Properties of Matter 4 Question 14
16. If the piston is pushed at a speed of $5 mms^{-1}$, the air comes out of the nozzle with a speed of
(a) $0.1 ms^{-1}$
(b) $1 ms^{-1}$
(c) $2 ms^{-1}$
(d) $8 ms^{-1}$
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Answer:
Correct Answer: 16. $\frac{\lambda a}{2 y}$
Solution:
- Terminal velocity $v _T=\frac{2 r^{2} g}{9 \eta}\left(\rho _S-\rho _L\right)$
and viscous force $F=6 \pi \eta r v _T$
Rate of production of heat (power) : as viscous force is the only dissipative force.
Hence,
$$ \begin{aligned} \frac{d Q}{d t}=F v _T & =\left(6 \pi \eta r v _T\right)\left(v _T\right)=6 \pi \eta r v _T^{2} \\ & =6 \pi \eta r{\frac{2}{9} \frac{r^{2} g}{\eta}\left(\rho _S-\rho _L\right) }^{2} \\ & =\frac{8 \pi g^{2}}{27 \eta}\left(\rho _S-\rho _L\right)^{2} r^{5} \text { or } \frac{d Q}{d t} \propto r^{5} \end{aligned} $$
