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Gravitational potential energy at large distances

Learn how to calculate gravitational potential energy when we can no longer assume the gravitational field is uniform.

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  • blobby green style avatar for user Sadist
    I read somewhere that aboslute gravitational potential energy is negative because it is workdone against gravity and that energy of an object in bound state is negative so is it the reason or we just take it as negative for our own convenience?
    (11 votes)
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  • blobby green style avatar for user paul jordan
    confused: r^2 in newtonian definition is earth radius. but r @ is height or distance. So these are differnt things--how do they cancel out??
    (3 votes)
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  • blobby green style avatar for user Ryunah Kang
    Energy can't have a negative value. How come Ug= -(G*m1*m2)/r ?
    (2 votes)
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    • blobby green style avatar for user deepak
      Hello there, Ryunah Kang,
      It is a common misconception that energy cannot have a negative value. You see, when we refer to the object that Sal talked about at we do so by refering to its to total gravitational energy with respect to the space—time continuum.
      Ideally, Space-time when there exists no mass around a space-time continuum, there should be no deformation. If we consider mapping this plane, we can make its displacement (height or in the case of Sal's example at , r) 0. So at this point r=0.
      But when mass is introduced on to the continuum, it bends the fabric of space time. This allows for a gravity well to be produced. When an object experiences the force of this gravity well, it moves in its direction: downwards (with respect to the point where r=0) hence the Ug value is negative with respect to space—time.
      (5 votes)
  • mr pink red style avatar for user starryowl
    the value of g=9.8 m/s2, does it include any air resistance?
    (1 vote)
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  • purple pi teal style avatar for user Asmitha krishnakumar
    Does variation of g happen with respect to altitude ?
    (1 vote)
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    • duskpin ultimate style avatar for user DanX
      Yes, but the change in the gravitational acceleration is so miniscule and very rarely do you need to be more precise than 3 significant figures with this value. It'll take heights bigger than the peak of Mt. Everest to change the value of g even by 1e-2m/s^2.
      (3 votes)
  • female robot ada style avatar for user Hitaishi✨
    g=GM/R^2

    here g is inversely proportional to the R square. That means if radius will increase then acceleration due to gravity will decrease. But how is it possible?
    (1 vote)
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  • blobby green style avatar for user paul jordan
    @ Is mgy considered a negative force?
    (1 vote)
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    • blobby green style avatar for user deepak
      First off, mgy is not a force. It is the energy of gravitational potential on the object. This value is not negative as it is in reference to the height above the planet. Usually, the value of Ug is negative because the height of an object suspended in space-time is always negative due to the bending of space-time due to mass
      (2 votes)
  • blobby green style avatar for user ulatowskaj23
    So is the 'y'- which is the height, irrelevant then?
    (1 vote)
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    • starky ultimate style avatar for user Divyae Arya
      Well, the variables were kinda confusing but essentially the r=y. In the newtons law of gravitation equation the variable r it is normally used for radius of the Earth but now because we are dealing with these long distances, it is important to note that the r is actually the distance between the 2 objects. Hope, it helped:)
      (1 vote)
  • starky sapling style avatar for user Tashu Gupta
    Is there a proof for this formula?
    (1 vote)
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  • aqualine ultimate style avatar for user Tyler Mullens
    How can you find out what energy they have and how to find their value? I'm just really confused.
    (1 vote)
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Video transcript

- [Tutor] Let's do a little bit of a review of potential energy and especially gravitational potential energy, because in this video, we're going to get a little bit more precise. So let's say that I have an object here, it has a mass of M and I were to change its position in the vertical direction, we're assuming that we are on Earth, where the gravitational field is G and we have a change in the vertical direction of delta y, so this is its new position, the object of mass, M, so what is its change in gravitational potential energy? Well, we have seen this before, our change in potential energy, due to gravity is going to be the mass of our object times the gravitational field times our change in position in the vertical direction. Now this is all good, but there's a couple of things, that we can get a little bit more precise, that we can think about in this video, one is we assumed a constant gravitational field and that's fine if we're near the surface of a planet and we also are just doing everything on a relative basis, but is there a way to come up with an absolute number on what is the gravitational potential energy? Well, to think about that, let's just go back to Newton's Law of Gravity, Newton's Law of Gravity tells us that if I have two objects, so this object has mass, M1, this object has mass, M2 and the distance between their center of masses is r, that the magnitude of the force of gravity and if the force of gravity's acting in this direction on that object, it would have an equal and opposite direction on that object right over there, so the magnitude of the force of gravity is going to be equal to the gravitational constant times the product of the two objects' masses divided by the radius squared or another way to think about it, if we wanna know the gravitational field created by M2, well then you would just divide both sides by M1, so if you divide both sides by M1, you get that the gravitational field actually isn't constant, it is equal to the gravitational constant times, these cancel out, M2, the mass of the object generating the gravitational field divided by r squared and so this can actually help us come up with an absolute formula for gravitational potential energy. We can say that gravitational potential energy and this is typically denoted with a capital G over here is equal to, let's say the mass of the object and let's just call that M1, the mass of the object times the gravitational field, so that's G times the mass of the object creating the field divided by r squared times the height that this is above the center of mass of the object generating the gravitational field, well, what's that? Well, if this was Earth, you would view r as, well, how far am I above the center of mass of Earth? And so you would multiply that times r, times r and so this r and one of the rs in the r squared will cancel out, so that will just go away and then this will just become an r and now one thing to think about is what should the potential energy be as r goes to infinity? Well, what is the gravitational field as we go to infinity, as r gets really, really, really large? Well, as r gets really, really, really large, the gravitational field goes to zero, but at the same time, the further you get away, the higher the potential energy you should have and so the way that we've thought about this is we say, hey look, let's just put a negative out front, you could just put it right over here and then it all works out, the gravitational potential energies are always gonna be negative, but they get less negative, the further out you get to and at infinity, it gets to zero. So let me rewrite this. The potential energy, due to gravity can be written as the negative of the gravitational constant times the mass of the first object time the mass of the second object divided by r and you can see this looks an awful lot like Newton's Law of Gravity, what the difference is, we have a negative in front of the gravitational constant, instead of dividing by r squared, we only divide by r, but this is really, really, really valuable now, because now we have an absolute sense of what gravitational potential energy is and it meets all the properties we need, it would still meet this idea that the further you get away, you will have an increase in gravitational potential energy, because it's going to become less negative, but at the same time as you go to infinity, as r gets infinitely far, your potential energy is going to approach zero. So I will leave you there, in future videos, we will apply this.