What is the intensity of pressure in the ocean at a depth of 5000 ft, assuming that (a) salt water is incompressible and (b) salt water is compressible and weighs 64.0 1b/ft3 at the surface? E = 300,000 psi (constant).

Question

What is the intensity of pressure in the ocean at a depth of 5000 ft, assuming that (a) salt water is incompressible and (b) salt water is compressible and weighs 64.0 1b/ft³ at the surface? E = 300,000 psi (constant).

Accepted Answer

(a) Intensity of pressure p = [latex]\gamma [/latex]h = (64.0)(5000) = 320,000 psf gage.

(b) Since a given mass does not change its weight as it is compressed, dW = 0; then

[latex]dW=d(\gamma v)=\gamma dv+vd\gamma =0[/latex] or [latex]dv/v=-d\gamma /\gamma [/latex] (A)

From equations (4) , [latex]R=\frac{R_{0}}{Mg}=\frac{universal gas constant}{molar weight} [/latex] , and (11), [latex]E=\frac{dp}{-dv/v} [/latex] , the latter from Chapter 1, [latex]dp=-\gamma dh [/latex] and [latex]dv/v=-dp/E [/latex].

Substituting in (A),

[latex]dp/E =d\gamma /\gamma [/latex] (B)

Integrating, [latex]p=E ln \gamma +C[/latex]. At the surface, [latex]p=p_{0} [/latex] , [latex]\gamma =\gamma _{0} [/latex] ; then [latex]C=p_{0} [/latex] — E ln [latex]\gamma _{0} [/latex] and

[latex]p=E 1n \gamma +p_{0}-E ln\gamma _{0}[/latex] or [latex]p-p_{0}=Eln(\gamma /\gamma _{0})[/latex] (C)

Putting [latex]dp=-\gamma dh[/latex] in (B) , [latex]\frac{-\gamma dh}{E}=\frac{d\gamma }{\gamma } [/latex] or [latex]dh=-\frac{Ed\gamma }{\gamma ^2} [/latex] . Integrating,

[latex]h=E/\gamma +C_{1} [/latex] (D)

At the surface, [latex]h=0[/latex] , [latex]\gamma =\gamma _{0} [/latex] ; then [latex]C_{1}=-E/\gamma _{0} [/latex] , [latex]h=(E/\gamma )-(E/\gamma _{0}) [/latex] and hence

[latex]\gamma =\frac{\gamma _{0}E}{\gamma _{0}h+E}=\frac{(64.0)(300,000 imes 144)}{(64.0)(-5000)+(300,000 imes 144)}=64.5 1b/ft^3 [/latex]

remembering that h is positive upwards and using psf units for E. From (C),

[latex]p=(300,000 imes 144)1n(64.5/64.0)=336,000 [/latex] psf gage

Question 2.23: What is the intensity of pressure in the ocean at a depth of...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions