|MadSci Network: Physics|
Marisa, This is an interesting thought experiment, and definitely a great lead-in to the concept of buoyancy and Archimedes' Principle. Archimedes' Principle says that an object emmersed in a fluid will experience a force (called the buoyant force) equal in magnitude to the weight of the displaced fluid. We typically think of fluids as liquids, however gasses are also fluids, so Archimedes' Principle is a powerful observation that allows us to explain not only why boats float on water, but also why hot air balloons and other dirigibles float in the air. There are three forces acting on the ball: gravity, drag, and buoyancy. Note that I call the force "drag" because it is comprised both of surface friction and also pressure forces associated with the motion of an object through a fluid. Without going into much detail, I'll just point out that (as you might expect) the drag force will be zero when the object is not moving relative to the fluid, and will increase in magnitude the faster the object is travelling relative to the fluid. Gravity is pulling the object down and buoyancy is trying to hold it up, with drag acting opposite to the direction of motion (in this case, also up). Since you want to know if the object will ever stop, we only need to consider if the buoyant force will ever equal the force due to gravity, because if that happens, the ball will stop accelerating downward and the drag force will cause it to eventually stop. If the buoyant force gets larger than the force of gravity, the ball will stop moving downward and eventually accelerate back up. So, how do we know if the buoyant force will ever equal the force due to gravity? You guessed it: Archimedes' Principle. Again, it says that the buoyant force will equal the weight of the displaced fluid, and in order to make the buoyant force equal the force due to gravity, the displaced fluid would have to weigh as much as the object itself. Since the displaced fluid would take up the same volume as the object, what we are really asking is: will the fluid ever have the same density as the object? In the last reference I've listed, there is a simple graph of water density versus depth in a typical ocean on earth. You'll note that the density increases until you get to a certain depth and then it stays about the same. If the density of your object is greater than this maximum density of the ocean water, the object will sink to the bottom of the ocean. If it is less than the minimum density (the density at the surface), the object will float on the surface. If it is somewhere in between, the object will sink to the point where the density of the object equals the density of the water, and it will "float" there. This is called being "neutrally buoyant." But you haven't really asked about a typical ocean on earth, you've proposed a hypothetical situation where you have an infinitely deep ocean. Honestly I'm not sure if an infinitely deep ocean would behave like an ocean on earth, where the density levels off at some maximum value, or if the density could somehow keep increasing because of the huge pressure forces that would be acting at an infinitely deep level (I guess they'd be infinitely strong). Trying to think about what an infinite force would do to water makes my head hurt, and that's why I said at the beginning that this is a very interesting thought experiment to me. The fact is though that the oceans on earth are deep enough that the density stays roughly constant through the majority of the ocean. The layers near the surface where the density of the water changes are relatively thin when compared to the total depth. This leads me to believe that even if you have a very very deep ocean (let's leave out the infinite part since that will keep my head from hurting), the density profile will be similar to an ocean on earth. I hope this has been a helpful discussion and good luck in your continued quest for knowledge. David Coit References: http://www.lerc.nasa.gov/WWW/K-12/WindTunnel/Activities/buoy_Archimedes.html http://www.infoplease.com/ce6/sci/A0804583.html http://hyperphysics.phy-astr.gsu.edu/hbase/pbuoy.html http://www.windows.ucar.edu/cgi-bin/tour.cgi/earth/Water/density.html [note added by MadSci reviewer: Using the Google search engine to search on "density pressure water" reveals density-pressure graphs at London South Bank University. There it is shown that the density of water can go as high as 2.4 g/cm^3 when the pressure is at about 10^12 Pascals near "room temperature", but at that temperature and pressure the phase is one of the exotic forms of solid ice. At terrestrial temperatures water is some form of solid when the pressure is above about 10^10 Pascals.]
Try the links in the MadSci Library for more information on Physics.