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  4. Two tubes of radii r1 and r2, and lengths l1 and l2, respectively, are connected in series and a liquid flows through each of them in stream line conditions. P1 and P2 are pressure differences across the two tubes. If P2 is 4P1 and l2 is (l1/4), then the radius r2 will be equal to :

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Q. Two tubes of radii $r_1$ and $r_2$, and lengths $l_1$ and $ l_2$, respectively, are connected in series and a liquid flows through each of them in stream line conditions. $P_1$ and $P_2$ are pressure differences across the two tubes.
If $P_2$ is $4P_1$ and $l_2$ is $\frac{l_1}{4}$, then the radius $r_2$ will be equal to :

JEE MainJEE Main 2017Mechanical Properties of Fluids

Solution:

$\frac{\phi v}{dt}=\frac{\pi}{8} \frac{pr^{4}}{nL}$
$\frac{p_{1}r^{4}_{1}}{L_{1}}=\frac{p_{2}r^{4}_{2}}{L_{2}}$
$\frac{p_{1}r^{4}_{1}}{l_{2}}=\frac{4p_{1}r^{4}_{2}}{l_{1/4}}=r_{2}^{4}=\frac{r_{1}^{4}}{16}$
$r_{2}=\frac{r_{1}}{2}$