What does buoyant force mean?
Have you ever dropped your swimming goggles in the deepest part of the pool and tried to swim down to get them? It can be frustrating because the water tries to push you back up to the surface as you're swimming downward. The name of this upward force exerted on objects submerged in fluids is the buoyant force.
So why do fluids exert an upward buoyant force on submerged objects? It has to do with differences in pressure between the bottom of the submerged object and the top. Say someone dropped a can of beans in a pool of water.
Bean pollution is a crime. If you see someone throwing beans into a pool or ocean call the Society for Bean Free Waterways immediately.
Because pressure
Essentially it's that simple. The reason there's a buoyant force is because of the rather unavoidable fact that the bottom (i.e. more submerged part) of an object is always deeper in a fluid than the top of the object. This means the upward force from water has to be greater than the downward force from water.
OK, so it doesn't completely follow. After all, what if we considered an object where the area of the bottom was smaller than the area of the top (like a cone). Since
It's fun to try and think of more examples of other shapes, and then try to figure out why they won't make the buoyant force point downward.
Knowing conceptually why there should be a buoyant force is good, but we should also be able to figure out how to determine the exact size of the buoyant force as well.
We can start with the fact that the water on the top of the can is pushing down
We can relate these forces to the pressure by using the definition of pressure
We can use the formula for hydrostatic gauge pressure
Notice that each term in this equation contains the expression
Now this term
So we can replace
Here's the interesting part. Since
Since there is no water left in the region of space where the can is now, all that water went somewhere else in the fluid.
So we are definitely going to replace the term
Well, imagine the can was floating with half of its volume submerged beneath the surface of the fluid.
There would no longer be any downward force from the water pressure on the top of the can. And the depth
But the term
You could of course choose to write the formula in terms of the volume of the can
That pretty much does it. This formula gives the buoyant force on a can of beans (or any other object) submerged wholly or partially in a fluid. Let's take stock of what we have now. Notice how the buoyant force only depends on the density of the fluid
Surprisingly the buoyant force doesn't depend on the overall depth of the object submerged. In other words, as long as the can of beans is fully submerged, bringing it to a deeper and deeper depth will not change the buoyant force. This might seem strange since the pressure gets larger as you descend to deeper depths. But the key idea is that the pressures at the top and bottom of the can will both increase by the same amount and therefore cancel, leaving the total buoyant force the same.
Something might strike you as being wrong about all this. Some objects definitely sink, but we just proved that there is an upward force on every submerged object. How can an object sink if it has an upward force on it? Well, there is definitely an upward buoyant force on every submerged object, even those that sink. It's just that for sinking objects, their weight is greater than the buoyant force. If their weight was less than their buoyant force they would float. It turns out that it's possible to prove that if the density of a fully submerged object (regardless of its shape) is greater than the density of the fluid it's placed in, the object will sink.
The net vertical force (including gravity now) on a submerged object will be the buoyant force on the object minus the magnitude of the weight of the object.
We can use the formula we derived for buoyant force to rewrite
We can make that second term in the formula look a whole lot more like the first term if we use the rearranged definition of density to write the mass of the object
If the object is fully submerged the two volumes
So there it is! If the density of the object is greater than the density of the fluid the net force will be negative which means the object will sink if released in the fluid.
What is Archimedes' principle?
The way you will normally see the buoyant force formula written is with the
When you rearrange the formula in this way it allows you to notice something amazing. The term
But look at that! The mass of the displaced fluid times the magnitude of the acceleration due to gravity is just the weight of the displaced fluid. So remarkably, we can rewrite the formula for the buoyant force as,
This equation, when stated in words, is called Archimedes' principle. Archimedes' principle is the statement that the buoyant force on an object is equal to the weight of the fluid displaced by the object. The simplicity and power of this idea is striking. If you want to know the buoyant force on an object, you only need to determine the weight of the fluid displaced by the object.
Since there is no water left in the region of space where the can is now, all the water that was in that volume must have been displaced elsewhere in the fluid.
The fact that simple and beautiful (yet not obvious) ideas like this result from a logical progression of basic physics principles is part of why people find physics so useful, powerful, and interesting. And the fact that it was discovered by Archimedes of Syracuse over 2000 years ago, before Newton's laws, is impressive to say the least.
What's confusing about the buoyant force and Archimedes' principle?
Sometimes people forget that the density
People often forget that the volume in the buoyancy formula refers to the volume of the displaced fluid (or submerged volume of the object), and not necessarily the entire volume of the object.
Sometimes people think the buoyant force increases as an object is brought to deeper and deeper depths in a fluid. But the buoyant force does not depend on depth. It only depends on volume of the displaced fluid
Many people, when asked to state Archimedes' principle, usually give a look of confused exasperation before launching into a wandering discussion about people jumping naked out of bathtubs. So, make sure you understand Archimedes' principle well enough to state it clearly: "Every object is buoyed upwards by a force equal to the weight of the fluid the object displaces."
What do solved examples involving buoyant force look like?
Example 1: (an easy one)
A
What is the buoyant force on the gnome?
Example 2: (a slightly harder one)
A cube, whom you have developed a strong companionship with, has a total mass of
What must be the minimum side length of the cube so that it floats in sea water of density
We know that in order to float the buoyant force when the object is submerged must be equal to the magnitude of the weight of the cube. So we put this in equation form as,
Example 3: (an even harder one)
A huge spherical helium filled balloon painted to look like a cow is prevented from floating upward by a rope tying it to the ground. The balloon plastic structure plus all the helium gas inside of the balloon has a total mass of
What is the tension in the rope?
This one is a little harder so we should first draw a free body diagram (i.e. force diagram) for the balloon. There are lots of numbers here too so we could include our known variables in our diagram so that we can see them visually. (Note that in this case, the fluid being displaced is the air.)
Since the spherical cow balloon is not accelerating, the forces must be balanced (i.e. no net force). So we can start with a statement that the magnitudes of the total upward and downward forces are equal.