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Why ships don't sinkBecause the weight of the volume of water it displaces is bigger than the weight of the ship.

A ship will float when the weight of the water it displaces equals the weight of the ship. This is the Achimedes Principle.

Water exerts a pressure on the hull of the ship with a net force pushing it upwards.

If the area of the hull is large enough, then at some point the weight of the ship will be balanced by the force pushing up on the ship.

This is why heavy solid objects sink, but keep making the surface area larger and it will eventually float.

Metal shipsMetal is denser than water. But if you consider the full volume of the ship (including the metal hull and the space inside it), on average the ship is less dense than water. Why does a helium balloon float into the air? The rubber of the balloon does not float into the air by itself. It's the same concept.

Ships float because of buoyancy not because of what the material is. During WWII Japan and the US tested making ships out of CONCRETE due to metal shortage and they worked just fine.
A ship will not sink if its gross tonnage is less than that of its water displacement. For example, RMS Titanic's tonnage was 46,328 gross register tons (GRT) whilst its displacement was 52,310 tons. It should be noted that this rule does not stop ships from sinking when their hulls are breached and/or taking on water, as the excess weight of the water will upset the balance between tonnage and water displacement.

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