A horizontal tube has different cross sections at point A and B- The areas of cross section are a 1 and a 2- respectively- and pressures at these points are p 1 - g h 1 and p 2 - g h 2- where - is the density of liquid flowing in the tube and h 1 and h 2 are heights of liquid column in vertical tubes connected at A and B- If h 1 - h 2 - h- then the flow rate of the liquid in the horizontal tube is -

The correct option is A a _ 1 a _ 2 √( 2gh a _ 1 ^ 2 a _ 2 ^ 2 ) By Bernoulli's equation at A and B P_0+ρ gh_1+12ρ V_1^2=P_0+ρ gh_2+12ρ ...

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Standard XII

Physics

SHM expression

A horizontal ...

Question

A horizontal tube has different cross sections at point $A$ and $B$ . The areas of cross section are $a_{1}$ and $a_{2}$ , respectively, and pressures at these points are $p_{1} = ρ g h_{1}$ and $p_{2} = ρ g h_{2}$ , where $ρ$ is the density of liquid flowing in the tube and $h_{1}$ and $h_{2}$ are heights of liquid column in vertical tubes connected at $A$ and $B$ . If $h_{1} - h_{2} = h$ , then the flow rate of the liquid in the horizontal tube is :

a_{1} a_{2} \sqrt{\frac{2 g h}{a_{1}^{2} - a_{2}^{2}}}

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a_{1} a_{2} \sqrt{\frac{2 g}{h (a_{1}^{2} - a_{2}^{2})}}

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a_{1} a_{2} \sqrt{\frac{(a_{1}^{2} + a_{2}^{2}) h}{2 g (a_{1}^{2} - a_{2}^{2})}}

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\frac{2 a_{1} a_{2} g h}{\sqrt{a_{1}^{2} - a_{2}^{2}}}

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Solution

The correct option is A $a_{1} a_{2} \sqrt{\frac{2 g h}{a_{1}^{2} - a_{2}^{2}}}$
By Bernoulli's equation at $A$ and $B$

$P_{0} + ρ g h_{1} + \frac{1}{2} ρ V_{1}^{2} = P_{0} + ρ g h_{2} + \frac{1}{2} ρ V_{2}^{2} ρ g h_{1} + \frac{ρ V_{1}^{2}}{2} = ρ g h_{2} + \frac{ρ V_{2}^{2}}{2} ρ g (h_{1} - h_{2}) = \frac{ρ}{2} (V_{2}^{2} - V_{1}^{2}) \frac{2 ρ g h}{ρ} = V_{2}^{2} - V_{1}^{2} = 2 g h \to (1)$

Flow rate $= a_{1} v_{1} = a_{2} v_{2}$

$\Rightarrow v_{1} = \frac{a_{2} v_{2}}{a_{1}} \to (2)$

From equation $(1)$ and $(2)$

$v_{2}^{2} - (\frac{a_{2} v_{2}}{a_{1}})^{2} = 2 g h v_{2}^{2} [1 - \frac{a_{2}^{2}}{a_{1}^{2}}] = 2 g h v_{2}^{2} [\frac{a_{1}^{2} - a_{2}^{2}}{a_{1}^{2}}] = 2 g h v_{2}^{2} = \frac{a_{1}^{2} 2 g h}{a_{1}^{2} - a_{2}^{2}} v_{2} = \sqrt{\frac{a_{1}^{2} (2 g h)}{a_{1}^{2} - a_{2}^{2}}}$
Flow rate $= a_{2} v_{2} = a_{1} a_{2} \sqrt{\frac{2 g h}{a_{1}^{2} - a_{2}^{2}}}$

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