A solid sphere, of radius R acquires a terminal velocity $v_{1}$ when falling (due to gravity) through a viscous fluid having a coefficient of viscosity $\eta$. The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, $v_{2}$, when falling through the same fluid, the ratio ( $v_{1} / v_{2}$ ) equals :

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Fluid Mechanics

A solid sphere, of radius R acquires a terminal velocity

v_{1}

when falling (due to gravity) through a viscous fluid having a coefficient of viscosity

η

. The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity,

v_{2}

, when falling through the same fluid, the ratio (

v_{1} / v_{2}

) equals :

(A)27
(B) $1/27$
(C)9✓ Correct
(D) $1/9$

Answer: (C)

Solution

Terminal velocity scales as the square of the radius,

v_{T} \propto r^{2}

. Breaking one sphere into

27

equal spheres conserves volume,

\frac{4}{3} π R^{3} = 27 \cdot \frac{4}{3} π r^{3}

, so

r = \frac{R}{3}

. The ratio of terminal speeds is therefore

\frac{v _{1}}{v _{2}} = \frac{R ^{2}}{( R /3 ) ^{2}} = 9

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Single correct option

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