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Current time:0:00Total duration:8:16

Welcome back. So I was trying to rush and
finish a problem in the last two minutes of the video, and
I realize that's just bad teaching, because I
end up rushing. So this is the problem we were
going to work on, and you'll see a lot of these. They just want you to become
familiar with the variables in Newton's law of gravitation. So I said that there's two
planets, one is Earth. Now I have time to draw things,
so that's Earth. And then there's Small Earth. And Small Earth-- well, maybe
I'll just call it the small planet, so we don't
get confused. It's green, showing that
there's probably life on that planet. Let's say it has 1/2 the radius,
and 1/2 the mass. So if you think about
it, it's probably a lot denser than Earth. That's a good problem
to think about. How much denser is it, right? Because if you have 1/2 the
radius, your volume is much less than 1/2. I don't want to go into that
now, but that's something for you to think about. But my question is what
fraction, if I'm standing on the surface of this-- so the
same person, so Sal, if I'm on Earth, what fraction is the pull
when I'm on this small green planet? So what is the pull
on me on Earth? Well, it's just going to be-- my
weight on Earth, the force on Earth, is going to be equal
to the gravitational constant times my mass, mass of me. So m sub m times the mass of
Earth divided by what? We learned in the last video,
divided by the distance between me and the center
of the mass of Earth. Really, my center of mass and
the center of mass of Earth. But this is between the surface
of the Earth, and I'd like to think that I'm not
short, but it's negligible between my center of mass and
the surface, so we'll just consider the radius
of the Earth. So we divide it by the radius
of the Earth squared. Using these same variables,
what's going to be the force on this other planet? So the force on the other
planet, this green planet-- I'll do it in green-- and we're
calling it the small planet, it equals what? It equals the gravitational
constant again. And my mass doesn't change when
I go from one planet to another, right? Its mass now is what? We would write it m
sub s here, right? This is the small planet. And we wrote right here that
it's 1/2 the mass of Earth, so I'll just write that. So it's 1/2 the mass of Earth. And what's its radius? What's the radius now? I could just write the radius
of the small planet squared, but I'll say, well, we know. It's 1/2 the radius of Earth,
so let's put that in there. So 1/2 radius of Earth. We have to square it. Let's see what this
simplifies to. This equals-- so we can take
this 1/2 here-- 1/2G mass of me times mass of Earth over--
what's 1/2 squared? It's 1/4. Over 1/4 radius of
Earth squared. And what's 1/2 divided by 1/4? 1/4 goes into 1/2 two
times, right? Or another way you can think
about it is if you have a fraction in the denominator,
when you put it in the numerator, you flip it
and it becomes 4. So 4 times 1/2 is 2. Either way, it's just math. So the force on the small planet
is going to be equal to 1/2 divided by 1/4 is 2 times
G, mass of me, times mass of Earth, divided by the radius
of Earth squared. And if we look up here,
this is the same thing as this, right? It's identical. So we know that the force that
applied to me when I'm on the surface of the small planet is
actually two times the force applied on Earth, when
I go to Earth. And that's something interesting
to think about, because you might have said
initially, wow, you know, the mass of the object matters
a lot in gravity. The more massive the object,
the more it's going to pull on me. But what we see here is
that actually, no. When I'm on the surface of
this smaller planet, it's pulling even harder on me. And why is that? Well, because I'm actually
closer to its center of mass. And as we talked about earlier
in this video, this object is probably a lot denser. You could say it's only 1/2 the
mass, but it's much less than 1/2 of the volume, right? Because the volume is the cube
of the radius and all of that. I don't want to confuse you, but
this is just something to think about. So not only does the mass
matter, but the radius matters a lot. And the radius is actually the
square, so it actually matters even more. So that's something
that's pretty interesting to think about. And these are actually very
common problems when they just want to tell you, oh, you go to
a planet that is two times the mass of another planet, et
cetera, et cetera, what is the difference in force
between the two? And one thing I want you to
realize, actually, before I finish this video since I do
have some extra time, when we think about gravity, especially
with planets and all of that, you always
feel like, oh, it's Earth pulling on me. Let's say that this is the
Earth, and the Earth is huge, and this is a tiny spaceship
right here. It's traveling. You always think that
Earth is pulling on the spaceship, right? The gravitational
force of Earth. But it actually turns out, when
we looked at the formula, the formula is symmetric. It's not really saying one
is pulling on the other. They're actually saying
this is the force between the two objects. They're attracted
to each other. So if the Earth is pulling on
me with the force of 500 Newtons, it actually turns out
that I am pulling on the Earth with an equal and opposite
force of 5 Newtons. We're pulling towards
each other. It just feels like the Earth is,
at least from my point of view, that the Earth
is pulling to me. And we're actually both being
pulled towards the combined center of mass. So in this situation, let's say
the Earth is pulling on the spaceship with the force
of-- I don't know. I'm making up numbers
now, but let's say it's 1 million Newtons. It actually turns out that the
spaceship will be pulling on the Earth with the same force
of 1 million Newtons. And they're both going to be
moved to the combined system's center of mass. And the combined system's center
of mass since the Earth is so much more massive is
going to be very close to Earth's center of mass. It's probably going to
be very close to Earth's center of mass. It's going to be like
right there, right? So in this situation, Earth
won't be doing a lot of moving, but it will be pulled
in the direction of the spaceship, and the spaceship
will try to go to Earth's center of mass, but at some
point, probably the atmosphere, or the rock that it
runs into, it won't be able to go much further and
it might crash right around there. Anyway, I wanted just to give
you the sense that it's not necessarily one object just
pulling on the other. They're pulling towards
each other to their combined center of masses. It would make a lot more sense
if they had just two people floating in space, they actually
would have some gravity towards each other. It's almost a little romantic. They would float
to each other. And actually, you could
figure it out. I don't have the time to do
it, but you could use this formula and use the constant,
and you could figure out, well, what is the gravitational
attraction between two people? And what you'll see is that
between two people floating in space, there are other forms
of attraction that are probably stronger than their gravitational attraction, anyway. I'll let you ponder that
and I will see you in the next video.