Problem 16 - College Physics

Question

What gauge pressure must a pump produce to pump water from the bottom of the Grand Canyon (elevation 730 \(\mathrm{m} )\) to Indian Gardens (elevation 1370 \(\mathrm{m} ) ?\) Express your result in pascals and in atmospheres.

Step-by-Step Solution

Answer

The pump must produce 6,278,400 Pa or about 61.96 atm of gauge pressure.

Step 1: Understand the Problem

We need to calculate the gauge pressure required for a pump to move water from the bottom of the Grand Canyon to Indian Gardens, which are at different elevations. This involves understanding the change in height and how it influences the pressure needed.

Step 2: Calculate Height Difference

The change in elevation is calculated by subtracting the elevation at the bottom of the Grand Canyon from that of Indian Gardens. \[\text{Height difference} = 1370\,\mathrm{m} - 730\,\mathrm{m} = 640\,\mathrm{m}\]

Step 3: Apply the Hydrostatic Pressure Formula

Gauge pressure due to a fluid column is given by the formula:\[ P = \rho gh \]where \( \rho = 1000\,\mathrm{kg/m^3} \) is the density of water, \( g = 9.81\,\mathrm{m/s^2} \) is the acceleration due to gravity, and \( h = 640\,\mathrm{m} \) is the height difference.

Step 4: Perform the Calculation (Pascals)

Substitute to find the gauge pressure in pascals:\[P = 1000\,\mathrm{kg/m^3} \times 9.81\,\mathrm{m/s^2} \times 640\,\mathrm{m} = 6,278,400\,\mathrm{Pa}\]

Step 5: Convert Pascals to Atmospheres

Convert the pressure from pascals to atmospheres, knowing that 1 atmosphere is equivalent to 101,325 pascals:\[\text{Pressure in atm} = \frac{6,278,400\,\mathrm{Pa}}{101,325\,\mathrm{Pa/atm}} \approx 61.96\,\mathrm{atm}\]

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid in equilibrium due to the gravitational pull. In the context of this exercise, it refers to the pressure exerted by water when it is being pumped from a lower elevation to a higher one.
A core principle in hydrostatics is that the pressure at a point in a fluid at rest is due to the weight of the fluid above it. This weight creates a pressure increase, calculated with the formula:\[ P = \rho gh \]
where:

• \(P\) is the hydrostatic pressure
• \(\rho\) is the fluid's density
• \(g\) is the acceleration due to gravity
• \(h\) is the height of the fluid column

The formula allows us to see how directly the pressure is linked to the height of the fluid column. It's why an increase in elevation leads to an increase in hydrostatic pressure. This relationship is linear, meaning the pressure rises proportionally with height.
Understanding hydrostatic pressure and its computations is crucial for scenarios like pumping water uphill, as it dictates the amount of force needed to overcome the gravitational force on the water.

Height Difference

The height difference between two points in a fluid system significantly affects the gauge pressure required to move the fluid across those points. In our exercise, we're interested in the height difference between the bottom of the Grand Canyon and Indian Gardens.
To determine the height difference, simply subtract the lower elevation from the higher: 1370 meters - 730 meters = 640 meters. This value represents the additional distance the water must be pushed vertically, requiring increased pressure.
Height difference plays a key role in determining how much effort (i.e., pressure) is necessary to move fluids against gravity. This is especially relevant in environments such as mountain regions, where elevation changes can be substantial.
Thus, when tasked with such problems, calculating the height difference is always an essential step. It directly influences the amount of applied force required to achieve fluid movement from one point to another.

Pascal to Atmosphere Conversion

Pressure can be measured in various units, with pascals and atmospheres being two common ones. Pascal is the standard unit of pressure in the International System of Units (SI), while atmospheric pressure (atm) is another convenient unit representing the pressure exerted by Earth's atmosphere at sea level.
This conversion becomes necessary when a problem's requirements specify different units for the final answer.
To convert from pascals to atmospheres, use the conversion factor:

• 1 atm = 101,325 pascals

For example, if you have a pressure of 6,278,400 pascals, converting to atmospheres involves dividing by 101,325. Thus, the pressure in atmospheres is approximately 61.96 atm.
Understanding conversions is crucial in science and engineering, as it ensures that calculations and interpretations are consistent with the expectations and standards of various fields. In practice, being comfortable with these conversions can help when comparing results across different systems and regions.