PYQ NEET- Mechanical Properties Of Fluids L-3

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) $r^5$

B) $r^2$

C) $r^3$

D) $r^4$

Answer: $r^5$#

Sol:#

Key Concept The rate of heat generation is equal to the rate of work done by the viscous force which in turn is equal to its power. Rate of heat produced, $\frac{d Q}{d t}=F \times v_T$ where, $F$ is the viscous force and $v_T$ is the terminal velocity. As, $$ \text { As, } \quad \begin{aligned} \quad F & =6 \pi \eta r v_{\mathrm{T}} \ \Rightarrow \quad \frac{d \underline{d}}{d t} & =6 \pi \eta r v_{\mathrm{T}} \times v_{\mathrm{T}} \ & =6 \pi \eta r v_{\mathrm{T}}^2 \end{aligned} $$

From the relation for terminal velocity, $$ \begin{gathered} v_T=\frac{2}{9} \frac{r^2(\rho-\sigma)}{\eta} g \text {, we get } \ v_T \propto r^2 \end{gathered} $$

From Eq. (ii), we can rewrite Eq. (i) as

$$ \begin{aligned} \frac{d Q}{d t} & \propto r \cdot\left(r^2\right)^2 \ \text { or } \quad & \frac{d Q}{d t} \propto r^5 \end{aligned} $$