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Water enters a house through a pipe with 2.0cm inside diameter, at an absolute pressure of $4 \times 10^{5} Pa$ . The pipe leading to the second floor bathroom 5m above is 1.0cm in diameter. When the flow velocity at the inlet pipe is $4 m s^{- 1}$ find the flow velocity and pressure in the bathroom in $m s^{- 1}$ and Pa respectively.

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$16, 2.3 \times 10^{5}$

$16, 3.2 \times 10^{5}$

$132, 2.3 \times 10^{5}$

$32, 2.3 \times 10^{5}$

answer is A.

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Detailed Solution

By continuity equation, the flow velocity

$v_{2} = \frac{A_{1} v_{1}}{A_{2}} = \frac{π (0.01)^{2}}{π (0.005)^{2}} \times 4 = 16 {ms}^{- 1}$

Using Bernoulli’s equation between 1 and 2, we have

$p_{1} + \frac{1}{2} ρ v_{1}^{2} + ρ g h_{1} = p_{2} + \frac{1}{2} ρ v_{2}^{2} + ρ g h_{2}$

$p_{2} = p_{1} - \frac{1}{2} ρ (v_{2}^{2} - v_{1}^{2}) - ρ g (h_{2} - h_{1})$

$= 4 \times 10^{5} - \frac{1}{2} \times 1000 (16^{2} - 4^{2})$

$- 1000 \times 9.8 \times 5 = 2.3 \times 10^{5} Pa$

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