[fgt55 (OP)]

When a mass move down it lost energy because it never move up with the same gravity (it move down).

[MikeS]

No that's not it.  There are two masses about a fulcrum.  Assuming the fulcrum to remain an equidistant from the Earth.  The system weighs more when the weights are in vertical alignment than when they are in horizontal alignment. 
When they rotate one mass moves down the other moves up so they balance and the energy of the system should remain constant.  In the vertical alignment the system weighs more which represents an increase in energy of the system.  The gravitational potential energy of the system remains the same.  As far as I understand it there should not be an increase of weight when the masses are vertically aligned.  The discrepancy, I believe is caused by not taking into account gravitational time dilation.

[fgt55 (OP)]

When they rotate one mass moves down the other moves up so they balance and the energy of the system should remain constant.

why ?

In the vertical alignment the system weighs more which represents an increase in energy of the system. 

it's normal for me, because one mass is lower and other upper. Don't forget there is a torque.

[MikeS]

Let me re-phrase the question to get rid of problems to do with torque and inertia.
Lets have your original system but two identical ones.  One vertical and one horizontal, both with the fulcrum the same height above the Earth.
Why is the vertical one heavier using Newtonian gravity?

[fgt55 (OP)]

I don't understand could you make drawings ? and explain step by step ?

[MikeS]

Just imagine your original drawing and one rotated through 90 degrees.
Both having the fulcrum points the same distance above the Earth.
The system where the weights are in vertical alignment weighs more as pointed out by yourself (at least that's what I thought you meant) and confirmed by imatfaal (at least that's what I think he meant).

I believe that they should weigh the same but the maths appear to say differently.  That is the vertical system weighs more than the horizontal system.

I believe the discrepancy arises because Newtonian laws of gravity are being used where relativistic laws of gravity should have been used.  I may be wrong.

[fgt55 (OP)]

I believe that they should weigh the same

for me no, the distance of the center is the same (center of masses) but the center of gravity is different (and physics say it can be different). Think with energy, the sum is always 0. Make calculations and maybe you'll find your error.

[MikeS]

You assume I have an error, interesting.
When you say center of gravity what do you mean?  The center of the gravitational potential or the center of gravitational potential energy.

[fgt55 (OP)]

You assume I have an error, interesting.

In fact, I don't understand all your problem...Have you a problem with sum of energy or what ? For me, the system has more weight when the masses are vertical. And what ? If you move down you lost potential energy. If you turn and move down you lost energy too and win, the sum is always 0. Make a calculation and I can help you to find the error.

When you say center of gravity what do you mean?  The center of the gravitational potential or the center of gravitational potential energy.

I'm not fluently in english. I don't understand these 2 differences.

Like you seem to like this problem. Maybe you'll like this another. Take a fixed container full with water. An object O like the drawing show is empty (air inside for example with small pressure). There isn't other mass around water or object O. The pressure at right is bigger than at left, so the object must turn with torque and give energy ? I'm sure not, but where is the solution ? How to calculate pressure in water even it's evidence the pressure at right is bigger than at left ?

[MikeS]

I think what you are suggesting above is similar to an experiment carried out by Richard Feynman.  He was trying to find out if the thrust of a fluid emitted through a nozzle worked in reverse.  That is if fluid were admitted by the nozzle would it generate thrust.  The result of the experiment was it did not but he blew up a carboy in the process.

See
http://en.wikipedia.org/wiki/Feynman_sprinkler

[MikeS]

Going back to the original problem as I understand it.
With the two weights horizontal they both weigh the same and both have the same gravitational potential energy.

With the weights vertical about the same fixed fulcrum point.  The higher weight now weighs less but has more GPE.  The lower weight now weighs more but has less GPE.

As I understand it the total weight of the system as obtained by using the Newtonian law of gravity when the weights are horizontal is less than the same system when the weights are vertical.  That's what I understand.

My question is, if you weighed the two systems would they weigh the same or not?  If not, would taking a relativistic approach predict that they should weigh the same?

[imatfaal]

Mike - bit rushed, but this is worth bearing in mind.- I know maths is not your favourite but here goes

Force due to gravity falls of inverse square

Ignoring all the constants and units etc - just the maths
Initial State with horizontal bar

Final state with vertical bar

sorry gonna have to go - will try and pickit up

[fgt55 (OP)]

My question is, if you weighed the two systems would they weigh the same or not? 

not the same

If not, would taking a relativistic approach predict that they should weigh the same?

I don't know, but relativistic is not here for me

With the weights vertical about the same fixed fulcrum point.  The higher weight now weighs less but has more GPE.  The lower weight now weighs more but has less GPE.

You're right, there is a problem with GPE and weight. Example with masses at radius of 3m from the center. The center is at 10 m from the object which attrack (like Earth, but closed). The weight in vertical position is 1/13²+1/7²=0.0263 and the weight in horizontal position is 2*1/(10²+3²)=0.0183. (I cancel constant). The weight in vertical position is bigger than in horizontal position.

For the GPE (from 1 to the position of the mass), we have mgh, in vertical position this is: (1-1/13)*13+(1-1/7)*10 = 18 compared with horizontal position (1-1/13.34)*13.34+(1-1/7.6)*7.6 = 18.94, the GPE in horizontal position is bigger, that's normal but not in this proportion !

This is strange for me.

Edit: it's ok, it's because we can move with the same length so energy is the same.

[fgt55 (OP)]

edit more later

[MikeS]

I think I have got my head around it at last. 
With the weights in vertical alignment the center of the gravitational potential is below the fulcrum point.  This is because the gravitational potential and hence weight varies by the square rule.  The bottom weight has gained more weight than the top weight has lost.  So, in vertical alignment the system weighs more than in horizontal alignment.

My thinking was wrong.  It's obvious, I don't know how I missed it. I was stuck in a rut and considering it in terms of energy, in which case I believe I would have been correct and it would need to be considered from a relativistic viewpoint to make the energies balance.

fgt55 sorry this has been such a long and hazardous uphill trek. ;-)

[fgt55 (OP)]

I'm thinking about a problem with pressure when I calculate the pressure under water. I put a conic volume under pressure under gravity. The pressure at left or at right is lower thn the pressure in the center, so if we remove parts of the volume for have a helicoid like the drawings show, the volume turn in water ? because the pressure seems to be differents ! The vertical surface is closer to the center than the horizontal surface, so each surface see a different torque ?