Assume the bulk modulus to be constant and find an expression for the pressure as a function of depth h in the ocean. Use this expression and estimate the pressure assuming ρ0 = 2.00 slug/ft^3. Now, assume a constant density of 2.00 slug/ft^3 and calculate the pressure and the percent error,

Question

Assume the bulk modulus to be constant and find an expression for the pressure as a function of depth h in the ocean. Use this expression and estimate the pressure assuming [latex]ρ_0 = 2.00 slug/ft^3[/latex]. Now, assume a constant density of 2.00 [latex]slug/ft^3[/latex] and calculate the pressure and the percent error, assuming the estimate in the first calculation to be correct. Use depths of (a) 1500 ft, (b) 5000 ft, and (c) 15 000 ft.

Accepted Answer

Eq. 1.5.11 gives [latex]B= ho \frac{dp}{d ho }\mid _{T}[/latex] But, dp = ρgdh. Therefore,

[latex]ρ gdh= \frac{B}{ρ}dρ or \frac{dρ }{ρ^2}=\frac{g}{B}dh [/latex]

Integrate, using [latex]ρ_0 = 2.00 slug/ft^3, and B = 311,000 lb/in^2[/latex]:

[latex]\int\limits_{2}^{ ho }{\frac{dρ }{ρ^2}} =\frac{g}{B}\int\limits_{0}^{h}{dh}[/latex] . ∴

-[latex]\left\lgroup\frac{1}{ ho }-\frac{1}{2} ight group =\left\lgroup\frac{32.2 ft/s^2}{311,000 (lb/in^2) imes 144 in^2/ ft^2} ight group h= 7.2 imes 10^{-7}h[/latex]

This gives [latex] ho =\frac{1}{0.5- 7.2 imes 10^{-7} h}[/latex]

Now, [latex]p=\int\limits_{0}^{h}{ρ gdh}= \int\limits_{0}^{h}{\frac{g}{0.5- 7.2 imes 10^{-7} h}dh }=\frac{g}{- 7.2 imes 10^{-7}} ln(0.5 -7.2 imes 10^{-7}h )[/latex]

If we assume ρ = const: p = ρ gh = 2.0× 32.2× h = 64.4h

b) For h = 5000 ft: [latex]p_{accurate} =\underline{ 323,200 psf} and p_{estimate} = \underline{322,000 psf} [/latex].

% error =[latex]\frac{322,000 -323,200}{323,200} imes 100=\underline{-0.371 \%}[/latex]

Question 2.22: Assume the bulk modulus to be constant and find an expressio...

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