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

• A. $r^{5}$
• B. $r^{2}$
• C. $r^{3}$
• D. $r^{4}$

Answer 1

Answer: (A)

The rate of heat generation is equal to the rate of work done by the viscous force which in turn is equal to its power.
Rate of heat produced, $\frac{d Q}{d t}=F \times v_{T}$
where, $F$ is the viscous force and $v_{T}$ is the terminal velocity.
As, $ F=6 \pi \eta r v_{T}$
$\Rightarrow \frac{d Q}{d t}=6 \pi \eta r v_{T} \times v_{T}=6 \pi \eta r v_{T}^{2}\,\,\,$...(i)
From the relation for terminal velocity,
$v_{T}=\frac{2}{9} \frac{r^{2}(\rho-\sigma)}{\eta} g$, we get
$v_{T} \propto r^{2}$
From Eq. (ii), we can rewrite Eq. (i) as
$\frac{d Q}{d t} \propto r \cdot\left(r^{2}\right)^{2} $
or$ \frac{d Q}{d t} \propto r^{5}$
Hence, the rate of production of heat is proportional to $r^{5}$.