 of 192

Given below are two statements: one is labelled as
Assertion A and the other is labelled as Reason R
Assertion A : A spherical body of radius (5±0.1)mm(5\pm 0.1)mm having a particular density is falling through a liquid of constant density. The percentage error in the calculation of its terminal velocity is 44%.
Reason R : The terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius. In the light of the above statements, choose the correct answer from the options given below

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Answer: (C) A is true but R is false

The terminal velocity is
VT=29r2g(ρ−ρ′)η⇒VT∝r2{V}_{T}=\dfrac{2}{9}{r}^{2}g\dfrac{\left(\rho -{\rho }^{'}\right)}{\eta } \Rightarrow {V}_{T}\propto {r}^{2}
Differentiating and dividing by VT{V}_{T},
⇒dVTVT=2drr=2×0.15\Rightarrow \dfrac{d{V}_{T}}{{V}_{T}}=2\dfrac{dr}{r}=2\times \dfrac{0.1}{5}
So, the percentage error is
dVTVT×100=0.25×100=4\dfrac{d{V}_{T}}{{V}_{T}}\times 100=\dfrac{0.2}{5}\times 100=4%
As can be seen the terminal velocity is directly proportional to the square of the radius. Hence AA is true but RR is false.

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