[Physics] Better explanation of why an object floats: density? or buoyancy in equilibrium with object’s weight

forcesgravitywater

Across StackExchange, you see two explanations for why an object floats:

  1. The buoyant force, equal to weight of displaced water, is in equilibrium with the object's weight.
  2. The density of the object is less than the density of water.

Explanation #1 is great, but #2 strikes me as incomplete/imprecise for non-convex objects? My question: Is there a sensible, concise definition of an object's volume and mass such that an object floats if and only if its density (i.e. mass/volume) is less than the density of water? If so, are explanation #1 and #2 of equal quality?

Example where density explanation looks problematic:

(i) identical objects except orientation. (ii) red object floats, blue object sinks

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  • The red and blue objects are identical except the blue object is
    rotated approx. 95 degrees.

    • The mass of the red and blue objects are identical.
    • I would think a definition of volume would be invariant to rotation? Hence, red and blue objects have identical volume, hence identical density?
  • There exists a mass for the identical objects such that the red object floats while the blue object sinks if placed on top of a body of water.

The orientation of the red object leads to greater water displacement than the blue object when placed on water. What's going on is an obvious application of Archimedes Principle. On the other hand, an explanation involving density seems rather more complicated? For some density explanation to work, an object's volume must include all air in the interior of the object below waterline (which will depend on the orientation of the object). Fine definition of volume and density? Should explanation #1 be preferred or #2? Or entirely equivalent?

Best Answer

Buoyancy is equal to the weight of the displaced liquid. If you insist on a density formulation, you have to define "instantaneous density" as total mass divided by the submerged volume. When an object is first in contact with the water this number will be very high and the object initially "sinks". As it gets more submerged, the "volume" increases although the mass remains constant. Thus it may reach equilibrium depending on shape and orientation.

That simple definition covers all your cases, and even explains things like capsizing.

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