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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 6.7.66
Chapter 6, Problem 6.7.66

Orientation and force A plate shaped like an equilateral triangle 1 m on a side is placed on a vertical wall 1 m below the surface of a pool filled with water. On which plate in the figure is the force greater? Try to anticipate the answer and then compute the force on each plate.
Two submerged plates, one diamond-shaped and one square, each 1m below water surface, with equal side lengths of 1m.

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Step 1: Understand the problem context. We have two plates submerged 1 meter below the water surface. One plate is diamond-shaped (a square rotated 45 degrees) and the other is a square, both with side lengths of 1 meter. We need to determine which plate experiences a greater force due to water pressure and then calculate the force on each plate.
Step 2: Recall the formula for hydrostatic force on a submerged vertical surface: \(F = \rho g A \bar{h}\), where \(\rho\) is the density of water, \(g\) is acceleration due to gravity, \(A\) is the area of the plate, and \(\bar{h}\) is the depth of the centroid of the plate below the water surface.
Step 3: Calculate the area \(A\) of each plate. For the square plate, \(A = 1 \times 1 = 1\) m\(^2\). For the diamond-shaped plate, since it is a square rotated by 45 degrees with side length 1 m, its area is also \(1\) m\(^2\) (area of a square is side squared, rotation does not change area).
Step 4: Determine the depth of the centroid \(\bar{h}\) for each plate. For the square plate, the centroid is at the midpoint, so \(\bar{h} = 1 + \frac{1}{2} = 1.5\) m below the surface. For the diamond-shaped plate, the vertical height of the diamond is \(\sqrt{2}\) times the side length, so the centroid is at half this height below the surface plus the 1 m depth to the top vertex. Calculate \(\bar{h}\) accordingly.
Step 5: Use the hydrostatic force formula to compute the force on each plate by substituting \(\rho\), \(g\), \(A\), and \(\bar{h}\) for each plate. Compare the forces to determine which plate experiences the greater force.

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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 given by the formula P = ρgh, where ρ is the fluid density, g is gravitational acceleration, and h is the depth below the surface.

Force on a Submerged Surface

The force exerted by a fluid on a submerged surface is the product of the pressure and the area of the surface. Since pressure varies with depth, the total force is found by integrating the pressure over the surface area, often simplified by using the average pressure at the centroid depth.
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Centroid and Orientation of Submerged Plates

The centroid of a submerged plate determines the average depth at which pressure acts. The orientation affects the shape and position of the plate relative to the fluid surface, influencing the centroid depth and thus the hydrostatic force on the plate.
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