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Under a constant pressure head, the rate of flow of orderly volume flow of liquid through a capillary tube is $V$. If the length of the capillary is doubled and the diameter of the bore is halved, then the rate of flow would become
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The correct answer is:
$V / 32$
Given, $V_{1}=V, l_{1}=l, r_{1}=r$
$$
\begin{aligned}
&l_{2}=2 l_{1}=2 l \\
&r_{2}=\frac{r_{1}}{2}=\frac{r}{2} \\
&V_{2}=?
\end{aligned}
$$
According to Poiseuille's formula, the rate of flow of liquid through a narrow tube is given as
$$
\begin{aligned}
& V=\frac{\pi p r^{4}}{8 n l} \\
\Rightarrow \quad & V \propto \frac{r^{4}}{l} \\
\Rightarrow \quad & \frac{V_{2}}{V_{1}}=\left(\frac{r_{2}}{r_{1}}\right)^{4} \cdot \frac{l_{1}}{l_{2}}
\end{aligned}
$$
$$
\begin{aligned}
&=\left(\frac{r / 2}{r}\right)^{4}\left(\frac{l}{2 l}\right)=\left(\frac{1}{2}\right)^{4} \cdot \frac{1}{2}=\frac{1}{32} \\
\Rightarrow \quad \quad V_{2} &=\frac{V_{1}}{32}=\frac{V}{32}
\end{aligned}
$$
$$
\begin{aligned}
&l_{2}=2 l_{1}=2 l \\
&r_{2}=\frac{r_{1}}{2}=\frac{r}{2} \\
&V_{2}=?
\end{aligned}
$$
According to Poiseuille's formula, the rate of flow of liquid through a narrow tube is given as
$$
\begin{aligned}
& V=\frac{\pi p r^{4}}{8 n l} \\
\Rightarrow \quad & V \propto \frac{r^{4}}{l} \\
\Rightarrow \quad & \frac{V_{2}}{V_{1}}=\left(\frac{r_{2}}{r_{1}}\right)^{4} \cdot \frac{l_{1}}{l_{2}}
\end{aligned}
$$
$$
\begin{aligned}
&=\left(\frac{r / 2}{r}\right)^{4}\left(\frac{l}{2 l}\right)=\left(\frac{1}{2}\right)^{4} \cdot \frac{1}{2}=\frac{1}{32} \\
\Rightarrow \quad \quad V_{2} &=\frac{V_{1}}{32}=\frac{V}{32}
\end{aligned}
$$
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