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Vapour Pressure

Two tubes of ...

Question

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 $4 P_{1}$ and $l_{2}$ is $\frac{l_{1}}{4}$ , then the radius $r_{2}$ will be equal to :

A

2 r_{1}

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B

4 r_{1}

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C

r_{1}

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D

\frac{r_{1}}{2}

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Solution

The correct option is A $2 r_{1}$
Given, $P_{2} = {4 P}_{1}$ & $l_{2} = \frac{l_{1}}{4}$
By the equation of continuity:
$A_{1} v_{1} = A_{2} v_{2}$
$\frac{A_{1}}{A_{2}}$ = $\frac{v_{1}}{v_{2}}$
Pressure is inversely proportional to velocity.
So, $\frac{P_{1}}{P_{2}}$ = $\frac{v_{2}}{v_{1}}$
So, $\frac{v_{2}}{v_{1}}$ = $\frac{1}{4}$
Thus,
$\frac{A_{1}}{A_{2}}$ = $\frac{1}{4}$
$\frac{r_{1}^{2}}{r_{2}^{2}}$ = $\frac{1}{4}$
$r_{2} = 2 r_{1}$

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