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## Physics library

### Course: Physics library > Unit 9

Lesson 2: Buoyant Force and Archimedes' Principle# Archimedes principle and buoyant force

Explore Archimedes' principle and the buoyant force in fluid dynamics. Understand how pressure varies with depth and acts in all directions, leading to the upward net force on submerged objects. This principle explains why objects float and is key to fluid displacement and the concept of buoyancy. Created by Sal Khan.

## Want to join the conversation?

- I am confused because I thought at the beginning of this lecture Sal said that the pressure around a submerged object was equal from all directions but at the end of his lecture he says that the pressure is greater underneath the object than on top and that is the buoyant force. Could someone help clear this up for me? What is Sal actually saying?(55 votes)
- His first example was a point, which has a volume of zero. Around this point the pressure is equal because no volume of liquid is displaced. Also, a point has no "height", hence no difference between pressure up and bottom. In the cube example, it has volume, which generates difference in pressure up and bottom.(2 votes)

- does that mean the pressure in me helps me not to be squashed by the atmosphere(31 votes)
- This question appeared in my physics final paper and there is confusion amongst students over its answer: "A wooden block is lying on the bottom of the tank sticking (with glue) to it. When water is poured into the tank, water does not enter below the block. Is there a buoyant force acting on the block? Explain."

Please help.(13 votes)- Short answer, no. Remeber how suction cups works. because the rubber doesn't let any air under the cup, the pressure of atmosphere is what making it stick to a surface. Same rules apply here(3 votes)

- Does an object (such as a hot air ballooon) float because it weighs less than a volume of normal air equivalent to the volume of space it takes up? Also, as the air is heated up by the balloon, it becomes less dense and it should float. But what exactly is pushing it up--is it the hot air itself pushing upwards on the inner walls of the balloon?(7 votes)
- There are two explanations as to what is pushing up the balloon. One is that atmosphere (which is a fluid in static equilibrium) cannot distinguish between the balloon and an equivalent amount of normal air in its place. Therefore, it provides an upward force due to difference in hydrostatic pressure at the bottom of balloon and the top, which is equal in magnitude to the weight of normal air the size of the balloon. Had there been normal air there it would have been static as the upward force would have been equal to its weight but since the weight of the balloon is less than the upward force acting on it, it will move up. This is what concept of floating and upthrust is. The second explanation is considering the tendency of the entire system to lower its gravitational potential energy which can be done if the balloon were replaced with air (due to its greater mass) and therefore all elements of air above the balloon, try to reduce the net energy by coming down in place of the balloon and in the process providing an upward push on it.(3 votes)

- if I exhale completely, I sink to the bottom of a pool. If my lungs are full of air, I float. Is this simply due to the fact that the volume of my body is greater when my lungs are full of air, and thus my overall density is less? Is there not some other buoyant property afforded by the fact that my lungs are full of a low-density gas?(3 votes)
- When you inhale, you increase your volume, which makes you displace more water, which increases the buoyant force on you. The air you inhale has very little mass, so it doesn't really add anything to your weight. The net upward force increases.(5 votes)

- I've seen people write things like "Archimedes' principle says that the buoyant force acting on an object is equal to the weight of the liquid displaced. This simply means that if something is denser than the liquid, it will sink." I've tried figuring out how they came to that conclusion and did a considerable amount of research on it, but I could never figure it out. How did they come up with that?(2 votes)
- This is something difficult to visualize. But here is how to get there:

The force of water above the object is given by rho*g*h, and the buoyant force underneath the object is equal to the (pressure at the bottom of the object)*(surface area of the bottom of the object). Let's look a little closer at that surface area. The surface area is related to the volume; generally, the greater the total surface area of an object, the greater the object's total volume. For example, an empty balloon has a much smaller surface area than a balloon filled with air. Why did the surface area change? Well, that's because we increased the balloon's volume!

Now, if an object has a greater density, that means that, per amount of surface area, that object also has a greater mass for that given area. If that amount of mass on the surface of the object is greater than the mass of the area of the water (or any liquid) underneath it, then the gravitational force pulling downwards on the mass of the object will cause the object to "push aside" the liquid in its way.(7 votes)

- At8:00, to summarize Archimedes principle- for every submerged object the weight of water displacement equals the object's weight? Is that right? Thanks.(0 votes)
- No. Nothing was talked about the object's weight. It summarizes saying that the buoyancy force acting on the submerged object is equal to weight of displaced liquid, which depends only on the volume of the object. That is not the only force acting in the object though, there is also the object's weight but it was not mentioned so far.(6 votes)

- how does he know that the pressure of the cube at the bottom is higher than the pressure on top?(3 votes)
- Pressure is directly proportional to the depth below the surface of the liquid. The deeper we go down, the higher the pressure. The larger the cube, the more the pressure difference between the top and bottom.(3 votes)

- At8:08, Sal says the net upward force of submerged objects equals the weight of
**liquid**displaced. Would it be accurate to say this applies to**gases**too since we're talking about weight, and possible compression wouldn't change weight?(2 votes)- That is correct. Archimedes' principle is a law of ideal fluids. Fluids aren't just liquids, they also include gases.(3 votes)

- How can water exert a force, creating a greater pressure on the bottom of an object than the top, if the object (take a cube for example) has been placed on the bottom of the container holding the water, so that there is no water below the object (the object is directly against the container), and thus the only force from the water (besides the pressure resulting from the water's force acting on the sides of the object) would be that acting downwards. If this were the case, even if the object's density were such that the object had the potential to float, i.e. if its density were less than that of water, making the force of gravity on the object lesser than that of the buoyant force that would be produced by the volume of the water displaced, how the water be able to fulfill this desire to displace the cube so that only part of it would be submerged. What I am trying to say is that, if an object that should float is placed on the bottom of a container, how could water create the buoyant force needed to accelerate the object, if the water is unable to create a larger pressure on the bottom of the object, due to a lack of contact between the bottom of the object and the water itself. Would the object still float, or would it remain at rest on the bottom, with the force of gravity on the object being cancelled by the normal force provided by the bottom of the container?(2 votes)
- That's a good question. My initial response is if there was a vacuum between the cube and the bottom surface, then you would be correct, the cube would stay pinned to the bottom even though it is less dense than water. However, in reality, there is no vacuum there even with the cube touching the bottom surface and as a result there is still greater water pressure pushing up on the bottom surface of the cube than pushing down on the top surface causing the cube to float.

Not sure if anyone has done this experiment but I would guess that if you take a very flat cube and place it against a very flat surface under water, that it would take more force to initially lift the cube when it is flat on the bottom versus when it is off the bottom.(2 votes)

## Video transcript

Let's say we have
a cup of water. Let me draw the cup. This is one side of the cup,
this is the bottom of the cup, and this is the other
side of the cup. Let me say that it's
some liquid. It doesn't have to be water,
but some arbitrary liquid. It could be water. That's the surface of it. We've already learned that the
pressure at any point within this liquid is dependent
on how deep we go into the liquid. One point I want to make before
we move on, and I touched on this a little bit
before, is that the pressure at some point isn't just acting
downwards, or it isn't just acting in one direction. It's acting in all directions
on that point. So although how far we go down
determines how much pressure there is, the pressure is
actually acting in all directions, including up. The reason why that makes sense
is because I'm assuming that this is a static system,
or that the fluids in this liquid are stationary, or you
even could imagine an object down here, and it's
stationary. The fact that it's stationary
tells us that the pressure in every direction must be equal. Let's think about a
molecule of water. A molecule of water, let's say
it's roughly a sphere. If the pressure were different
in one direction or if the pressure down were greater than
the pressure up, then the object would start accelerating
downwards, because its surface area
pointing upwards is the same as the surface area pointing
downwards, so the force upwards would be more. It would start accelerating
downwards. Even though the pressure is a
function of how far down we go, at that point,
the pressure is acting in every direction. Let's remember that, and now
let's keep that in mind to learn a little bit about
Archimedes' principle. Let's say I submerge a cube into
this liquid, and let's say this cube has dimensions
d, so every side is d. What I want to do is I want to
figure out if there's any force or what is the net
force acting on this cube due to the water? Let's think about what the
pressure on this cube is at different points. At the depths along the side of
the cube, we know that the pressures are equal, because
we know at this depth right here, the pressure is going to
be the same as at that depth, and they're going to offset each
other, and so these are going to be the same. But one thing we do know, just
based on the fact that pressure is a function of depth,
is that at this point the pressure is going to be
higher-- I don't know how much higher-- than at this point,
because this point is deeper into the water. Let's call this P1. Let's call that pressure on top,
PT, and let's call this point down here PD. No, pressure on the
bottom, PB. What's going to be the net
force on this cube? The net force-- let's call that
F sub N-- is going to be equal to the force acting
upwards on this object. What's the force acting
upwards on the object? It's going to be this pressure
at the bottom of the object times the surface area at the
bottom of the object. What's the surface area at
the bottom of the object? That's just d squared. Any surface of a cube is d
squared, so the bottom is going to be d squared minus--
I'm doing this because I actually know that the pressure
down here is higher than the pressure here, so this
is going to be a larger quantity, and that the net force
is actually going to be upwards, so that's why I can
do the minus confidently up here-- the pressure
at the top. What's the force at the top? The force at the top is going to
be the pressure on the top times the surface area of
the top of the cube, right, times d squared. We can even separate out the d
squared already at that point, so the net force is equal to
the pressure of the bottom minus the pressure of the top,
or the difference in pressure times the surface area of either
the top or the bottom or really any of the
sides of the cube. Let's see if we can figure
what these are. Let's say the cube is submerged
h units or h meters into the water. So what's the pressure
at the top? The pressure at the top is
going to be equal to the density of the liquid-- I keep
saying water, but it could be any liquid-- times how
far down we are. So we're h units down, or maybe
h meters, times gravity. And what's the pressure
the bottom? The pressure at the bottom
similarly would be the density of the liquid times the depth,
so what's the depth? It would be this h and then
we're another d down. It's h plus d-- that's our total
depth-- times gravity. Let's just substitute both of
those back into our net force. Let me switch colors to keep
from getting monotonous. I get the net force is equal to
the pressure at the bottom, which is this. Let's just multiply it out, so
we get p times h times g plus d times p times g. I just distributed this out,
multiplied this out. That's the pressure at the
bottom, then minus the pressure at the top, minus phg,
and then we learned it's all of that times d squared. Immediately, we see something
cancels out. phg, phg subtract. It cancels out, so we're
just left with-- what's the net force? The net force is equal to dpg
times d squared, or that equals d cubed times
the density of the liquid times gravity. Let me ask you a question:
What is d cubed? d cubed is the volume
of this cube. And what else is it? It's also the volume of
the water displaced. If I stick this cube into the
water, and the cube isn't shrinking or anything-- you
can even imagine it being empty, but it doesn't have to be
empty-- but that amount of water has to be moved out
of the way in order for that cube to go in. This is the volume of
the water displaced. It's also the volume
of the cube. This is the density-- I keep
saying water, but it could be any liquid-- of the liquid. This is the gravity. So what is this? Volume times density is the mass
of the liquid displaced, so the net force is
also equal to the mass of liquid displaced. Let's just say mass times
gravity, or we could say that the net force acting on this
object is-- what's the mass of the liquid displaced
times gravity? That's just the weight
of liquid displaced. That's a pretty interesting
thing. If I submerge anything, the net
force acting upwards on it, or the amount that I'm
lighter by, is equal to the weight of the water
being displaced. That's actually called
Archimedes' principle. That net upward force due to
the fact that there's more pressure on the bottom than
there is on the top, that's called the buoyant force. That's what makes
things float. I'll leave you there to just to
ponder that, and we'll use this concept in the next couple
of videos to actually solve some problems.
I'll see you soon.