Note on average density and why ships do not sink

Floating on the dead sea

Let’s ask a very generic question: I hand you an object and ask you to predict whether the object would float or sink. How would you go about doing that ? Well, you can measure the mass of the object and the volume of the object and can derive this quantity called Average Density (\rho_{avg} )

\rho_{avg} = m_{object}/V_{object}

It is the average density of the entire object as a whole. If this object is submerged in a fluid of density \rho_f , then we can draw the following force diagram:

If \rho_{avg} > \rho_{f} , we note that this generic object would sink and if \rho_{avg} < \rho_{f} it would float!. Therefore in order to make any object float in water, you need to ensure its average density is less than the density of the fluid its submerged in!

Why does a ship stay afloat in sea?

A ship is full of air! Although it is made from iron which sinks in water but with all the air that it is full of, it’s average density (m_{ship}/V_{ship} ) drops down such that \rho_{avg-ship} < \rho_{sea-water} .

Fun Experiment:

If you drop some raisins in soda, you will notice that they raise up and fall down like so (Try it out!):


This is because air bubbles that form on the top of the raisin decrease its average density to the point that its able to make the raisin raise all the way from the bottom to the top. BUT once it reaches the top all the air bubbles escape into the atmosphere (its average density increases) and the raisin now falls down.

Questions to ponder:

  • Why do people not sink in the dead sea ?
  • How are submarines/divers able to move up and down the ocean ? How would you extend the average density argument in this case.
  • Why do air bubbles in soda always want to raise up ?
  • If the total load that needs to on a ship is 25 tons. What should be the total volume of the ship in order to remain afloat if the density of sea water is 1029 kg/m3,

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