Buoyant force | AP Physics | Khan Academy

- [Instructor] Beach balls float on water,

icebergs float on water,
certain things float on water,

whereas others, like a steel ball, sinks.

Why? Why do certain things
float and certain things sink?

And also, if you consider
the things that are floating,

sometimes, most of the
object is above the water,

like the beach ball, but
when it comes to the iceberg,

look, most of the object is
submerged below the water.

So what controls how much
of that object is above

and how much of it is submerged?

Now, you probably know the answer

has something to do with
density, but why density?

Why does density matter?

What is the deeper reason behind this?

That's what we wanna
figure out in this video,

so let's begin.

So, let's start by submerging
an object inside a fluid.

Let's consider water to be our fluid,

and let's submerge a cubicle object.

A cube because it's a nice
shape. Easier to analyze, okay?

We know that the water
is going to start pushing

on this cube in all the
directions perpendicular

to the surface.

And if we calculate that force per area,

we call that as the pressure.

Now, the important thing

is that the pressure increases with depth.

So, let's see how that affects the cube.

First, let's consider
the horizontal forces.

For example, consider
the forces on the left

and the right side of the cube.

The forces would look somewhat like this.

Look, the pressure is
increasing with the depth,

but notice the forces
pretty much cancel out.

And the same thing is gonna happen

with the front and back
surface of the cube as well,

so all the horizontal forces cancel out.

But what about the top
and the bottom surface?

Well, let's see.

The pressure on the top is smaller

than the pressure from the bottom.

And because the surface
area is exactly the same

on both the top and the bottom,

the forces from the bottom will be larger

than the force from the top.

And therefore, now the
forces don't cancel out.

Instead, there is a net
force acting upwards.

And that net force that's acting upwards

is what we call the buoyant force.

This is the force that
tends to make things float.

It tends to make things buoyant.

That's what the b stands for over here.

And so, notice where does
the buoyant force come from?

It ultimately comes from the
fact that as you go deeper,

the pressure increases.

That is where it all comes from.

Now, although we did
this analysis for a cube,

this would be true for any
object of any arbitrary shape.

Even if you take some kind of a rock,

which has some random shape,

the fact is the pressure from the top

is smaller than the
pressure from the bottom.

And so, when you look at all the forces,

eventually, there will be
a net force acting upward,

the buoyant force.

Now comes the big question,

because of this buoyant
force, how do we know

whether this object is
going to float or sink?

Well, it depends upon the
weight of this object.

If the weight of this object

is smaller than the buoyant force,

well, then the buoyant force wins.

The net force will now be upwards,

and as a result, the object
will accelerate upwards,

making it float.

On the other hand, if the
weight of the object is larger

than the buoyant force,
then the weight wins

and therefore, the object
will accelerate down,

in other words, the object will sink,

which means all we need to do

is figure out what the buoyant force is.

If we can do that, we can predict

whether an object is
going to float or sink.

But how do we figure out
what the buoyant force

is going to be, especially
when the objects

have irregular shapes like this?

Well, here's a way to think about it.

Imagine that instead of having a rock,

if I had a styrofoam over here,

but of exactly the same shape

and volume submerged in the
same liquid, in the same fluid,

the question to think about is,

would the buoyant force
now be the same as before

or would it be different?

Why don't you pause the
video and think about this?

All right, at first, it might feel like,

"Hey, I have a different
substance altogether,

so the buoyant force must
be different, obviously."

But think about it,

the buoyant force,
where does it come from?

It comes from the pressure
differences, isn't it?

And what does pressure depend on?

Well, the pressure only
depends upon the depth.

And so if you have the
exact same shape as before,

then all the forces will be
exactly the same, and therefore,

the pressure would be
exactly the same as before,

and therefore, the buoyant force

would be exactly the same as before.

So, this is the key insight.

This means that the buoyant force

has nothing to do with what
material you have submerged.

All that matters is the
shape of the material.

If the shape remains the same,

then regardless of what material it is,

the buoyant force should be the same.

Okay, the buoyant force does
not depend upon the material,

how can we use that insight?

Well, now, in this exact same shape,

let's fill water. (laughs)

Well, the buoyant force is
gonna be the same as before,

but this time, we know
that this particular piece

of water is in equilibrium
because, remember,

this is just water.

It's static. It's not moving at all.

This piece of water is neither
sinking nor it's floating.

It's not accelerating upwards,
which means it's static.

And therefore, this
means this piece of water

is in equilibrium.

So, the weight of this piece of water

must be exactly equal

to the buoyant force.

And there we have it.

We have figured out what
the buoyant force must be.

For this particular shape,

it doesn't matter what
you put inside this,

the buoyant force should equal the weight

of this fluid.

In other words, when you fill this object,

when you fill this space with some object,

that fluid got displaced somewhere,

and the weight of that fluid
that got displaced literally

is the buoyant force.

Isn't it amazing?

Long story short, the buoyant
force acting on any object

will always equal the weight
of the fluid that it displaces.

And this is what we call
the Archimedes' principle.

So now let's see if we
can use this insight

to figure out when will an object float

and when it'll sink, okay?

So, we know that if the
weight of the object

is larger than the buoyant
force, in other words,

larger than the weight of
the fluid it displaces,

the object is going to sink.

So this is a sinking case,
but when will this happen?

When will the object have more
weight compared to the fluid

that it's displacing?

Well, what is weight?

Weight is just mass times gravity.

So we can plug mg over here.

This represents the mass of the object,

and this represents the mass of the fluid

that got displaced.

But we know that mass is
the density times volume.

So we can replace masses

with density of the object
times the volume of the object,

and the density of the fluid displaced

times the volume of the fluid displaced.

But here's the key thing,

the volume of the fluid
displaced is exactly the same

as the volume of the object, right?

And therefore, these things cancel out,

and look what the condition becomes.

The condition for sinking

is the density of the
object should be larger

than the density of the
fluid it's submerged in.

When that happens, the object will sink.

But can we now understand why?

Well, because if the density of the object

is larger than that of the fluid,

then the weight of the object

will be larger than
the weight of the fluid

that it displaces when
it's completely submerged.

And as a result, its weight wins.

Its weight will be larger
than the buoyant force

and that's why it ends up sinking.

And this is why a steel
ball sinks in water

because it has a higher
density than water.

But what if an object has a lower density

than the fluid it's submerged in?

Well, then its weight would be lower

than the weight of the
fluid that it displaces,

and therefore, the buoyant
force will be larger

and the whole object will accelerate up.

In other words, this is the
condition for flotation.

This is the reason why beach balls

and icebergs float in water

because they have a density
that's smaller than water.

And now, look, we have
the complete equation

for the buoyant force.

The buoyant force is the
weight of the fluid displaced,

which is the density of the
fluid times the volume times g.

And so this is the expression
for the buoyant force.

And again, this helps us see

why it's the density that matters

because, look, the weight
of the displaced fluid

will have the exact same volume

as the weight of the submerged object.

The volumes cancel out,

and that's why it's eventually
the density that decides

whether something's
going to sink or float.

So it all comes from the
Archimedes' principle,

which eventually comes from
the pressure differences.

Amazing, right?

But now, let's think about
what's going to happen

to this object.

We know it's going to
start accelerating up,

but when will it stop?

Well, let's see.

As long as the whole
thing is submerged inside,

the buoyant force stays the same.

But now, as it starts
coming above the surface,

it starts displacing lesser fluid.

Can you see that?

It now only displaces this
much amount of the fluid,

and therefore, the buoyant
force will become smaller.

But as long as it's still
larger than the weight,

the object continues accelerating upwards,

and eventually, at some point,

the weight of the fluid displaced

will exactly match the
weight of the object,

and that's when equilibrium is reached

and the object pretty
much stops at that point.

So look, for an object to float,

it needs to be submerged
enough so that the weight

of the fluid displaced exactly
equals the object's weight.

Does that make sense?

Now, consider the beach ball.

It has a very low density,
so the amount of water needed

to match the weight of the
beach ball is very little.

So only a small portion
needs to be submerged

because by then, the weight of the water

that has been displaced
already equals the weight

of the entire beach ball

because the beach ball has a
much smaller density compared

to the water.

That's the whole idea.

On the other hand, if
you consider an iceberg,

it has a density very, very
close to that of water.

And so now, to displace the
water equal to its weight,

you need to submerge a lot more.

Pretty much the entire
thing needs to be submerged

because its density is very,
very close to that of water.

It's still less, that's why it's floating,

but it's very close to that of water.

So, higher the density of the object,

more it needs to be submerged

so that it can be in equilibrium.

So, if you put it all together,

we can say that when
the density of an object

is smaller than density of the
fluid, it will be floating.

And the amount of the
objects submerged depends

on how small the density is.

If the density of the
object is very tiny compared

to the density of the fluid,

it'll be submerged just a little bit.

On the other hand, if
the density is very close

to the density of the
fluid, but still smaller,

it needs to be smaller
for it to be floating,

but if it's close enough, then
most of it will be submerged.

On the other hand, if the
density of the object is higher

than the density of the fluid itself,

then the object will sink.

And by the way, in all the examples,

we have taken water as our fluid,

but this would work for any fluid, okay?

For example, air is also a fluid,

so it can also put a buoyant force.

But since air has a very
low density compared

to everyday objects, we
don't usually notice it.

But if you consider a
helium balloon, its density,

helium density is smaller
than that of the air.

And now, as a result of
that, helium tends to float,

and that's why helium
balloons tend to rise up.

But now is an interesting question,

what if the density of the
object exactly equals the density

of the fluid?

What happens then?

Well, now, if you completely submerge it,

it will neither float nor sink.

We call this the neutral buoyancy.

That object will just stay
at that particular depth,

and that's exactly how submarines

can stay at a particular depth.

They do that by changing
their average density.

If they want to sink,

they will increase their average density

by allowing water to flood their tanks.

On the other hand,

if it now needs to come towards
the surface of the water,

then it'll decrease its average density.

It does so by now using compressed air

to force the water out.

And by carefully balancing
the amount of water

and air inside its tanks,

it can also make sure that
its density exactly equals out

of the surrounding water,
maintaining neutral buoyancy,

and in which case, it'll
stay at a particular depth.

That's amazing, right?

Now, instead of submarine,
imagine you were wearing a suit

which could also maintain
neutral buoyancy in water.

Then you would be, just
like the submarine,

stay at a particular location in water,

not going down nor going up.

In other words, you would
kind of feel weightless,

which means now you can train

for the weightless environments
that you would face

in the space stations.

And that's exactly

what the NASA's Neutral Buoyancy Lab does.

It's a giant swimming pool inside

which astronauts can
maintain neutral buoyancy

and train for the weightless environment.

It's not perfect, but it's way better

and more realistic than training
on the ground, for example.