Fluid Mechanics

JEE Main level, single correct, medium

188 of 192

Question 188 of 192

A solid sphere, of radius R acquires a terminal velocity v1{v}_{1} when falling (due to gravity) through a viscous fluid having a coefficient of viscosity η.\eta . The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, v2,{v}_{2}, when falling through the same fluid, the ratio (v1v2)\left(\dfrac{{v}_{1}}{{v}_{2}}\right) equals:

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Answer: (D) 99

If RR is the radius of initial drop and rr is the radius of new drop, then
43πR3=27×43πr3⇒r=R3\dfrac{4}{3}\pi {R}^{3}=27\times \dfrac{4}{3}\pi {r}^{3}\Rightarrow r=\dfrac{R}{3}
Terminal velocity,
vT∝r2{v}_{T}\propto {r}^{2}
⇒v1v2=R2(R3)2=9\Rightarrow \dfrac{{v}_{1}}{{v}_{2}}=\dfrac{{R}^{2}}{{\left(\dfrac{R}{3}\right)}^{2}}=9

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