Water is pumped up from the Colorado River to supply Grand...

Water is pumped up from the Colorado River to supply Grand Canyon Village, located on the rim of the canyon. The river is at an elevation of $564 \mathrm{~m},$ and the village is at an elevation of $2096 \mathrm{~m} .$ Imagine that the water is pumped through a single long pipe $15.0 \mathrm{~cm}$ in diameter, driven by a single pump at the bottom end. (a) What is the minimum pressure at which the water must be pumped if it is to arrive at the village? (b) If $4500 \mathrm{~m}^{3}$ of water is pumped per day, what is the speed of the water in the pipe? Note: Assume the free-fall acceleration and thedensity of air are constant over this range of elevations. The pressures you calculate are too high for an ordinary pipe. The water is actually lifted in stages by several pumps through shorter pipes.

Water is pumped up from the Colorado River to supply Grand Canyon Village, located on the rim of the canyon. The river is at an elevation of $564 \mathrm{~m},$ and the village is at an elevation of $2096 \mathrm{~m} .$ Imagine that the water is pumped through a single long pipe $15.0 \mathrm{~cm}$ in diameter, driven by a single pump at the bottom end.
(a) What is the minimum pressure at which the water must be pumped if it is to arrive at the village? (b) If $4500 \mathrm{~m}^{3}$ of water is pumped per day, what is the speed of the water in the pipe? Note: Assume the free-fall acceleration and thedensity of air are constant over this range of elevations. The pressures you calculate are too high for an ordinary pipe. The water is actually lifted in stages by several pumps through shorter pipes.

Key Concepts

Hydrostatic Pressure

This concept refers to the pressure exerted by a fluid at a given depth due to the force of gravity. It is calculated using the equation pressure = density × gravitational acceleration × height difference, which in this context determines the minimum pressure needed to lift water from a lower elevation to a higher elevation.

Continuity Equation and Volumetric Flow Rate

In fluid dynamics, this principle asserts that the mass flow rate through a pipe remains constant for an incompressible fluid. It relates the volumetric flow rate to the cross?sectional area of the pipe and the speed of the fluid (Q = A × v). This concept is used to determine the velocity of the water in the pipe given the flow volume per unit time.

Transcript

00:01 Hi there.

00:02 So for this problem with the information that we are given for part a we are asked about what is the minimum pressure at which the water must be pumped if it is to arrive of the village so we need to find that in that pressure okay, so we know that the cross -sectional area is the same everywhere so the speed is the same everywhere so for this we can apply the following equation that the pressure p plus one divided by 2 times the density times the speed to the square, this plus the density times the acceleration due to gravity, times the height.

00:39 This corresponds to the quantities for the river.

00:43 And the quantities for the rim is the pressure, plus 1 divided by 2, the density, the speed to the square, and this plus the density times the acceleration due to gravity times y, and this for the rim.

01:00 Once we know this, we just need to substitute all of these values in here.

01:07 So we know that the pressure at the river is the one that we want to determine.

01:11 So we leave it just like that.

01:13 The first expression in here is zero because the speed is zero.

01:18 And this is also zero.

01:20 So we are left with just the density, the acceleration to the gravity, and the height of the river.

01:26 Then this is equal to the pressure at the rim that is one at the river.

01:32 Atmosphere, this plus the density, the acceleration to gravity, and the height of the rim.

01:40 So the pressure then is going to be equal to one atmosphere plus the density, the acceleration to gravity, and the difference between the height of the rim minus the height of the river.

01:58 So we now need to substitute the values in here.

02:00 So we will have that the pressure is equal to one atmosphere plus the density of water that we know is 1 ,000 kilogram per cubic meter.

02:12 This time the acceleration due to the gravity that is 9 .8 meters per second square.

02:17 The height of the rim is given and that is 2096 meters minus the height of the river that we are also given.

02:25 That is 564.

02:27 So from this, we obtain a pressure of one atmosphere plus 15 mega, where mega means 10 to 6, pascal...

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