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Let's say that I have some
object, and when it's outside

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of water, its weight
is 10 newtons.

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When I submerge it in water-- I
put it on a weighing machine

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in water-- its weight
is 2 newtons.

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What must be going on here?

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The water must be exerting some
type of upward force to

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counteract at least
8 newtons of the

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object's original weight.

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That difference is the
buoyant force.

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So the way to think about is
that once you put the object

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in the water-- it could be a
cube, or it could be anything.

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We know that we have a downward
weight that is 10

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newtons, but we know that once
it's in the water, the net

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weight is 2 newtons, so there
must be some force acting

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upwards on the object
of 8 newtons.

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That's the buoyant force that
we learned about in the

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previous video, in the video
about Archimedes' principle.

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This is the buoyant force.

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So the buoyant force is equal
to 10 minus 2 is equal to 8.

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That's how much the water's
pushing up.

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And what does that
also equal to?

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That equals the weight of the
water displaced, so 8 newtons

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is equal to weight of
water displaced.

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What is the weight of
the water displaced?

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That's the volume of the water
displaced times the density of

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water times gravity.

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What is the volume of
water displaced?

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It's just the volume of water,
divide 8 newtons by the

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density of water,
which is 1,000

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kilograms per meter cubed.

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A newton is 1 kilogram meter
per second squared.

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Then, what's gravity?

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It's 9.8 meters per
second squared.

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If we look at all the units,
they actually do turn out with

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you just ending up having
just meters cubed, but

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let's do the math.

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We get 8 divided by 1,000
divided by 9.8 is equal to 8.2

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times 10 to the negative 4.

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V equals 8.2 times 10 to the
minus 4 cubic meters.

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Just knowing the difference in
the weight of an object-- the

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difference when I put
it in water-- I can

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figure out the volume.

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This could be a fun game
to do next time your

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friends come over.

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Weigh yourself outside of water,
then get some type of

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spring or waterproof weighing
machine, put it at the bottom

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of your pool, stand on it, and
figure out what your weight

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is, assuming that you're dense
enough to go all the

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way into the water.

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You could figure out somehow
your weight in water, and then

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you would know your volume.

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There's other ways.

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You could just figure out how
much the surface of the water

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increases, and take
that water away.

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This was interesting.

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Just knowing how much the
buoyant force of the water was

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or how much lighter we are when
the object goes into the

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water, we can figure out the
volume of the object.

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This might seem like a very
small volume, but just keep in

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mind in a meter cubed, you
have 27 square feet.

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If we multiply that number
times 27, it equals 0.02

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square feet roughly.

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In 0.02 square feet, how many--
in a square foot,

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there's actually-- 12 to the
third power times 12 times 12

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is equal to 1,728 times 0.02.

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So this is actually
34 square inches.

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The object isn't as small as you
may have thought it to be.

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It's actually maybe a little bit
bigger than 3 inches by 3

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inches by 3 inches, so it's
a reasonably sized object.

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Anyway, let's do another
problem.

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Let's say I have some balsa
wood, and I know that the

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density of balsa wood is 130
kilograms per meter cubed.

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I have some big cube of balsa
wood, and what I want to know

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is if I put that-- let
me draw the water.

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I have some big cube of balsa
wood, which I'll do in brown.

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So I have a big cube of balsa
wood and the water should go

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on top of it, just so
that you see it's

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submerged in the water.

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I want to know what percentage
of the cube goes below the

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surface of the water?

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Interesting question.

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So how do we do that?

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For the object to be at rest,
for this big cube to be at

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rest, there must be zero net
forces on this object.

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In that situation, the buoyant
force must completely equal

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the weight or the force
of gravity.

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What's the force of gravity
going to be?

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The force of gravity is just the
weight of the object, and

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that's the volume of the balsa
wood times the density of the

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balsa wood times gravity.

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What's the buoyant force?

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The buoyant force is equal to
the volume of the displaced

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water, but that's also the
volume of the displaced water

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and it's the volume of the cube
that's been submerged.

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The part of the cube that's
submerged, that's volume.

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That's also equal
to the amount of

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volume of water displaced.

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We could say that's the volume
of the block submerged, which

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is the same thing, remember,
as the volume of the water

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displaced times the density
of water times gravity.

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Remember, this is density
of water.

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So remember, the buoyant force
is just equal to the weight of

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the water displaced and that's
just the volume of the water

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displaced times the density
of water times gravity.

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Of course, the volume of the
water displaced is the exact

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same thing as the volume
of the block

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that's actually submerged.

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Since the block is stationary,
it's not accelerating upwards

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or downwards, we know that
these two quantities must

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equal each other.

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So V, the volume of the wood,
the entire volume, not just

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the amount that's submerged,
times the density of the wood

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times gravity must equal the
volume of the wood submerged,

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which is equal to the volume of
the water displaced times

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the density of water
times gravity.

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We have the acceleration
of gravity.

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We have that on both sides,
so we can cross it out.

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Let me switch colors to
ease the monotony.

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What happens if we divide both
sides by the volume of the

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balsa wood?

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I'm just rearranging
this equation.

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I think you'll figure it out.

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We divide both sides by that,
and you get the volume

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submerged divided by the volume
of the balsa wood-- I

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just divided both sides by VB
and switched sides-- is equal

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to the density of the balsa
wood divided by

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the density of water.

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Does that make sense?

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I just did a couple of quick
algebraic operations, but

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hopefully that got rid of
the g, and that should

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make sense to you.

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Now we're ready to solve
our problem.

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My original question is
what percentage of

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the object is submerged?

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That's exactly this number.

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If we say this is the volume
submerged over the total

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volume, this is the
percent submerged.

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That equals the density of
balsa wood, which is 130

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kilograms per meter cubed,
divided by the density of

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water, which is 1,000 kilograms
per meter cubed, so

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130 divided by 1,000 is 0.13.

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Vs over VB is equal to
0.13, which is the

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same thing as 13%.

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So, exactly 13% percent of this
balsa wood block will be

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submerged in the water.

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That's pretty neat to me.

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It actually didn't have
to be a block.

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It could have been shaped
like a horse.

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I'll see you in the
next video.

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