A solid sphere, of radius R acquires a terminal velocity v1 when falling (due to gravity) through a viscous fluid having a coefficient of viscosity h. The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, v2, when falling through the same fluid, the ratio (v1/v2) equals :

Question

A solid sphere, of radius R acquires a terminal velocity v1 when falling (due to gravity) through a viscous fluid having a coefficient of viscosity h. The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, v2, when falling through the same fluid, the ratio (v1/v2) equals : (A) 1/27 (B) 1/9 (C) 27 (D) 9

Tardigrade Admin · Accepted Answer

We have VT = (2/9) (r2/η) (ρ0 - ρ1)g ⇒ vT propto r2 since mass of the sphere will be same ρ (4/3) π R3 = 27. (4/3) π r3 ρ ⇒ = (R/3) ∴ (v1/v2) = (R2/r2) = 9

1/27

1/9

27

9

We have VT​=92​ηr2​(ρ0​−ρ1​)g⇒vT​∝r2 since mass of the sphere will be same ρ34​πR3=27.34​πr3ρ⇒=3R​ ∴v2​v1​​=r2R2​=9

We have
$V_{T} = \frac{2}{9} \frac{r ^{2}}{η} (ρ_{0} - ρ_{1}) g \Rightarrow v_{T} \propto r^{2}$
since mass of the sphere will be same
$ρ \frac{4}{3} π R^{3} = 27. \frac{4}{3} π r^{3} ρ \Rightarrow= \frac{R}{3}$
$∴ \frac{v _{1}}{v _{2}} = \frac{R ^{2}}{r ^{2}} = 9$