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  4. 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

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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

NEETNEET 2018Mechanical Properties of Fluids

Solution:

Let's consider, $r =$ radius of a small sphere
$V _{ T }=$ Terminal velocity Power $=$ rate of production of heat
$P = F V _{ T } \ldots \ldots . . .(1)$
From Stoke's formula,
$F =6 \pi \eta V _{ T } r \ldots \ldots \ldots . .(2)$
substitute equation (2) in equation (1), we get
$P =6 \pi \eta r V _{ T }^{2} \ldots \ldots \ldots(3)$
we know that the, the terminal velocity is given by $V _{ T }=\frac{2 r ^{2}(\rho-\sigma)}{9 \eta}$
$V _{ T } \propto r ^{2} \ldots \ldots(4)$
from equation (3) and equation (4), we get $P \propto r ^{5}$