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Welcome back.

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So I was trying to rush and
finish a problem in the last

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two minutes of the video, and
I realize that's just bad

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teaching, because I
end up rushing.

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So this is the problem we were
going to work on, and you'll

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see a lot of these.

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They just want you to become
familiar with the variables in

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Newton's law of gravitation.

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So I said that there's two
planets, one is Earth.

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Now I have time to draw things,
so that's Earth.

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And then there's Small Earth.

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And Small Earth-- well, maybe
I'll just call it the small

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planet, so we don't
get confused.

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It's green, showing that
there's probably

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life on that planet.

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Let's say it has 1/2 the radius,
and 1/2 the mass.

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So if you think about
it, it's probably a

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lot denser than Earth.

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That's a good problem
to think about.

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How much denser is it, right?

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Because if you have 1/2 the
radius, your volume is much

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less than 1/2.

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I don't want to go into that
now, but that's something for

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you to think about.

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But my question is what
fraction, if I'm standing on

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the surface of this-- so the
same person, so Sal, if I'm on

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Earth, what fraction is the pull
when I'm on this small

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green planet?

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So what is the pull
on me on Earth?

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Well, it's just going to be-- my
weight on Earth, the force

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on Earth, is going to be equal
to the gravitational constant

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times my mass, mass of me.

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So m sub m times the mass of
Earth divided by what?

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We learned in the last video,
divided by the distance

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between me and the center
of the mass of Earth.

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Really, my center of mass and
the center of mass of Earth.

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But this is between the surface
of the Earth, and I'd

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like to think that I'm not
short, but it's negligible

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between my center of mass and
the surface, so we'll just

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consider the radius
of the Earth.

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So we divide it by the radius
of the Earth squared.

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Using these same variables,
what's going to be the force

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on this other planet?

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So the force on the other
planet, this green planet--

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I'll do it in green-- and we're
calling it the small

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planet, it equals what?

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It equals the gravitational
constant again.

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And my mass doesn't change when
I go from one planet to

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another, right?

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Its mass now is what?

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We would write it m
sub s here, right?

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This is the small planet.

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And we wrote right here that
it's 1/2 the mass of Earth, so

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I'll just write that.

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So it's 1/2 the mass of Earth.

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And what's its radius?

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What's the radius now?

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I could just write the radius
of the small planet squared,

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but I'll say, well, we know.

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It's 1/2 the radius of Earth,
so let's put that in there.

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So 1/2 radius of Earth.

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We have to square it.

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Let's see what this
simplifies to.

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This equals-- so we can take
this 1/2 here-- 1/2G mass of

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me times mass of Earth over--
what's 1/2 squared?

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It's 1/4.

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Over 1/4 radius of
Earth squared.

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And what's 1/2 divided by 1/4?

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1/4 goes into 1/2 two
times, right?

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Or another way you can think
about it is if you have a

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fraction in the denominator,
when you put it in the

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numerator, you flip it
and it becomes 4.

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So 4 times 1/2 is 2.

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Either way, it's just math.

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So the force on the small planet
is going to be equal to

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1/2 divided by 1/4 is 2 times
G, mass of me, times mass of

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Earth, divided by the radius
of Earth squared.

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And if we look up here,
this is the same

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thing as this, right?

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It's identical.

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So we know that the force that
applied to me when I'm on the

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surface of the small planet is
actually two times the force

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applied on Earth, when
I go to Earth.

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And that's something interesting
to think about,

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because you might have said
initially, wow, you know, the

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mass of the object matters
a lot in gravity.

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The more massive the object,
the more it's

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going to pull on me.

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But what we see here is
that actually, no.

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When I'm on the surface of
this smaller planet, it's

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pulling even harder on me.

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And why is that?

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Well, because I'm actually
closer to its center of mass.

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And as we talked about earlier
in this video, this object is

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probably a lot denser.

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You could say it's only 1/2 the
mass, but it's much less

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than 1/2 of the volume, right?

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Because the volume is the cube
of the radius and all of that.

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I don't want to confuse you, but
this is just something to

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think about.

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So not only does the mass
matter, but the

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radius matters a lot.

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And the radius is actually the
square, so it actually

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matters even more.

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So that's something
that's pretty

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interesting to think about.

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And these are actually very
common problems when they just

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want to tell you, oh, you go to
a planet that is two times

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the mass of another planet, et
cetera, et cetera, what is the

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difference in force
between the two?

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And one thing I want you to
realize, actually, before I

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finish this video since I do
have some extra time, when we

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think about gravity, especially
with planets and

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all of that, you always
feel like, oh, it's

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Earth pulling on me.

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Let's say that this is the
Earth, and the Earth is huge,

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and this is a tiny spaceship
right here.

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It's traveling.

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You always think that
Earth is pulling on

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the spaceship, right?

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The gravitational
force of Earth.

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But it actually turns out, when
we looked at the formula,

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the formula is symmetric.

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It's not really saying one
is pulling on the other.

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They're actually saying
this is the force

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between the two objects.

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They're attracted
to each other.

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So if the Earth is pulling on
me with the force of 500

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Newtons, it actually turns out
that I am pulling on the Earth

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with an equal and opposite
force of 5 Newtons.

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We're pulling towards
each other.

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It just feels like the Earth is,
at least from my point of

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view, that the Earth
is pulling to me.

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And we're actually both being
pulled towards the combined

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center of mass.

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So in this situation, let's say
the Earth is pulling on

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the spaceship with the force
of-- I don't know.

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I'm making up numbers
now, but let's say

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it's 1 million Newtons.

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It actually turns out that the
spaceship will be pulling on

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the Earth with the same force
of 1 million Newtons.

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And they're both going to be
moved to the combined system's

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center of mass.

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And the combined system's center
of mass since the Earth

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is so much more massive is
going to be very close to

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Earth's center of mass.

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It's probably going to
be very close to

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Earth's center of mass.

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It's going to be like
right there, right?

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So in this situation, Earth
won't be doing a lot of

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moving, but it will be pulled
in the direction of the

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spaceship, and the spaceship
will try to go to Earth's

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center of mass, but at some
point, probably the

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atmosphere, or the rock that it
runs into, it won't be able

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to go much further and
it might crash

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right around there.

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Anyway, I wanted just to give
you the sense that it's not

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necessarily one object just
pulling on the other.

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They're pulling towards
each other to their

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combined center of masses.

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It would make a lot more sense
if they had just two people

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floating in space, they actually
would have some

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gravity towards each other.

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It's almost a little romantic.

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They would float
to each other.

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And actually, you could
figure it out.

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I don't have the time to do
it, but you could use this

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formula and use the constant,
and you could figure out,

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well, what is the gravitational
attraction

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between two people?

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And what you'll see is that
between two people floating in

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space, there are other forms
of attraction that are

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probably stronger than their

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gravitational attraction, anyway.

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I'll let you ponder that
and I will see

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you in the next video.

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