# 17.6 Floating

Before you read this post, I suggest that you read posts 16.44 and 17.5.

In post 16.44, I wrote that stuff that has a lower density than water will float on water. Why? Let’s think of an object that sinks suspended in water by a thin thread, as shown on the left-hand side of the picture above. Gravity pulls the object downwards. The gravitational force exerted is mg, where g is the modulus of the gravitational field (post 16.16) and m is the mass of the object; from the definition of density this can be written as Vρg where V is the volume of the object and ρ is its density (post 16.44). Although forces are vectors, we can treat them as scalars in this post because we are considering only forces that act in a straight line – downwards or upwards. Then we represent the downwards direction by a positive scalar and the upwards direction by a negative scalar. (We could , of course, use vector notation to derive the same results but the unit vector that defined direction would simply cancel out in all our equations.)

However, there is also an upwards force acting on the object. To understand why, let’s think of the forces acting on the same volume of water when the object isn’t there (right-hand side of the picture above). We don’t expect this volume of water to move (see post 17.5), even though gravity is pulling it downwards. Why doesn’t it move? There must be a force acting on it that is equal and opposite to the force exerted by gravity. The force that gravity exerts on the water is mog = Vρog where mo and ρo are the mass and density of the water, respectively. So the upward force acting on the volume of water is
Vρog.

This force must be exerted by the pressure in the surrounding water (see post 17.5) and so must also act when the volume of water is replaced by the sinking object. So, the total force acting on the sinking object is

F = VρgVρog = Vg(ρρo)

which is less than the force that would be exerted on it, by gravity, if it were surrounded by air. This result is called Archimede’s principle after the ancient Greek philosopher who first described it. We can use it to find the density of an object (post 17.7).

Now suppose ρ is greater than ρo. Then F is positive (it acts downwards) so the object sinks.

But if ρ is less than ρo, F is negative (it acts upwards) so the object floats.

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