Question

A small sphere of radius ${}^{\prime }{r}^{\prime }$ 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. $r^4$
• B. $r^3$
• C. $r^5$
• D. $r^2$

Answer 1

Answer: (C)

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