Why will the water not overflow when ice melts

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This is an old question but still there's something I could not understand. Here it goes:
A glass of water has an ice cube floating in it.The water level just touches the rim of the glass. will the water overflow when the ice melts?

This is how I imagined the scenario:
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Now everywhere I see its explanation is given using Archimedes principle like this "Volume of the Ice will be equal to volume of water displaced". But from what I know Archimedes principle states that volume of water displaced is equal to volume of object immersed in water and here the object is NOT completely immersed.

And now wish to get a proper explanation of why when the ice melts , the water won't overflow.

Best Answer

Archimedes' principle says that the buoyant force on any object (partially or fully submerged) is equal to the weight of the water it displaces. It doesn't just apply to fully immersed objects. The only major difference in the application of Archimedes's principle to partially immersed objects is that the buoyant force is determined by the portion of the object's volume that is immersed, rather than the full volume.

So the argument goes like this:

  • The ice cube is in equilibrium, so the buoyant force on the ice cube must equal the weight of the ice cube.
  • Thus, the weight of the ice cube is equal to the weight of the liquid water it displaces.
  • When the ice cube melts, it will turn into liquid water with the same weight.
  • Thus, the weight of the melted ice cube is equal to the weight of the water it displaced when it was solid.
  • This means that the volume of the ice cube, once it melts, is equal to the volume that the ice displaced when it was solid.
  • Thus, the water level does not change.
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