Question

A small sphere initially at rest, falls in to a viscous liquid. Due to drag, heat is produced. Then the relation between rate of production of heat and the radius of the sphere at terminal velocity, is best represented by:

• A. $\propto r^2$
• B. $\propto r^5$
• C. $\propto r^6$
• D. $\propto r^3$

Answer 1

The correct option is B $\propto r^5$


We have that
Terminal velocity, $V_T=\frac{2r^{2}g}{9\eta }({\rho }_s-{\rho }_L)...\left(i\right)$
$r\to$ radius of sphere
$g\to$ acceleration due to gravity
$y\to$ coefficient of viscosity
${\rho }_s\to$ density of sphere
${\rho }_L\to$ density of fluid.

$\&$ viscous Force (Drag force):
$F=6\pi \eta r{V}_T...\left(ii\right)$ for [$v=V_T$]
$\because$rate of production of heat$\frac{dQ}{dt}=$Rate of energy production
$\Rightarrow \frac{dQ}{dt}=\text{Power(P)}$
Hence we can write as:
$P=F\times V_T$
$P=6\pi \eta r{V}_T\times V_T$
Substituting from Eq $\left(i\right)\&\left(ii\right)$,
$P=6\pi r\eta {\left[\frac{2r^{2}g}{9\eta }\right({\rho }_s-{\rho }_L\left)\right]}^2$
Or, $P=6\pi \eta r^5{\left[\frac{2g}{9\eta }\right({\rho }_s-{\rho }_L\left)\right]}^2$
$\therefore P\propto r^5$