Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.7.7

Chapter 6, Problem 6.7.7

What is the pressure on a horizontal surface with an area of 2 m² that is 4 m underwater?

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Identify the physical principle involved: Pressure at a certain depth underwater is given by the hydrostatic pressure formula, which accounts for the pressure due to the water column above the surface.

Write down the formula for pressure at depth: \(P = P_0 + \rho g h\), where \(P\) is the total pressure at depth, \(P_0\) is the atmospheric pressure on the surface, \(\rho\) is the density of the water, \(g\) is the acceleration due to gravity, and \(h\) is the depth underwater.

List the known values: \(h = 4\) m (depth), \(\rho \approx 1000\) kg/m³ (density of water), \(g \approx 9.8\) m/s² (acceleration due to gravity), and \(P_0 \approx 101325\) Pa (atmospheric pressure at sea level).

Calculate the hydrostatic pressure component: multiply \(\rho\), \(g\), and \(h\) to find the pressure due to the water column.

Add the atmospheric pressure \(P_0\) to the hydrostatic pressure to find the total pressure on the horizontal surface. Note that the area of the surface does not affect the pressure value, as pressure is force per unit area.

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Key Concepts

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

Pressure in Fluids

Pressure in a fluid at a certain depth is the force exerted per unit area due to the weight of the fluid above. It increases with depth because more fluid is stacked above, creating greater force on the surface below.

Hydrostatic Pressure Formula

Hydrostatic pressure is calculated using the formula P = ρgh, where ρ is the fluid density, g is acceleration due to gravity, and h is the depth. This formula helps determine the pressure exerted by a fluid at a given depth.

Area and Pressure Relationship

Pressure is independent of the surface area on which it acts; it depends only on the fluid properties and depth. The total force on the surface can be found by multiplying pressure by area, but pressure itself remains constant regardless of area size.

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