[T] A rectangular dam is 40 ft high and 60 ft wide. Compute the total force F on the dam when a.

Name: Problem 247, Chapter 2: Calculus Volume 2
Uploaded: 2020-05-26T17:06:13-08:00
Duration: 3 min 4 s
Channel: Chloe Hacker
Description: [T] A rectangular dam is 40 ft high and 60 ft wide. Compute the total force F on the dam when a. the surface of the water is at the top of the dam and b. the surface of the water is halfway down the dam.

step 1 So this question is asking us to figure out the pressure on a dam when the dam is full of water

Question

[T] A rectangular dam is 40 $\mathrm{ft}$ high and 60 $\mathrm{ft}$ wide. Compute the total force $F$ on the dam when a. the surface of the water is at the top of the dam and b. the surface of the water is halfway down the dam.

   [T] A rectangular dam is 40 $\mathrm{ft}$ high and 60 $\mathrm{ft}$ wide. Compute the total force $F$ on the dam when
a. the surface of the water is at the top of the dam and
b. the surface of the water is halfway down the dam.

Calculus Volume 2

Gilbert Strang, 1st Edition

Chapter 2, Problem 247

Instant Answer

Solved by Expert Chloe Hacker

05/26/2020

Step 1

The formula is given by $F = \int_{c}^{d} P \cdot h(y) \cdot l(y) \, dy$, where $P$ is the pressure of the water, $h(y)$ is the height function, and $l(y)$ is the length function. Show more…

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[T] A rectangular dam is 40 $\mathrm{ft}$ high and 60 $\mathrm{ft}$ wide. Compute the total force $F$ on the dam when a. the surface of the water is at the top of the dam and b. the surface of the water is halfway down the dam.

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Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the gravitational force acting on it. This pressure increases linearly with depth and is commonly expressed as P = ?gh, where ? is the fluid density, g is the acceleration due to gravity, and h is the depth below the free surface. Understanding this principle is essential when determining forces on submerged surfaces, such as the dam face in the problem.

Integration of Pressure Forces

Integration of pressure forces is the process of summing the variable pressure acting over a submerged surface to obtain the total force. Since the pressure varies with depth, we use calculus to integrate the pressure distribution over the area of the surface. This technique is crucial in fluid statics to calculate forces exerted on structures, accounting for changes in pressure with depth.

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