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 where n is the outer unit normal. Use the result of Exercise 31 to show that F = - Wk, 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 show: The expression is equals to the expression .
A solid occupies a region with surface and is immersed in a liquid with constant density .
The pressure at the depth is given as follows.
is the constant density and
is the acceleration due to the gravity.
Write the given surface integral that has to be evaluated as follows.
Write the expression to find flux of the vector field across the surface .
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.
is the volume of the solid occupied region .
Consider an arbitrary constant vector as follows.
Consider the vector field as follows.
Substitute for ,
Calculation of :
Substitute for , for , and for in equation (4),
Substitute for in equation (3),
If , then the expression is written as follows.
Consider and rewrite the expression as follows.
If , then the expression is written as follows
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