Answers to the buoyancy questions PDF

Preview

Summary

123456...

Description

Answers to the buoyancy questions Archimedes (287-212 BC) a mathematician, physicist, engineer, inventor and astronomer Fbuoyancy = Weight of fluid displaced = ρlfluid g Vdisplaced fluid First question Based on common sense, one expects that once the load (stone) is removed the boat will rise. This might be sufficient for a correct answer. However, it is instructive to undertake a rigorous analysis. Proof by a detailed analysis

When the stone is in the boat:

ρboatVboat g + ρstoneVstone g = ρwaterVwater displaced g

ρboat a l (1) g + ρstone b l (1) g = ρwater h l (1) g dividing through by l (1) g we have,

ρboat a + ρstone b = ρwater h h =

ρboat a + ρstone b ρwater

When the stone is thrown into the pool

ρboatVboat = ρwaterVwater displaced

ρboat a l (1) g = ρwater h' l (1) g

ρboat a = ρwater h' or

h' =

ρboat a ρwater

ρboat a + ρstone b h ρ water ρ a + ρstone b ρwater = boat = ρ a ρwater ρboat a h' boat ρwater

then h ρ b =1+ stone ρboat a h' Because

ρstone b >0 ρboat a Therefore,

h > h' or the boat will rise. Second question When the stone is in the boat it displaces water equal to its weight

ρwaterVwater displaced g = ρstoneVstone g → ρwaterVwater displaced = ρstoneVstone buoyancy weight Then, Vwater displaced =

ρstone Vstone since ρstone > ρwater → Vwater displaced > Vstone ρwater

Then, when the stone is thrown into the pool, the volume displaced by the stone is equal to its own actual volume (Vstone ) , which is less than when it is inside the boat

(V

stone

< Vwater displaced ) . Therefore, the water level of the pool will drop (H ' < H ) . It is

claimed that none of the famous scientists answered the second question correctly!

More detailed proof Mass ( M ) of water inside the pool before and after remains the same M = ρ water L H (1) − ρ water l h (1) = ρ water L H ' (1) − ρ water l h' (1) − ρ water l b (1)

L H − l h = L H ' − l h' − l b H − H ' = ( h − h' − b)

l L

% l " ρ a + ρ stone b ρ boat a − −b' H − H ' = $ boat ρwater ρ water # &L "ρ %l b H − H ' = $ stone − 1' # ρ water &L "ρ %lb H = H ' + $ stone − 1' # ρ water & L

ρstone >1 , then ρwater

"ρ %l b >0 $ stone − 1' # ρ water & L

or

H'...