Properties of Matter 4 Question 12
14. Two spheres $P$ and $Q$ for equal radii have densities $\rho _1$ and $\rho _2$, respectively. The spheres are connected by a massless string and placed in liquids $L _1$ and $L _2$ of densities $\sigma _1$ and $\sigma _2$ and viscosities $\eta _1$ and $\eta _2$, respectively. They float in equilibrium with the sphere $P$ in $L _1$ and sphere $Q$ in $L _2$
and the string being taut (see figure). If sphere $P$ alone in $L _2$ has terminal velocity $\mathbf{v} _P$ and $Q$ alone in $L _1$ has terminal velocity $\mathbf{v} _Q$, then
(2015 Adv.)
(a) $\frac{\left|\mathbf{v} _P\right|}{\left|\mathbf{v} _Q\right|}=\frac{\eta _1}{\eta _2}$
(b) $\frac{\left|\mathbf{v} _P\right|}{\left|\mathbf{v} _Q\right|}=\frac{\eta _2}{\eta _1}$
(c) $\mathbf{v} _P \cdot \mathbf{v} _Q>0$
(d) $\mathbf{v} _P \cdot \mathbf{v} _Q<0$
Assertion and Reason
Mark your answer as
(a) If Statement I is true, Statement II is true; Statement II is the correct explanation for Statement I
(b) If Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I
(c) If Statement I is true; Statement II is false
(d) If Statement I is false; Statement II is true
Show Answer
Answer:
Correct Answer: 14. 3
Solution:
- Terminal velocity is given by
$$ \begin{aligned} v _T & =\frac{2}{9} \frac{r^{2}}{\eta}(d-\rho) g \\ \frac{v _P}{v _Q} & =\frac{r _P^{2}}{r _Q^{2}} \times \frac{\eta _Q}{\eta _P} \times \frac{\left(d-\rho _P\right)}{\left(d-\rho _Q\right)} \\ & =\left(\frac{1}{0.5}\right) \times\left(\frac{2}{3}\right) \times \frac{(8-0.8)}{(8-1.6)} \\ & =4 \times \frac{2}{3} \times \frac{7.2}{6.4}=3 \end{aligned} $$
