<p>Your approach of adding all the forces together is correct. You have a mistake in in the solution of the $F_{oc}$ integral, though.
You forgot to integrate the terms $\rho_{w}gh_{w}$ and $-\rho_{oc}gh_{w}$</p>

<p>It should read:</p>

<p>$F_{oc} = ... = \rho_{w}gh_{w}(h_{w}+h_{oc}) - \rho_{w}gh_{w}h_{w} + 0.5\rho_{oc}g(h_{w}+h_{oc})^{2} - 0.5\rho_{oc}gh_{w}^{2} - \rho_{oc}gh_{w}(h_{w}+h_{oc}) + \rho_{oc}gh_{w}h_{w} $</p>

<p>You have to keep this in mind for $F_{M}$ as well.</p>

CC BY-SA 3.0
2016-09-15T17:06:13.447Z
2016-09-15T17:11:38.197Z
2016-09-15T17:11:38.197Z
ye-ti-800
<p>In the book of <a href="https://books.google.com/books?id=-nCHlVuJ4FoC&amp;pg=PA80&amp;source=gbs_toc_r&amp;cad=4#v=onepage&amp;q&amp;f=false" rel="nofollow">Turcotte and Schubert, Geodynamics</a> I'm trying to solve problem 2.8.</p>

<p>The pressure in the water, in the oceanic crust, and in the mantle beneath the oceanic crust are (from equation 2.19, page 78) is</p>

<p>$$p_w = \rho_w g y , \quad  0 \le y \le h_{w}$$
$$p_{oc} = \rho_w g h_{w} + \rho_{oc} g (y - h_{w}) , \quad  h_{w} \le y \le h_{w}+h_{oc}$$</p>

<p>$$p_m = \rho_w g h_w + \rho_{oc} g h_{oc} + \rho_m g (y-h_w-h_{oc}), \quad  h_w+h_{oc} \le y \le h_{cc}$$
I should integrate that to find the force at the base of the oceanic crust. </p>

<p>we know that </p>

<p>$$F=∫_0^b pdy=ρg∫_0^by dy=0.5 ρgb^2$$</p>

<p>so</p>

<p>$$F_m=∫_{h_w+h_{oc}}^{h_{cc}} p_mdy$$</p>

<p>then I tried this</p>

<p>$$F_m=∫_{h_w+h_{oc}}^{h_{cc}}\rho_w g h_w + \rho_{oc} g h_{oc} + \rho_m g (y-h_w-h_{oc})dy$$</p>

<p>and it didn't provide the proper result, so i tried to integrate all the pressures:</p>

<p>$$F_w=∫_0^{h_{w}}\rho_w g ydy = 0.5 \rho_w g h_{w}^2$$
$$F_{oc} = ∫_{h_w}^{h_{w}+h_{oc}} \rho_w g h_{w} + \rho_{oc} g (y - h_{w})dy= \rho_w g h_{w}+0.5 \rho_{oc} g (h_{w}+h_{oc})^2 - 0.5 \rho_{oc} g h_{w}^2 - \rho_{oc} g h_{w}$$</p>

<p>I did the same for the last integral and added them all together: $$F_o = F_w+F_{oc}+F_m$$ </p>

<p>but I still don't get the same final expression as equation 2.20, page 78.</p>

<p>Can someone give me a hint to solve the integral? are the limits of my integral correct? Am I adding the right forces together? </p>

CC BY-SA 3.0
2015-07-24T19:32:52.110Z
I deleted the previous post because I thought I had found the answer, but actually not !
user1166251
CC BY-SA 3.0
2015-07-26T10:19:20.843Z
I wish I had brought the book with me, and my solutions, as I've definitely done this problem.  I will try and get a book from the library of the university I am visiting and answer this tomorrow.
Neo
CC BY-SA 3.0
2015-07-28T00:36:59.110Z
That would be great !
user1166251
<geophysics><geodynamics><isostasy>
How to define the force at the base of an oceanic crust
user1166251
# How to define the force at the base of an oceanic crust

(Asked by: username_0 on 2015-07-24)

In the book of [Turcotte and Schubert, Geodynamics](https://books.google.com/books?id=-nCHlVuJ4FoC&pg=PA80&source=gbs_toc_r&cad=4#v=onepage&q&f=false) I'm trying to solve problem 2.8.

The pressure in the water, in the oceanic crust, and in the mantle beneath the oceanic crust are (from equation 2.19, page 78) is

$$p\_w = \rho\_w g y , \quad 0 \le y \le h\_{w}$$
$$p\_{oc} = \rho\_w g h\_{w} + \rho\_{oc} g (y - h\_{w}) , \quad h\_{w} \le y \le h\_{w}+h\_{oc}$$

$$p\_m = \rho\_w g h\_w + \rho\_{oc} g h\_{oc} + \rho\_m g (y-h\_w-h\_{oc}), \quad h\_w+h\_{oc} \le y \le h\_{cc}$$
I should integrate that to find the force at the base of the oceanic crust.

we know that

$$F=∫\_0^b pdy=ρg∫\_0^by dy=0.5 ρgb^2$$

so

$$F\_m=∫\_{h\_w+h\_{oc}}^{h\_{cc}} p\_mdy$$

then I tried this

$$F\_m=∫\_{h\_w+h\_{oc}}^{h\_{cc}}\rho\_w g h\_w + \rho\_{oc} g h\_{oc} + \rho\_m g (y-h\_w-h\_{oc})dy$$

and it didn't provide the proper result, so i tried to integrate all the pressures:

$$F\_w=∫\_0^{h\_{w}}\rho\_w g ydy = 0.5 \rho\_w g h\_{w}^2$$
$$F\_{oc} = ∫\_{h\_w}^{h\_{w}+h\_{oc}} \rho\_w g h\_{w} + \rho\_{oc} g (y - h\_{w})dy= \rho\_w g h\_{w}+0.5 \rho\_{oc} g (h\_{w}+h\_{oc})^2 - 0.5 \rho\_{oc} g h\_{w}^2 - \rho\_{oc} g h\_{w}$$

I did the same for the last integral and added them all together: $$F\_o = F\_w+F\_{oc}+F\_m$$

but I still don't get the same final expression as equation 2.20, page 78.

Can someone give me a hint to solve the integral? are the limits of my integral correct? Am I adding the right forces together?

**Tags:** geophysics, geodynamics, isostasy

> **Comments:**
> **username_0**: I deleted the previous post because I thought I had found the answer, but actually not !  
> **username_1**: I wish I had brought the book with me, and my solutions, as I've definitely done this problem.  I will try and get a book from the library of the university I am visiting and answer this tomorrow.  
> **username_0**: That would be great !  

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## 1 answers

### Answer by username_2 on 2016-09-15. Score: 5 | Accepted answer

Your approach of adding all the forces together is correct. You have a mistake in in the solution of the $F\_{oc}$ integral, though.
You forgot to integrate the terms $\rho\_{w}gh\_{w}$ and $-\rho\_{oc}gh\_{w}$

It should read:

$F\_{oc} = ... = \rho\_{w}gh\_{w}(h\_{w}+h\_{oc}) - \rho\_{w}gh\_{w}h\_{w} + 0.5\rho\_{oc}g(h\_{w}+h\_{oc})^{2} - 0.5\rho\_{oc}gh\_{w}^{2} - \rho\_{oc}gh\_{w}(h\_{w}+h\_{oc}) + \rho\_{oc}gh\_{w}h\_{w} $

You have to keep this in mind for $F\_{M}$ as well.


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