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Most physics books will tell
you that the acceleration

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due to gravity near the
surface of the Earth

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is 9.81 meters per
second squared.

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And this is an approximation.

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And what I want to
do in this video

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is figure out if this is the
value we get when we actually

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use Newton's law of
universal gravitation.

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And that tells us that the
force of gravity between two

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objects-- and let's just
talk about the magnitude

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of the force of gravity
between two objects--

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is equal to the universal
gravitational constant times

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the mass of one of the
bodies, M1, times the mass

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of the second body divided by
the distance between the center

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of masses of the bodies squared.

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So let's use this, the
universal law of gravitation

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to figure out what the
acceleration due to gravity

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should be at the
surface of the Earth.

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And I have a g right over here.

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I have the mass of the Earth,
which I've looked up over here.

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And we also have the
radius of the Earth.

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And for the sake of
this, we're going

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to assume that the distance
between the body, if we're

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at the the surface of the
Earth, the distance between that

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and the center of
the Earth is just

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going to be the
radius of the Earth.

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And so this will give us
the magnitude of the force.

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If we want to figure out the
magnitude of the acceleration,

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which this really is-- I
actually didn't write this

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is a vector.

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So this is just the magnitude
of the acceleration.

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If you wanted the acceleration,
which is a vector,

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you'd have to say downwards or
towards the center of the Earth

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in this case.

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But if you want
the acceleration,

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we just have to
remember that force

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is equal to mass
times acceleration.

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And if you wanted to
solve for acceleration

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you just divide both
sides times mass.

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So force divided by mass
is equal to acceleration.

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Or if you take the
magnitude of your force

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and you divide by
mass, you're going

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to get the magnitude
of your acceleration.

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This is a scalar quantity.

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This is a scalar
quantity right over here.

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So if you want the acceleration
due to gravity, you divide.

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Let's write this in terms of
the force of gravity on Earth.

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So the magnitude of
the force of gravity

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on Earth, this one
right over here.

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So this will be in
the case of Earth.

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I just wrote Earth
really, really small.

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So one of these masses
is going to be Earth.

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It's going to be this
mass right over here.

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And so if you wanted
the acceleration

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due to gravity at the
surface of the Earth,

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you would just have to
divide by the mass that

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is being accelerated
due to that force.

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And in this case, it
is the other mass.

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It is the mass that's
sitting on the surface.

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So let's divide both
sides by that mass.

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Let's divide both
sides by that mass.

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And this will give
us the magnitude

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of the acceleration on
that mass due to gravity.

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So this is equal to the
magnitude of acceleration,

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due to gravity.

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And the whole reason why this
is actually a simplifying thing

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is that these two, this M2
right over here and this M2

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cancels out.

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And so the magnitude
of our acceleration

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due to gravity using Newton's
universal law of gravitation

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is just going to be this
expression right over here.

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It's going to be the
gravitational constant times

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the mass of the Earth
divided by the distance

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between the object's
center of mass

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and the center of the
mass of the Earth.

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And we're going to
assume that the object is

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right at the surface,
that its center of mass

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is right at the surface.

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So this is actually going to
be the radius of the Earth

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squared, so divided
by radius squared.

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Sometimes this is also viewed
as the gravitational field

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at the surface of the Earth.

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Because if you
multiply it by a mass,

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it tells you how much force
is pulling on that mass.

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But with that out of
the way, let's actually

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use a calculator to
calculate what this value is.

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And then what I want to do
is figure out, well, one,

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I want to compare
it to the value

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that the textbooks
give us and see, maybe,

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why it may or may
not be different.

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And then think
about how it changes

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as we get further
and further away

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from the surface of the Earth.

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And in particular, if
we get to an altitude

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that the space shuttle or the
International Space Station

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might be at, and this is at
an altitude of 400 kilometers

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is where it tends to
hang out, give or take

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a little bit, depending
on what it is up to.

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So first, let's just
figure out what this value

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is when we use a universal
law of gravitation.

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So let's get my calculator out.

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So we know what g is.

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It is 6.6738 times 10
to the negative 11.

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This EE button means, literally,
times 10 to the negative 11.

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So this is 6.6738 times
10 to the negative 11.

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And then I want to
multiply that times

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the mass of Earth, which
is right over here.

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That is 5.9722 times
10 to the 24th.

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So times 10 to the 24th power.

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And we want to divide that by
the radius of Earth squared.

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So divided by the
radius of Earth

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is-- so this is in kilometers.

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And I just want to make
sure that everything

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is the same units.

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So 6,371 kilometers--
actually, let me scroll over.

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Well, you can't see the
kilometers right now.

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But this is kilometers.

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It is the same thing
as 6,371,000 meters.

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If you just multiply
this by 1,000.

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Or you could even
write this as 6.371.

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6.371 times 10 to
the sixth meters.

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And we're going to square this.

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That's the radius of the Earth.

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The distance between the center
of mass of Earth and the center

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of mass of this object,
which is sitting

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at the surface of the Earth.

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And so let's get our drum roll.

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And we get 9.8.

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And if we round, we actually
get something a little bit

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higher than what the
textbooks give us.

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We get 9.82.

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Let's just round.

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So we get 9.82-- 9.82
meters per second squared.

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And so you might say,
well, what's going on here?

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Why do we have this
discrepancy between what

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the universal law of
gravitation gives us

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and what the average
measured acceleration

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due to the force of gravity
at the surface of the Earth.

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And the discrepancy here, the
discrepancy between these two

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numbers, is really
because Earth is not

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a uniform sphere
of uniform density.

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And that's what we have
to assume over here

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when we use the universal
law of gravitation.

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It's actually a little bit
flatter than a perfect sphere.

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And it definitely does
not have uniform density.

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The different layers of the
Earth have different densities.

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You have all sorts of
different interactions.

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And then you also, if you
measure effective gravity,

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there's also a little bit of a
buoyancy effect from the air.

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Very, very, very,
very negligible, I

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don't know if it would have
been enough to change this.

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But there's other minor,
minor effects, irregularities.

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Earth is not a perfect sphere.

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It is not of uniform density.

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And that's what accounts
for the bulk of this.

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Now, with that out of the
way, what I'm curious about

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is what is the
acceleration due to gravity

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if we go up 400 kilometers?

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So now, the main difference
here, g will stay the same.

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The mass of Earth
will stay the same,

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but the radius is now
going to be different.

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Because now we're placing the
center of mass of our object--

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whether it's a space station
or someone sitting in the space

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station, they're going to
be 400 kilometers higher.

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And I'm going to exaggerate
what 400 kilometers looks like.

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This is not drawn to scale.

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But now the radius is going
to be the radius of the Earth

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plus 400 kilometers.

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So now, for the case
of the space station,

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r is going to be not
6,371 kilometers.

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It's going to be 6,000--
we're going to add 400

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to this-- 6,771
kilometers, which

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is the same thing as
6,771,000 meters, which

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is the same thing as 6.771
times 10 to the sixth meters.

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This is-- 1, 2, 3, 4, 5,
6-- 10 to the sixth meters.

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So let's go back
to our calculator.

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So second entry, that's
the last entry we had.

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And instead of 6.371
times 10 to the sixth,

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let's add 400
kilometers to that.

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So then we get 6.7.

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So we're adding 400 kilometers.

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So it was 371.

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Now it's 771 times
10 to the sixth.

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And what do we get?

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We get 8.69 meters
per second squared.

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So now the acceleration here is
8.69 meters per second squared.

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And you can verify that
the units work out.

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Because over here,
gravity is in meters

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cubed per kilogram
second squared.

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You multiply that times
the mass of the Earth,

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which is in kilograms.

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The kilograms cancel out
with these kilograms.

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And then you're dividing
by meters squared.

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So you divide this
by meters squared.

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You're left with meters
per second squared.

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So the units work out as well.

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So there's an important
thing to realize.

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And this is a misconception.

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We do a whole video
on it earlier,

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when we talk about the
universal law of gravitation,

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is that there is gravity when
you are in orbit up here.

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The only reason why it feels
like there's not gravity

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or it looks like
there's not gravity

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is that this space
station is moving

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so fast that it's
essentially in free fall.

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But it's moving so fast that
it keeps missing the Earth.

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And in the next video,
we'll figure out

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how fast does it have to
travel in order for it to stay

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in orbit, in order for it to not
plummet to Earth due to this,

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due to the force of gravity,
due to the acceleration that

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is occurring, this centripetal,
this center-seeking

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00:10:10,200 --> 00:10:11,600
acceleration?

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00:10:11,600 --> 00:00:00,000