First slide

Viscosity

Question

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

Moderate

A

$r^{3}$
B

$r^{2}$
C

$r^{5}$
D

$r^{4}$

Solution

The viscous drag force, $F = 6 π η r v$
where v= terminal velocity
$∴$ The rate of production of heat = power
= force $\times$ times terminal velocity
$\Rightarrow Power = 6 π η r v \cdot v = 6 π η r v^{2}$
$∵ Terminal velocity v = \frac{2 r^{2} (ρ - σ) g}{9 η};$
$∴ v \propto r^{2}$
$Now, power = 6 π η r [\frac{4 r^{4} (ρ - σ)^{2}}{81 η^{2}} g^{2}] or Power \propto r^{5} .$

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