The Tilting Glass: Understanding the Dynamics

[Quinzio]

Today I had the sink filled with water and some dirty dishes in.
At the water surface, an empty glass was floating.
I noticed that it wasn't standing perpendicular to the surface but it was tilted at and angle.
I put i tperpendicular and gently let it free and it returned in the same position.

There has to be a torque that is trying to shift the glass from the vertical position.
I would like to have a clear understanding of the dynamics, if there's a flaw in my logic please tell me.
The glass has a center of mass CoM, which is fixed.
The buoyant force can be imagined in the CoM of the submerged volume filled with water.
Now, if the water CoM is below the glass CoM a torque will appear (unless the two CoM are perfectly one above the other, but that's nearly impossible).
As the glass tilts, the water CoM will move wrt the glass CoM
The movement stops when the two CoM stay one over the other, possibly with the water one above the glass CoM.

[PLAIN]http://img191.imageshack.us/img191/6923/glassk.jpg

 

[SteamKing]

There are two possibilities.
1. The c.g. of the glass may be offset to one side. This might be caused by the glass not having the same thickness around its circumference. There could also be something which is not uniform with the base of the glass.

2. The vertical c.g. of the glass might be located high enough above the bottom such that the glass is slightly unstable when floating. You would also need to check that the water is deep enough in the sink such that the glass is freely floating when tilted. Assuming the glass is freely floating and slightly unstable, any disturbance to the glass will cause it to try to find an equilibrium floating position, and it will tilt until this occurs. If the glass is unstable enough, no equilibrium position can be found, and the glass will capsize, or turn over completely.

 

[Bill_K]

Quinzio, Good question! There are two forces acting on the glass. One, the downward force due to gravity on the mass which is concentrated in the bottom of the glass. This force also produces a torque on the glass that tries to stand it upright. Two, the buoyancy force acting at the center of the submerged part of the glass. This force also produces a torque that tries to tip it over. The glass will reach an equilibrium position when two things happen: (a) the vertical forces balance, and (b) the torques from these forces balance.

 

[Quinzio]

There are two possibilities.
1. The c.g. of the glass may be offset to one side. This might be caused by the glass not having the same thickness around its circumference. There could also be something which is not uniform with the base of the glass.

I have no mean to check the perfect symmetry of the glass. I just tried to lay the glass on one side on a flat surface (the kitchen table). If the glass slightly rolls to find a preferred equilibrium point, then it means the CoM is not in the center. I have not found any evidence of that.

If you look at this picture, that I found on the web, even the legendary bottle in the sea stays tilted on one side, it doesn't stay vertical nor horizontal
[PLAIN]http://www.uk-mouse-mats.co.uk/images/welcome/floating_bottle_400.jpg

2. The vertical c.g. of the glass might be located high enough above the bottom such that the glass is slightly unstable when floating. You would also need to check that the water is deep enough in the sink such that the glass is freely floating when tilted. Assuming the glass is freely floating and slightly unstable, any disturbance to the glass will cause it to try to find an equilibrium floating position, and it will tilt until this occurs. If the glass is unstable enough, no equilibrium position can be found, and the glass will capsize, or turn over completely.

 

[Bill_K]

There has to be a torque that is trying to shift the glass from the vertical position.

On the contrary! The natural position of the floating glass is entirely horizontal, on its side. It's only the downward torque exerted by the mass at the bottom of the glass that makes it stand partially upright.

Why does it want to lie horizontal? Because that's the position with minimum energy. By Archimedes' Principle the glass displaces an amount of water whose weight is equal to the weight of the glass. And this weight will be the same whether the glass is tilted or not. But just as an air bubble rises to the surface, the air in the glass wants to be high up, as close to the surface as possible. In order to make the glass stand on end, you have to force the lower end of the glass down farther into the water, thus increasing its potential energy.