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Gravitation (part 2) : A little bit more on gravity
相關課程
- 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.
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