Mechanical Properties of Fluids - Result Question 12
12. 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
(a) $r^{3}$
(c) $r^{4}$
(b) $r^{2}$
(d) $r^{5}$
[2018]
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Answer:
Correct Answer: 12. (d)
Solution:
- (d) Power $=$ rate of production of heat $=F \cdot V$
$=6 \pi \eta r V_T \cdot V_T=6 \pi \eta r V_T^{2}$
$(\because F=6 \pi \eta V_T r.$ stoke’s formula)
$V_T \propto r^{2}$ $\because \quad V_T=\frac{2}{9} \frac{r^{2}(\rho-\sigma)}{\eta} g$
$\therefore \quad$ Power $\propto r^{5}$
