# Archimedes Principle to determine height of continental crust

1. Jul 5, 2011

### kmam

I'm taking an introductory geology class and am having big trouble with the math questions, as it's been many many years since I last took math classes.
I am to use Archimedes Principle to calculate the thickness of continental crust beneath a certain coastal geographical point (at sea level and not accounting for post glacial rebound effects).

Data given:
Water depth to the abyssal plain: 5000m
Thickness of abyssal oceanic crust: 10000m
Density of continental crust: 2800 kg/m3
Density of oceanic crust: 3000 kg/m3
Density of mantle: 3300 kg/m3
Density of water: 1000 kg/m3

Our course material does not include any formula for Archimedes Principle, just a brief description of how it works. So, I've googled to find a formula, but really am having a hard time understanding what it is I'm looking at. I have the basic formula, but I'm not seeing how I should use it to get what I need: Apparent immersed weight=weight-weight of displaced fluid.

I understand that the fluid here is the mantle, with density given for it, and of course the cont. crust density appears relevant, but that's about as far as I can get with this! And I don't understand why there is data for oceanic crust, water depth and density of water....

I'm assuming there are some other formulas involved in this calculation. I know thickness can be calculated from Volume=height*length*width. But nothing in the data seems to hint at length or width....?? aaah, this is so confusing, I don't know where to begin even. Any hints or tips will be greatly appreciated.

2. Jul 6, 2011

### Staff: Mentor

Why not take a volume which is 1m x 1m x height in metres? I'm guessing you should consider that floating on magma is a column of continent, and alongside that is a column of ocean and sea floor. If these columns were not displacing the same weight of magma, then one would sink and the other rise until equilibrium was achieved. Or something like that.

3. Jul 9, 2011

### jeannagui

well it's not clear for me but i'll try to help:
the one u’ve found(immersed weight=weight-weight of displaced fluid) is used in case the body is not floating totaly immersed well it’s not the case because
density of the body(continental crust: 2800 kg/m3) less than that of the fluid(mantle: 3300 kg/m3)
so in that case the right one is: Fg = Fp weight of floating body (↓) = boyant force (↑)
thickness means height so try using this:
Fg = Fp
rs g Vs = rl g Vi ……………….. V=Ah
rs g As hs = rl g Ai hi …………..As=Ai
rs g hs = rl g hi

where: - rs density of the floating body ( ~ continental crust)
- g accn due to gravity (9.8)
- Vs volume of floating body ( ~ continental crust) As its area hs its height
-------------------------------------------------------------------------------------------------------------------
- rl density of the liquid where the body floats
- Vi the volume under water of the water(~mantle)

4. Jul 29, 2011

### Gunilla

did you get to know how to solve the problem Kmam? I am doing the same course now and got stuck in the same question.
gunillaserin@hotmail.com

5. Nov 6, 2013

### Techivegg

Hello. I am on the same course, and was searching for the answer to the same question. In case anyone else is looking for it in the future, here is a picture and the equation to use:

https://dl.dropboxusercontent.com/u/8908063/equation.png [Broken]
[mirror: http://i.imgur.com/q5Ejoqe.png]

Note, that in this particular quetsion, zh does not exist as we're assuming we're at sea level. So just remove zh from the equation all together and you'll be fine :)

You begin with the equation:

zc.ρc = zw.ρw + zo.ρo + [zc-(zh + zw + zo)].ρm

First, you rearrange the equation so that zc is on its left hand side as follows:

zc.ρc = zw.ρw + zo.ρo + zc.ρm -(zh + zw + zo).ρm
zc.ρc - zc.ρm = zw.ρw + zo.ρo -(zh + zw + zo).ρm
zc.[ρc - ρm] = zw.ρw + zo.ρo -(zh + zw + zo).ρm
zc = [zw.ρw + zo.ρo -(zh + zw + zo).ρm]/[ρc - ρm]

Then you subsitute the values you were given for thickness and density.

Then you solve the equation!

Cheers!

Last edited by a moderator: May 6, 2017