A pump, submerged at the bottom of a well that is $ 35 $ $ \text{m} $ deep, is used to pump water uphill to a house that is $ 50 $ $ \text{m} $ above the top of the well, as shown to the right. The density of water is $ 1000 $ $ \text{kg/m}^3 $. All pressures are gauge pressures. Neglect the effects of friction, turbulence, and viscosity.

Question

A pump, submerged at the bottom of a well that is $ 35 $ $ \text{m} $ deep, is used to pump water uphill to a house that is $ 50 $ $ \text{m} $ above the top of the well, as shown to the right. The density of water is $ 1000 $ $ \text{kg/m}^3 $. All pressures are gauge pressures. Neglect the effects of friction, turbulence, and viscosity.(a) Residents of the house use $ 0.35 $ $ \text{m}^3 $ of water per day. The day’s pumping is completed in $ 2 $ $ \text{hours} $ during the day. Calculate the minimum work required to pump the water used per day.
, (b) Calculate the minimum power rating of the pump.
, (c) The average pressure the pump actually produces is $ 9.20 \times 10^5 $ $ \text{N/m}^2 $. Within the well the water flows at $ 0.50 $ $ \text{m/s} $ and the pipe has a diameter of $ 3.0 $ $ \text{cm} $. At the house, the pipe diameter is $ 1.25 $ $ \text{cm} $. Calculate the flow velocity when a faucet in the house is open.
, (d) Explain how you would calculate the minimum pressure at the faucet.
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Jason · Accepted Answer

$\boxed{W \approx 2.92 \times 10^5\,\text{J}}$
$\boxed{P \approx 40.5\,\text{W}}$
$\boxed{v_x \approx 2.88\,\text{m/s}}$
The minimum pressure at the faucet is calculated by equating the pressure energy per unit volume to the sum of the gravitational potential energy $\rho g\,\Delta x$ and the dynamic pressure $\frac{1}{2}\rho\,v_x^2$, i.e., $P_{\text{min}} = \rho g\,\Delta x + \frac{1}{2}\rho\,v_x^2$. In short: apply Bernoulli’s principle.

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