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, $\frac{dQ}{dt}=F\times v_T$
where, $F$ is the viscous force and $v_T$ is the terminal velocity.
As, $F=6\pi \eta r{v}_T$
$\Rightarrow \frac{dQ}{dt}=6\pi \eta r{v}_T\times v_T=6\pi \eta r{v}_T^2$...(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
$\frac{dQ}{dt}\propto r\cdot {\left(r^2\right)}^2$
or$\frac{dQ}{dt}\propto r^5$
Hence, the rate of production of heat is proportional to $r^5$.