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Q.

A spherical ball falls through a viscous medium with terminal velocity v. If this ball is replaced by another ball of the same mass but half the radius, then the terminal velocity will be [neglect the effect of buoyancy].

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a

2v

b

8v

c

v

d

4v

answer is B.

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Detailed Solution

$m = \frac{4}{3} π r^{3} ρ \Rightarrow ρ α \frac{1}{r^{3}} \Rightarrow \frac{ρ_{1}}{ρ_{2}} = {(\frac{r_{2}}{r_{1}})}^{3} \Rightarrow \frac{ρ_{1}}{ρ_{2}} = \frac{1}{8} \Rightarrow ρ_{2} = 8 ρ_{1}$
Terminal velocity $v α r^{2} ρ$
$\frac{v_{2}}{v_{1}} = \frac{r_{2}^{2}}{r_{1}^{2}} \times \frac{ρ_{2}}{ρ_{1}} \Rightarrow v_{2} = v [\frac{1}{4} \times 8] = 2 v$

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