Q. Two water pipes of diameters $2 \, cm$ and $4 \, cm$ are connected with the main supply line. The velocity of flow of water in the pipe of $2 \, cm$ diameter is
NTA AbhyasNTA Abhyas 2022
Solution:
Given, $d_{A}=2cm$ and $d_{B}=4cm$
$\therefore r_{A}=1 \, cm$ and $r_{B}=2 \, cm$
From the equation of continuity $aV=$ constant, where $a$ is area of cross-section of pipe and $V$ is speed of water in pipe.
$\therefore \frac{V_{A}}{V_{B}}=\frac{a_{B}}{a_{A}}=\frac{\pi \left(\right. r_{B} \left.\right)^{2}}{\pi \left(\right. r_{A} \left.\right)^{2}}=\left(\frac{2}{1}\right)^{2} \, $
$\Rightarrow V_{A}=4V_{B}$
Hence, speed of water in pipe of $2cm$ diameter is $4$ times that of pipe of $4cm$ diameter.
$\therefore r_{A}=1 \, cm$ and $r_{B}=2 \, cm$
From the equation of continuity $aV=$ constant, where $a$ is area of cross-section of pipe and $V$ is speed of water in pipe.
$\therefore \frac{V_{A}}{V_{B}}=\frac{a_{B}}{a_{A}}=\frac{\pi \left(\right. r_{B} \left.\right)^{2}}{\pi \left(\right. r_{A} \left.\right)^{2}}=\left(\frac{2}{1}\right)^{2} \, $
$\Rightarrow V_{A}=4V_{B}$
Hence, speed of water in pipe of $2cm$ diameter is $4$ times that of pipe of $4cm$ diameter.