JEE PYQ: Mechanical Properties Of Fluids Question 16
Question 16 - 2020 (05 Sep 2020 Shift 2)
In an experiment to verify Stokes law, a small spherical ball of radius $r$ and density $\rho$ falls under gravity through a distance $h$ in air before entering a tank of water. If the terminal velocity of the ball inside water is same as its velocity just before entering the water surface, then the value of $h$ is proportional to: (ignore viscosity of air)
(1) $r^4$
(2) $r$
(3) $r^3$
(4) $r^2$
Show Answer
Answer: (1) $r^4$
Solution
Using $v^2 - u^2 = 2gh$ $\Rightarrow v^2 - 0^2 = 2gh \Rightarrow v = \sqrt{2gh}$ Terminal velocity: $V_T = \frac{2}{9}\frac{r^2(\rho - \sigma)g}{\eta}$ After falling through $h$, the velocity should be equal to terminal velocity: $\therefore \sqrt{2gh} = \frac{2}{9}\frac{r^2(\rho - \sigma)g}{\eta}$ $\Rightarrow 2gh = \frac{4}{81}\frac{r^4 g^2(\rho - \sigma)^2}{\eta^2} \Rightarrow h \propto r^4$
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