Now that we've discussed some basic properties of fluids including the density and specific gravity
and surface tension and seen some examples of how to apply hydrostatic pressure,
both the gauge pressure and the total pressure, we're now ready to move on to a few key principles
that have been derived about fluids and the way they behave, Archimedes’ principle and Pascal’s law.
Let’s start with Archimedes’ principle. This principle basically says that when you put an object
under water, it will feel a buoyant force upwards. You might have noticed this. If you've ever been
in water, you feel much lighter. This is because there's a force acting upwards on you.
We call this force the buoyant force. The reason you have a buoyant force on you,
the reason there’s something pushing up on you is that you have a pressure from the water
on top of you in one direction. You also have a pressure on the bottom at the lower part of this mass
as you can see it here. But as we just saw, the pressure at one height and the pressure
at a different height are different pressures. So, the pressure at the bottom is actually more
than the pressure at the top of the object, meaning it has a net force acting on it upwards.
This object actually has displaced some water and we can use this idea of displaced water to figure out
how much pressure and how much buoyant force is lifting the object upwards. The displaced water
is always the exact same volume as an object completely under water. So, we could sort of imagine
this object having replaced that water. We took this amount of water out and then we stuck this object in.
So, we consider this displaced water which has some mass to it. The force upwards on you,
the buoyant force will be equal to the weight of this displaced water. This is Archimedes’ principle.
This buoyant force we can always find by considering how much water the object in the water
displaced by being put there. The force, the buoyant force acting upwards on the object
will be exactly equal to the force in newtons, the gravitational force that would be pulling the water downwards.
So, you can always find this buoyant force using this principle of Archimedes just by finding the mass
of the displaced water which you can find from the volume of the object that’s in the water
and multiplying it by g since you want the force that would normally be pulling this water down
by the force of gravity. So, the buoyant force in summary by Archimedes principle
is equal to the mass of the displaced water by your object times the gravitational acceleration, g.
We can also determine when an object will float and if it will float using this principle.
What we need for an object in the water like this one which has a gravitational force
pulling it downwards and a buoyant force trying to lift it upwards, for it to float, we need
the buoyant force to be greater than the gravitational force. If it’s exactly equal to
the gravitational force, the object would sort of stay still and not go up or down. What this means
if we need the buoyant force to be greater than the gravitational force is that we need the buoyant force
which we found as the mass of the displaced water times g to be greater than the mass of the object
times g since that’s the force of gravity pulling it downwards. We can divide both sides by the volume
since these volumes will be the same. We can see that this means that the mass of the object,
the mass times gravity of the water that’s displaced has to be greater than the mass of the object itself.
We rewrite mass as the density times the volume and then we divide it by the volume as I said.
So, the conclusion is that for an object to float, we simply need the density of water to be greater
than the density of that object. We can also determine if an object is floating, how much of it
will emerge from the top of the water. Again, this object will have a buoyant force
equal to the gravitational force pulling the object down. This is the condition for it to be floating
in equilibrium. This means that the mass of the water displaced also has to be equal to the mass
of the object just like we just saw. Therefore, once again rewriting the masses in terms of their densities
times their volumes, we can see that for a floating object, the volume that is submerged of your object
divided by the entire volume of the object has to be equal to the density of the object
divided by the density of the water. In other words, if you have an object that’s more dense,
it will have to submerge itself more and displace a little bit more water if it wants to float.
The ratio of the amount of the object that’s submerged, V submerged over V object
has got to be equal to the ratio of the density of your object to that of water.