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Gravitation (part 2)

A little bit more on gravity. Created by Sal Khan.

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  • leaf grey style avatar for user Aayush A.V
    well moon is smaller but its gravity isnt double much less more than earth's gravity.why?i dont understand.
    also what is the comparison between the two the planets? is it of same density but more compressed to get half the radius?
    (22 votes)
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    • leaf green style avatar for user Mark Zwald
      Gravitational acceleration at the surface of a body is:
      a = GM/R^2
      assuming a uniform density ρ and spherical body, the mass will be
      M = ρV = ρ * 4/3πR^3
      substitute that M into the acceleration equation...
      a = G/R^2 * ρ * 4/3πR^3
      a = 4/3GρπR
      so the gravitational acceleration at the surface increases linearly with the radius of the body.
      (15 votes)
  • marcimus pink style avatar for user Ilvânia
    gravitational field is the same as gravitational acceleration? if no, why its symbol is g?
    why is g=a ? accelaration is the variation of the velocity in a certain time, so the earth moves if someone, for example, falls from a building. please someont to explain, i am really confused
    (2 votes)
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    • male robot hal style avatar for user Andrew M
      Surely you've noticed that when you "convert" a mass to a weight (on earth) you use 9.8 N/kg, and you also know that the acceleration due to gravity on earth is 9.8 m/s^2. Is it coincidence that they are both 9.8? Nope.

      g stands for gravitational field strength AND acceleration due to gravity. They are the same thing. Let's calculate the acceleration of something if we drop it. The force on it is mg. Newton's second law tells us F = ma.
      mg = ma
      m cancels out
      g = a
      So the acceleration will be exactly equal to the gravitational field strength, and mass does not matter.
      This is why everything falls with the same acceleration (if we can neglect air resistance)
      Now let's look at the units.
      Gravitational field strength has units of Newtons/kg. That's why the force on a 10 kg mass is 98 N, because (9.8 N/kg)*10 kg = 98 N.
      The units of acceleration of course are m/s^2. So how can g be BOTH gravitation field strength AND acceleration due to gravity? Let's look more closely at the units:
      A newton is a kg*m/s^2
      gravitational field strength is in N/kg
      So g = 9.8 N/kg = (9.8 kg*m/s^2)/kg = 9.8 m/s^2
      In other words, N/kg is the same thing as m/s^2.
      (7 votes)
  • duskpin ultimate style avatar for user Injila Ahmed
    Why we call G as universal gravitational constant if it is derived by taking only earth's mass and acceleration due to gravity on earth 'g'.and distance between center of masses of masses of earth and object.If we have taken everything of earth here then why it is universal?
    as we know Fg=(Gm1.m2)/r^2
    then we find out G's value as G=(Fg.r^2)/(m1.m2).
    here Fg is gravitational force on earth,r is distance between object having mass m1 on earth's surface and center of mass of earth having mass m2.
    (3 votes)
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  • blobby green style avatar for user isha hassan
    i didn't get the concept of higher gravitational attraction of a smaller planet than earths(ok,if its the radius adjusting itself with the masses),then how come moons gravity less than that of earths?
    (1 vote)
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    • starky ultimate style avatar for user Astronomer-to-be
      It's not about size but about mass. In short: The heavier the object, the greater its gravity. Black holes, for example, are reeeally small (compared to the rest of the universe) but incredibly dense and thus incredibly heavy. This is what causes them to attract everything that comes too close to it - even light - and never let it go. So, if there's a planet that is smaller but heavier than Earth, its gravitational force will be stronger. And the moon's gravity is less than the Earth's because it is much lighter, as Andrew already said.
      (5 votes)
  • blobby green style avatar for user Julius Omondi
    If. The earth suddenly stopped in its orbit assumed to be circular how long it might take before it falls into the sun
    (1 vote)
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  • purple pi teal style avatar for user Nischal Chandur
    so.....now let's say a body is moving with uniform acceleration in space.......
    this means that since there is no friction in space it keeps accelerating and eventually reach the speed of light and probably travel faster that light........
    so.......can there be objects that are faster than light?
    (2 votes)
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    • male robot hal style avatar for user Andrew M
      No, because Newtonian physics does not apply as you get close to the speed of light.
      The object's mass increases, so it becomes harder to accelerate as it gets close to light speed
      The mass approaches infinity as speed approaches c
      Nothing with mass can travel at or above the speed of light
      (3 votes)
  • aqualine seed style avatar for user Anthoan Oroz
    If G is there to cancel the units, why int it then just a whole number like 1?
    (1 vote)
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  • winston default style avatar for user Jerry Zhu
    If a smaller planet has a larger gravitation, how does the moon has a smaller gravitational force than Earth?
    (2 votes)
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  • spunky sam blue style avatar for user Hemanth C.
    The distance between two electrons in an orbit ( I am using that concept just for this example) is very small, so the gravitational force between them should become immeasurably strong. So how come they are still repelled by the electromagnetic force? Shouldn't the gravitational force between them be stronger because of the very small distance, or at least be able to balance the Coulomb force?
    (1 vote)
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  • marcimus pink style avatar for user Soumya Kumar
    Is there any difference in the magnitude of inertial and gravitational mass?
    (2 votes)
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Video transcript

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.