# B Buoyancy problem

1. Dec 9, 2016

### farolero

theres one beam of wood of density 0.5 floating on water of density one

the beam is vertical and i assume it will go to horizontality

this force that moved the wood beam implies a work has been done and hence potential energy has been spent

but if the beam has density 0.5 it means it has the same volume underwater than over water either horizontally or vertically hence its cog hasnt varied height and on the same manner the level of water remains in the same level with respect to the cog of the beam of wood

so from where comes the potential gravitational energy that has applied the work on the wood to move it from vertically floating to horizontally floating

my conclusion from this thought experiment is that a 0.5 density beam will float indiferently horizontally or vertically to kep conservation of energy true

is this correct?

2. Dec 9, 2016

### Drakkith

Staff Emeritus
The center of mass of the beam is much further underwater when the beam is vertical than when it is horizontal. Hence it is more energetically favorable for it to float horizontally than vertically. If it helps, try to imagine cutting the beam into thin blocks. When oriented vertically, the blocks corresponding to the bottom end of the beam had to have much more work performed on them to counteract the buoyancy force than the blocks making up the top end and the water they are displacing is now at a higher level than it otherwise would be.

3. Dec 9, 2016

### farolero

well i dont see that what you say, check this diagram, the cog of the beam remains in same position weather the beam floats vertically or horizontally:

4. Dec 9, 2016

### Drakkith

Staff Emeritus
The beam will not float vertically that way. It will fall over into the horizontal position.

5. Dec 9, 2016

### farolero

im not very sure of that if the cog of the beam doesnt move nor the level of water moves

whay does it fall then? how has the beam spent potential energy as to move naturally?

maybe its imposible to get an exact 0.5 density and the slightest offset could make the beam lean?

after all ships as seen from in front are nearly vertical

6. Dec 9, 2016

Why?

7. Dec 9, 2016

### jbriggs444

Floating horizontally or floating vertically, the center of mass of the beam does not change. It is located exactly at the surface of the water. Half of the beam is submerged and half floating either way.

However, the center of mass of the water is displaced downward when the beam rotates to horizontal. The hole in the water where the beam resides has moved upward and water has moved downward to fill in that space.

8. Dec 9, 2016

### A.T.

Consider the torque from the buoyant force around the CoM for small perturbations of those two extremes.

9. Dec 9, 2016

### Drakkith

Staff Emeritus
I was under the impression that the beam was underwater.

10. Dec 9, 2016

### Drakkith

Staff Emeritus
That's right. A vertical orientation is an unstable position for the beam to be in, as even the slightest force will disturb it and send it rotating to the horizontal position.

The part of the beam underwater wants to move upwards, while the part above water wants to move downwards. Both of these can happen if the beam rotates even slightly.

Of course. They are bottom-heavy, so their center of masses are under the water. If they weren't, they'd flip upside down or onto their sides.

11. Dec 9, 2016

### farolero

i see so to solve the problem i take the cog of the half overwater and the cog of the half underwater

then potential energy is spent in both cogs, is this correct?

but this sound strange to me how can something has two cogs?

12. Dec 9, 2016

### jbriggs444

You are missing the point. The center of gravity of the beam is utterly irrelevant. The center of gravity of the water is the bit that matters.

13. Dec 9, 2016

### farolero

oh i see so potential energy to move the vertical beams comes from the water lowering its cog

thanks a lot i wouldnt have seen it without help

14. May 19, 2017

### Crowxe

consider the beam is 2 beams with half length attached together , and each has its own cog ...cog 1 above water and cog 2 under water, now the lower beam fully submerged and have a buoyancy force acting almost exactly at the same point of cog 2 upward with 0.5 , and the upper beam have have a downward force acting on cog 1 with amount of 1.0

2 forces acting oppositely, must tip over

15. May 20, 2017

### AlphaLearner

Yes, for a body to float vertically, center of gravity should remain below the surface of water. This beam (mass uniformly distributed) half immersed vertically in stagnant water will have its center of gravity at center shifted above the surface of water due to apparent reduce in weight due to buoyancy force faced at lower half. Still the beam can float vertically as long as the water and air above is absolutely stagnant. Once disturbed, even slightest of the forces acting at upper or lower half of beam perpendicularly create torque and result in collapsing horizontally. Consider the same beam with greater area of cross - section and lesser height, shift in center of gravity will reduce and tries to keep it closer to center but never at center or below center due to buoyancy, chances of flipping reduce as greater torque is required. In order to bring the resultant center of gravity to center of beam or below center, mass of lower part of beam should be increased, means as long as the beam has mass uniformly distributed, it is impossible to keep it floating vertical as long as water is not stagnant or air above water is not stagnant.

I thought on my own and wrote it But... after seeing above threads, All I have done was summarizing all the above threads. I was just late to give a reply to this post

Last edited: May 20, 2017
16. May 20, 2017

### AlphaLearner

Water don't lower cog. It pushes the cog of beam upwards. Even if the beam flips horizontally, the cog will be at the upper half of beam that's why it is easier to roll a beam floating horizontally. But when compared to vertical orientation, horizontal orientation will have cog at lesser height compared to vertical height and since water flipped the beam in such a way, Yes, water has lowered its cog to bring it to static equilibrium. Be clear.

17. May 20, 2017

### jbriggs444

This is a simplistic view that assumes that center of pressure does not move appreciably as the object rocks from side to side. That assumption may be justified in the case at hand, but cannot be justified in general. Take for instance a flat balsa raft floating in the mercury with a mast extending vertically above the center. The center of mass of the raft+mast can be above the surface of the mercury but the raft nonetheless floats in a stable horizontal orientation. This is because the center of pressure moves more dramatically under small deflections than the center of gravity.

That is a more correct condition for stability -- that the center of mass deflects less than the center of pressure under small rotations.

Last edited: May 20, 2017
18. May 20, 2017

### AlphaLearner

Means center of pressure shifts itself in order to minimize shift in center of gravity to maintain stability? Then should effectiveness of shift in center of pressure to reduce change in center of gravity should depend on area of cross - section of body floating on surface?

19. May 20, 2017

### jbriggs444

It's not purely area. Consider a catamaran or pontoon boat. You improve stability by moving the pontoons farther apart. But that does not change either the surface cross-section or the wetted surface.

20. May 20, 2017

### AlphaLearner

But, in case of this beam, area of cross section is less, no features to move Its ends farther like pontoons then it seems effect of center of pressure too less or considered negligible. I don't feel extending pontoons will effect center of pressure rather increase moment of inertia to prevent flipping on...