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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.7.56
Chapter 6, Problem 6.7.56

Force on the end of a tank Determine the force on a circular end of the tank in Figure 6.78 if the tank is full of gasoline. The density of gasoline is ρ = 737 kg/m³.

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Identify the physical principle involved: The force on the circular end of the tank due to the gasoline is caused by the hydrostatic pressure exerted by the fluid at different depths.
Recall the formula for hydrostatic pressure at a depth \(h\) in a fluid: \(P = \rho g h\), where \(\rho\) is the fluid density, \(g\) is the acceleration due to gravity, and \(h\) is the depth below the fluid surface.
Determine the shape and dimensions of the circular end of the tank to express the pressure distribution over its surface. Since pressure varies with depth, the force is found by integrating the pressure over the area of the circular end.
Set up the integral for the total force \(F\) on the circular end: \(F = \int P \ dA = \int \rho g h \ dA\), where \(dA\) is an infinitesimal area element on the circular surface and \(h\) is the depth corresponding to that element.
Express \(h\) in terms of the vertical coordinate on the circular end, set the limits of integration according to the radius of the circle, and perform the integration to find the total force. Remember to multiply by \(g = 9.81 \ \text{m/s}^2\) and use \(\rho = 737 \ \text{kg/m}^3\).

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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 increases linearly with depth and is calculated as P = ρgh, where ρ is the fluid density, g is gravitational acceleration, and h is the depth below the surface.

Force on a Surface in a Fluid

The force exerted by a fluid on a submerged surface is the product of the pressure at that surface and the area. For curved or vertical surfaces, the pressure varies with depth, so the total force is found by integrating pressure over the surface area.
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Density and Its Role in Fluid Statics

Density (ρ) is the mass per unit volume of a substance and directly affects the hydrostatic pressure in a fluid. A higher density fluid exerts greater pressure at a given depth, influencing the force on submerged surfaces like the tank end.
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Related Practice
Textbook Question

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Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

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Use the general slicing method to find the volume of the following solids.

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Textbook Question

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