raindrop
What factors are involved in how fast a buoyant object rises to to the surface? Say I released a half-full water bottle from the bottom of a pool, is there a formula to predict its speed of ascent? Would the speed remain constant throughout the length of travel?

Any feedback is much appreciated.

PJParent001
It might be more interesting to calculate the amount of force the rising object would exert on a cable tethered to an electric generator.

meBigGuy
Again, PJParent001 cannot answer the question, but feels the need to reply.

The buoyancy of an object in water in determined by the difference in water pressure between the bottom and top of the object. This causes a net force upward (assuming the object is light enough).

So, we have a gravitational force pushing down, and a buoyancy force pushing up.

http://en.wikipedia.org/wiki/Buoyancy

Essentially the upward force is the weight of water displaced, and the downward force is the weight of the object.

Based on the force and the mass, you can calculate the acceleration.

But, wait. As the object rises, it displaces water, and that has a significant effect on the total acceleration.

http://www.physicsmyths.org.uk/buoyancy.htm

Hope that helps. If so a positive feedback would be appreciated.
bm1957
QUOTE (meBigGuy+Feb 8 2008, 10:27 AM)
Hope that helps. If so a positive feedback would be appreciated.

raindrop can't post feedback after 1 post! (it doesn't befit you to beg, mbg)
mr_homm
The physicsmyths.org link is interesting, but even they do not get it completely right! They are correct that some of the force goes into accelerating liquid downward, so that the observed acceleration will be less than what you would expect from applying F=ma to an isolated mass. However, they make the assumption that the water accelerates downward with acceleration -a when the mass has +a. This assumption is not generally true.

The picture of a box of buoyant material going up and a box of water going down nicely shows the net effect if the object's rise: it swaps places with some water. However, the actual path taken by the object is a straight line upward, while the actual path taken by the water is a flow pattern involving the entire fluid. For instance, consider what happens to the surface of a cup of water if you remove an immersed object. The surface goes down, right? Now apply this to the case of a fully submerged object moving upwards. Draw an imaginary horizontal plane just above the object, and then let the object move up past it. There now must be more water below the plane than there was before, and so water must have crossed this plane surface, heading downward.

Now, how exactly is this velocity distributed throughout the water? It is clearly not enough just to fill in the hole left behind by the object, because in order to reach that hole, the water must flow around the object, which will disturb the fluid farther away, which must also therefore flow, and so on. One way to look at it is to imagine that the object is held steady and there is a downward current in the water. Neglecting viscosity, and assuming the flow speed is not too high, this is Stokes flow. For a simple shape such as a sphere, stokes flow gives a velocity pattern which resembles a vertical cross section of an onion (not the horizontal cross section that gives rings, the other one).

Returning to the point of view where the bulk of the water is stationary except for the distubance created by the rising object, you must subtract the object's velocity from the Stokes flow (to account for the change of reference frame). This leaves a flow pattern which looks remarkably like the magnetic field of an ordinary bar magnet. It is basically a standard dipole field, only this field shows how the water is flowing, not how a magnetic field is oriented. Water will exit from the space just above the object (of course, since it's being forced out by the rising object), flow out and around, and get sucked back into the hole beneath the object.

Using energy to analyze the problem, when the mass rises a distance dx, the amount of gravitational energy released is (m_d - m)gdx, where m_d is the mass of displaced fluid, and this goes into the kinetic energy of both the upward mass motion and the flowing water motion. If the flow of water is spread widely through the fluid, then the actual velocity of the water flow may be rather low in most places, so that the water takes only a small share of the energy. In that case, the buoyant mass keeps most of the energy, and its upwards acceleration will therefore be nearly what F=ma says. On the other hand, if the flow of the fluid is confined, for instance if the buoyant mass is a 10cm diameter disk in a vertical tube of diameter 11cm, then the water must squeeze by the sides of the mass to get below it, and the water will then have a quite high velocity, taking a lot of the energy for its flow and leaving the mass very little.

That means that there is a limiting case (very narrow tube) in which the acceleration is very close to zero, even with no viscous drag. On the other hand, there is a limiting case of a very broad tube (say, the Atlantic ocean), in which the flow spreads out quite far. It is NOT clear, without doing the exact calculation, what the object's acceleration would be in this case. It would involve deriving the flow field and computing the ratio of its total kinetic energy to that of the rising object. If the object has a complicated shape, even getting the flow field will be nearly impossible. For a perfect sphere, I could probably do the calculation, if I had the leisure, but it would still be substantial.

So! Things are complicated, and that's even without including the fact that flows in the fluid will not respond instantaneously to the motion of the mass, but instead will be delayed by the speed of sound in the fluid, or that the sound itself will dissipate energy, further reducing the mass's acceleration, or that real fluids have viscosity, or that fast motion produces turbulence (more energy loss) or that real objects aren't spheres. Messy!

Hope that helps, but in this case I'm rather afraid it doesn't....

--Stuart Anderson
PJParent001
QUOTE (meBigGuy+Feb 8 2008, 03:27 AM)
Again, PJParent001 cannot answer the question, but feels the need to reply.

The buoyancy of an object in water in determined by the difference in water pressure between the bottom and top of the object. This causes a net force upward (assuming the object is light enough).

So, we have a gravitational force pushing down, and a buoyancy force pushing up.

http://en.wikipedia.org/wiki/Buoyancy

Essentially the upward force is the weight of water displaced, and the downward force is the weight of the object.

Based on the force and the mass, you can calculate the acceleration.

But, wait. As the object rises, it displaces water, and that has a significant effect on the total acceleration.

http://www.physicsmyths.org.uk/buoyancy.htm

Hope that helps. If so a positive feedback would be appreciated.