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let's do a little bit of 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 mass of 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 or 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 M 1 this object has mass M 2 and the distance between their center of masses is our that the magnitude of the force of gravity and if the force of gravity is 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 wanted to know the gravitational field created by M 2 well then you would just divide both sides by M 1 so if you divide both sides by M 1 you get that the gafe rotational field actually isn't constant it is equal to the gravitational constant times these cancel out to 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 the don't denoted with a capital G over here is equal to let's say the mass of the object and let's just call that M 1 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 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 an and one of the R's 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 that gravitational potential energies are always going to 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 times 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 our but this is really 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

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