Buoyancy

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 of Syracuse (287–212 BCE) 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′ = ρobject − ρfluid

is the relative density.

When …