A small sphere of radius r falls from rest in a viscous liquid. As...

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 terminal velocity, is proportional to

Options

A ${r}^{5}$
B ${r}^{2}$
C ${r}^{3}$
D ${r}^{4}$

Correct answer

A. ${r}^{5}$

Step-by-step solution

Rate of heat produced = power of viscous force

terminal velocity

{V}_{T} = \dfrac{2{r}^{2} ( ρ - \sigma )g}{9η}

⇒ {V}_{T} ∝ {r}^{2}

\dfrac{dQ}{dt} = 6\text{\pi η}r{V}_{T} \times {V}_{T}

⇒ \dfrac{dQ}{dt} ∝ {r}^{5}

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