Buoyancy

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

Buoyant forces act on the foundations of buildings. Tokyo underground train stations need to be pinned down to avoid bobbing to the surface from the buoyant forces caused by increasing water levels.

B = Fbottom − Ftop

B = PbottomA − PtopA = (ρfluidghbottom − ρfluidghtop)A

B = ρfluidgΔhA = ρfluidgV = mfluidg

B = Wfluid

Archimedes' principle: The buoyant force (B) on an object immersed in a fluid is equal to the weight of the fluid displaced.

Archimedes (287-212 BCE) Greece (Syracuse) was commissioned by King Hiero II of Syracuse to determine if a golden crown made for him was made from pure gold or a low grade alloy as he suspected. The problem was, how to determine the gold content without damaging the crown. As the story goes, Archimedes decided a nice relaxing bath would help him think. He immersed himself in the water and felt a bit lighter -- an event experienced by billions of other people on countless occasions. What's unique about Archimedes' relaxing soak was that he realized, for possibly the first time ever in human history, that this buoyant force he was experiencing could be used to determine the quality of the king's crown.

Eureka -- perfect indicative form of ευρισκω [eurisko] -- to find. Related to the English word heuristics, referring to the methods used to find solutions to problems.

Buoy -- "Boiæ genus vinculorum tam ferreæ quam ligneæ." -- Festus (Webster's 1913 dictionary). something to do with tying it down with straps? (boiæ?)

The apparent weight (W′) of an object immersed in a fluid is given by …

W′ = W − B

W′ = mobjectg − mfluidg = (ρobjectV − ρfluidV)g

W′ = (ρobject − ρfluid)gV

W′ = ρ′gV

where

ρ′ = ρ_object − ρ_fluid

is the relative density.

When …

• ρ_object > ρ_fluid the apparent weight is positive but less than the actual weight.
• ρ_object = ρ_fluid the apparent weight is zero and the object has a neutral buoyancy.
• ρ_object < ρ_fluid the apparent weight is negative and the object is said to be "lighter than air".

Summary

• Buoyancy (also known as the buoyant force) is the force exerted on an object that is wholly or partly immersed in a fluid.
  • The symbol for the magnitude of buoyancy is B or F_B
  • As a vector it must be stated with both magnitude and direction.
    • Buoyancy acts upward for the kind of situations encountered in everyday experience.
  • As with other forces, the SI unit of buoyancy is the newton [N].
• Buoyancy is caused by differences in pressure acting on opposite sides of an object immersed in a static fluid.
  • A typical situation:
  1. The pressure on the bottom of an object is greater than the top
     (since pressure increases with depth).
  2. The force on the bottom pushes up and the force on the top pushes down
     (since force is normal to the surface).
  3. The direction of the net force due to the fluid is upward.
  • Pressure variations in a fluid are typically caused by gravity (since P = P₀ + ρgh), but in general buoyant forces act opposite the direction of the frame of reference acceleration.
    • Under conditions of apparent weightlessness there can be no buoyant forces.
• Archimedes' Principle
  • The magnitude of the buoyant force on an object is equal to the weight of the fluid it displaces.

  B = Wfluid displaced = ρfluidVdisplacedg

  • The factors that affect buoyancy are …
    • the density of the fluid,
    • the volume of the fluid displaced, and
    • the local acceleration due to gravity.
  • The buoyant force is not affected by …
    • the mass of the immersed object or
    • the density of the immersed object.
• Objects immersed in a fluid have an apparent weight that is …

  reduced by the buoyant force,	W′ = W − B
  less than their actual weight,	W′ < W
  directly proportional to the relative density (ρ′ = ρobject − ρfluid)	W′ = ρ′gV

Buoyancy and Density
densities	B > Wobject	B = Wobject	B < Wobject
ρobject < ρfluid	object rises (wholly immersed )	float on surface (partly immersed)
ρobject = ρfluid		neutral buoyancy (wholly immersed)
ρobject > ρfluid			object sinks

Problems

practice

1. Verify the rule of thumb that one cubic meter of helium can lift about one kilogram of load.
   • Answer it.
2. What fraction of an iceberg is above water?
   • Answer it.
3. Write something different.
   • Answer it.
4. Write something completely different.
   • Answer it.

conceptual

1. Two related questions.
   1. How does the buoyant force on an airplane compare to the buoyant force on a helium-filled, party balloon; that is, which is greater and why?
   2. Does the magnitude of the buoyant force alone explain why airplanes crash and balloons float? If not, why then do balloons float? (Explain your answer in either case.)
2. Which is heavier: a kilogram of feathers or a kilogram of lead?
3. Should footballs (American footballs that is — not soccerballs) be filled with helium to ensure longer passes and punts?

numerical

1. Some problems of this type would be nice.

Resources

• airship manufacturers
  • Aeros
  • Techsphere
  • World SkyCat

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