SATHEE: Properties of Matter 4 Question 14

Properties of Matter 4 Question 14

16. A small sphere falls from rest in a viscous liquid. Due to friction, heat is produced. Find the relation between the rate of production of heat and the radius of the sphere at terminal velocity.

(2004, 2M)

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Answer:

Correct Answer: 16. $\frac{d Q}{d t} \propto r^{5}$

Solution:

Terminal velocity $v_{T} = \frac{2 r^{2} g}{9 η} (ρ_{S} - ρ_{L})$

and viscous force $F = 6 π η r v_{T}$

Rate of production of heat (power) : as viscous force is the only dissipative force.

Hence,

$\begin{aligned} \frac{d Q}{d t} = F v_{T} & = (6 π η r v_{T}) (v_{T}) = 6 π η r v_{T}^{2} \\ = 6 π η r [\frac{2}{9} \frac{r^{2} g}{η} (ρ_{S} - ρ_{L})]^{2} \\ = \frac{8 π g^{2}}{27 η} {(ρ_{S} - ρ_{L})}^{2} r^{5} or \frac{d Q}{d t} \propto r^{5} \end{aligned}$

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Properties of Matter 4 Question 14

16. A small sphere falls from rest in a viscous liquid. Due to friction, heat is produced. Find the relation between the rate of production of heat and the radius of the sphere at terminal velocity.

Answer:

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