A house at the bottom of a hill is fed by a full tank of water 6.0 m deep and connected to the house by a pipe that is 75 m long at an angle of 61° from the horizontal (Fig. 13–53). Determine the water gauge pressure at the house.

Verified step by step guidance

Understand the problem: The water gauge pressure at the house is determined by the height of the water column above the house. Gauge pressure is the pressure due to the water column alone, ignoring atmospheric pressure. The height of the water column can be calculated using the geometry of the problem.

Step 1: Identify the height of the water column. The tank is 6.0 m deep, and the pipe is inclined at an angle of 61° to the horizontal. The vertical height of the pipe can be calculated using trigonometry. The total height of the water column is the sum of the depth of the tank and the vertical height of the pipe. Use the formula: \( h_{pipe} = L \sin(\theta) \), where \( L = 75 \ \text{m} \) and \( \theta = 61° \).

Step 2: Add the depth of the tank to the vertical height of the pipe to find the total height of the water column: \( h_{total} = h_{tank} + h_{pipe} \).

Step 3: Use the formula for gauge pressure due to a fluid column: \( P_{gauge} = \rho g h_{total} \), where \( \rho \) is the density of water (typically \( 1000 \ \text{kg/m}^3 \)), \( g \) is the acceleration due to gravity (\( 9.8 \ \text{m/s}^2 \)), and \( h_{total} \) is the total height of the water column calculated in the previous step.

Step 4: Substitute the values for \( \rho \), \( g \), and \( h_{total} \) into the formula to calculate the gauge pressure. Ensure the units are consistent throughout the calculation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It is calculated using the formula P = ρgh, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the height of the fluid column. In this scenario, the depth of the water tank (6.0 m) contributes to the pressure experienced at the house.

Gauge Pressure

Gauge pressure is the pressure relative to atmospheric pressure. It is measured using a gauge that does not account for atmospheric pressure, thus providing the pressure exerted by the fluid alone. In this case, the gauge pressure at the house will be determined by the hydrostatic pressure from the water column, minus the atmospheric pressure, which is typically not included in the calculation.

Fluid Dynamics in Pipes

Fluid dynamics in pipes involves the study of how fluids move through conduits, influenced by factors such as pipe length, diameter, and angle. The angle of the pipe (61°) affects the effective height of the water column and the pressure at the outlet. Understanding these dynamics is crucial for calculating the pressure at the house, as it determines how the water's potential energy is converted into pressure energy.

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Textbook Question

A house at the bottom of a hill is fed by a full tank of water 6.0 m deep and connected to the house by a pipe that is 75 m long at an angle of 61° from the horizontal (Fig. 13–53). How high could the water shoot if it came vertically out of a broken pipe in front of the house?

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