Chapter 16.9, Problem 32E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

A solid occupies a region *E* with surface *S* and is immersed in a liquid with constant density *ρ*. We set up a coordinate system so that the *xy*-plane coincides with the surface of the liquid, and positive values of *z* are measured downward into the liquid. Then the pressure at depth *z* is *p* = *ρgz*, where *g* is the acceleration due to gravity (see Section 8.3). The total buoyant force on the solid due to the pressure distribution is given by the surface integral
**F** = - *W***k**, where *W* is the weight of the liquid displaced by the solid. (Note that **F** is directed upward because *z* is directed downward.) The result is *Archimedes*’ *Principle:* The buoyant force on an object equals the weight of the displaced liquid.

To determine

**To show:** The expression

Explanation

**Given Data:**

A solid occupies a region

The pressure at the depth

Here,

Write the given surface integral that has to be evaluated as follows.

**Formula used:**

Write the expression to find flux of the vector field

Write the expression to find divergence of vector field

Write the expression for weight of the displace liquid when the solid immersed in the liquid.

Here,

Consider an arbitrary constant vector

Consider the vector field

Substitute

**Calculation of**
**:**

Substitute

Substitute

If

Consider

If

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