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LARGE Water Body 
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#1
Jan312, 01:37 PM

P: 47

We have heard of celestial bodies, but who heard of a large Water Body(made only of water)?
Considering that the water will not convert into other chemicals under pressure and enclosing it in a box full of 1 atm air and 300 K, How large/massful/volume should the water body be to get 9.81 m/s^2 at its surface? Consider:  Surface Tension  Gravitation Force  Atmospheric Pressure  Any other cohesive forces  Made of H2O, for H2O, and by H2O  If there is a variation in density, it is uniformally distributed through the body. I have tried to derive the formula for a given 'g', but the gravitational force and surface tension has to be considered for a pirticular 'area' and not a 'mass' to get the contraction/expansion of the water body. 


#2
Jan412, 06:14 AM

P: 882

You may neglect surface tension and similar forces. It counts for few milimeter droplets, but not for a large body.
If your body is not rotating (or the rotation is slow), you may assume spherical symmetry. You may probably assume that the density [itex]\rho [/itex] is uniform. So now the gravity on the surface is given as $$g(r) = \frac{4}{3}\pi \,r^3 \rho\cdot \frac{G}{r^2} = \frac{4\pi}{3}\,G\,\rho\, r$$ If you want to include the change of density with pressure you must write appropriate integrational equation and solve it, but as the [itex]\rho(P)[/itex] is nonlinear, it might be pretty difficult  you must use probably some iterative method or compute it numerically. 


#3
Jan412, 07:01 AM

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#4
Jan1112, 12:23 PM

P: 47

LARGE Water Body
And, if we do not consider the ρ to be uniform throughout the body at any instant, then the way to get to the answer will become much more complicated. And, density of the Earth varies with depth around many conditions and effects, as its not made of the same material. Here, we are considering that excessive pressure on water molecules down deep wont make different compounds. 


#5
Jan1112, 06:34 PM

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P: 15,201

You cannot with any validity assume that density for a "LARGE water body" (your words) will be uniform. Water is much more compressible than is iron, and a ball of solid iron at the center of the Earth is compressed to 3/5 its vacuum density due to the weight of the mass above it.



#6
Jan1212, 10:13 AM

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#7
Jan1212, 12:18 PM

P: 168

It makes radius of earth would be around 35 Km with all water mass. Almost 6 times larger than the current radius. Is it physically possible to have all liquid mass no rocky core? 


#8
Jan1212, 05:47 PM

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PF Gold
P: 12,253

Could you be sure that there would be no drastic molecular change near the centre under that pressure? (Or perhaps that's what's implied at the end of the previous post.)



#9
Jan1212, 05:51 PM

P: 418




#10
Jan1312, 04:41 AM

P: 47

Uhh, if we consider the 'practicality' of stuff, such as dissoctiation of h2o at high pressures, variation in density, etc. etc. etc. then please do derive the answer using the same, =D
Cos, in the short sight, if we consider too many stuff, we will get too complicated of an answer. 


#11
Jan1312, 05:35 AM

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PF Gold
P: 12,253

This thread needs some ground rules. Are we dealing with an incompressibke fluid (I.e. total theory) or a half way house involving the 'linear' modulus of water OR are we attempting to go further? We need to decide. There's no point arguing between different standpoints.



#12
Jan1312, 07:07 AM

P: 47

For an incompressible fluid, we already have the answer:
Once we get the above answer, only then we can talk about density variation and chemical changes. Its better to go step by step from rudimentary to advanced. So, can somebody give me an idea who/what to integrate for compressible fluid with uniform density. And we are considering bulk modulus. So, there will be a change in radius. 


#13
Jan1312, 07:28 AM

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P: 15,201

Hydrostatic equilibrium, [itex]\partial \rho/\partial r = \rho g[/itex].



#14
Jan1312, 07:49 AM

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PF Gold
P: 12,253

It strikes me that you could look at the equations for a gas atmosphere (and the derivations used) and find a way to include modulus in the formula. 


#15
Jan1312, 12:30 PM

P: 47

Uniform, i.e. if the liquid collapses, its net density increase uniformly everywhere, hence density is uniform. Its not like earth, where we have varying densities as we go deep. 


#16
Jan1312, 01:20 PM

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PF Gold
P: 12,253

"its net density increase uniformly everywhere"??
I think you mean its modulus is uniform or the substance is the same throughout. Uniform means the same everywhere and its density wouldn't be. It would increase in density towards the centre  which is not what 'uniform' means'. 


#17
Jan1312, 03:44 PM

P: 177

It turns out that the water in the central regions of the ball turns into a high pressure form of ice (ice IV assuming the ball is isothermal at 300K) long before the ball is massive enough to have a surface gravity of 9.81m/s^{2} (at least according to my preliminary calculations and what I have heard on TV). I am getting values of between ~2229Km and ~2360Km for the minimum radius to produce ice (assuming it forms at about 9500bar, the corresponding surface gravities are: 0.697 to 0.709m/s^{2}), depending on the value of the compressibility I use (assuming it is constant, the larger value corresponding to one that interpolates between the minimum and maximum densities, the smaller to that which obtains at everyday pressures). I might be able to extend this analysis to the necessary ball masses if I knew how we wanted to treat the density above 1GPa, for which I don't have any quantitative data (though I expect it exists somewhere on the internet). 


#18
Jan1412, 12:38 PM

P: 47




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