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Bernoulli's Principle

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Question

Water is flowing through a channel (lying in a vertical plane) as shown in the figure. Three sections $A, B$ and $C$ are shown. Section $B$ and $C$ have equal areas of cross- section. If $P_{A}, P_{B}$ and $P_{C}$ are the pressures at $A, B$ and $C$ respectively, then

P_{A} < P_{B} = P_{C}

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P_{A} < P_{B} < P_{C}

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P_{A} > P_{B} = P_{C}

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P_{A} > P_{B} > P_{C}

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Solution

The correct option is B $P_{A} < P_{B} < P_{C}$
Applying Bernoulli's principle and equation of continuity.
Comparing points $A$ and $B$ (the vertical elevation $h$ is same for both),
$A_{A} V_{A} = A_{B} V_{B}$ (equation of continuity)
$∵ A_{A} < A_{B}$
$\Rightarrow V_{A} > V_{B}$
$P_{A} + \frac{1}{2} ρ V_{A}^{2} + ρ g h = P_{B} + \frac{1}{2} ρ V_{B}^{2} + ρ g h$ (Bernoulli's equation)
$∵ V_{A} > V_{B}$
In accordance with Bernoulli's equation,
$\frac{1}{2} ρ V_{A}^{2} > \frac{1}{2} ρ V_{B}^{2}$
$∴ P_{A} < P_{B} . . . (1)$
Now comparing $C$ and $B$
Area of cross-section $A_{B} = A_{C}$
$\Rightarrow V_{B} = V_{C} = V$ , from equation of continuity
$P_{B} + \frac{1}{2} ρ V^{2} + ρ g h_{B} = P_{C} + \frac{1}{2} ρ V^{2} + ρ g h_{C}$
(Applying Bernoulli's equation between points $B$ and $C$ )
$\Rightarrow P_{B} + ρ g h_{B} = P_{C} + ρ g h_{C}$
$∵ h_{B} > h_{C}$
$\Rightarrow P_{B} < P_{C} . . . (2)$
Using Eq. $(1)$ and $(2)$ we get,
$P_{A} < P_{B} < P_{C}$

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Bernoulli's Principle

Standard XII Physics

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Water is flowing through a channel (lying in a vertical plane) as shown in the figure. Three sections A,B and C are shown. Section B and C have equal areas of cross- section. If PA,PB and PC are the pressures at A,B and C respectively, then

Water is flowing through a channel (lying in a vertical plane) as shown in the figure. Three sections $A, B$ and $C$ are shown. Section $B$ and $C$ have equal areas of cross- section. If $P_{A}, P_{B}$ and $P_{C}$ are the pressures at $A, B$ and $C$ respectively, then