A small sphere of radius falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity is proportional to [2018]

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Solution

Q.
A small sphere of radius $' r'$ falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity is proportional to [2018]

1 $r^{3}$

2 $r^{2}$

3 $r^{5}$

4 $r^{4}$

Ans.
(3)

The viscous drag force, $F = 6 π η r v;$

where $v =$ terminal velocity

$∴$ The rate of production of heat = power

$= force \times terminal velocity$

$\Rightarrow$ $Power = 6 π η r v \cdot v = 6 π η r v^{2}$

$∵$ Terminal velocity $v = \frac{2 r^{2} (ρ - σ) g}{9 η};$

$Now, power = 6 π η r [\frac{4 r^{4} {(ρ - σ)}^{2}}{81 η^{2}} g^{2}]$

or Power $\propto r^{5}$

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