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

$r^{4}$
B

$r^{3}$
C

$r^{5}$
D

$r^{2}$

Solution

Rate of heat produced = power loss against viscous force
$\Rightarrow \frac{dQ}{dt} = F_{v} \times v_{T}$
where the terminal velocity, $v_{T} = \frac{2 r^{2}}{9 η} (ρ - σ) g$
and viscous force $F_{v} = 6 {πηrv}_{T}$
hence $\frac{d Q}{d t} = 6 {πηrv}_{T} \times v_{T}$
$\Rightarrow \frac{dQ}{dt} \propto {rv}_{T}^{2} but v_{T} \propto r^{2}$
Hence $\frac{dQ}{dt} \propto r^{5}$

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