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 :-

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 3 (B) r 2 (C) r5 (D) r 4

Tardigrade Admin · Accepted Answer

Rate of heat produced ( d Q / dt )= F v × v T ∵ v T =(2 r 2/9 η)(ρ-σ) g propto r 2 =6 π η rv textrmT × v T ⇒ ( d Q / dt ) propto rv T 2 propto r 5

Q. 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 :-

$r^{3}$

$r^{2}$

$r^{5}$

$r^{4}$

Rate of heat produced
$\frac{d Q}{d t} = F_{v} \times v_{T}$
$∵ v_{T} = \frac{2 r ^{2}}{9 η} (ρ - σ) g \propto r^{2}$
$= 6 π ηr v_{T} \times v_{T}$
$\Rightarrow \frac{d Q}{d t} \propto r v_{T}^{2} \propto r^{5}$