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Reconciling equations for Gravitational Potential Energy

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louisaaa01:
Hi all,

So, in Physics, we are taught two equations for gravitational potential energy:

U = mgh, assuming that the gravitational field is constant, and

U = -GMm/r

Now, I've come across numerous questions in textbooks and Trial papers which say something to the effect: "If the orbital radius of a satellite is doubled, what happens to gravitational potential energy?". Using the first equation, you can conclude that gravitational potential energy doubles. However, for the second equation, since U is inversely proportional to r, this implies that U will actually halve (though this still ends up being an increase as U becomes less negative). Yet, most, if not all solutions I've seen to such questions (most often multiple choice), will align with the former - radius doubles, gravitational potential energy increases by a factor of 2.

How can we reconcile these two equations? Which one is correct?

Any assistance is very much appreciated!

DrDusk:

--- Quote from: louisaaa01 on November 08, 2019, 08:50:50 am ---Hi all,

So, in Physics, we are taught two equations for gravitational potential energy:

U = mgh, assuming that the gravitational field is constant, and

U = -GMm/r

Now, I've come across numerous questions in textbooks and Trial papers which say something to the effect: "If the orbital radius of a satellite is doubled, what happens to gravitational potential energy?". Using the first equation, you can conclude that gravitational potential energy doubles. However, for the second equation, since U is inversely proportional to r, this implies that U will actually halve (though this still ends up being an increase as U becomes less negative). Yet, most, if not all solutions I've seen to such questions (most often multiple choice), will align with the former - radius doubles, gravitational potential energy increases by a factor of 2.

How can we reconcile these two equations? Which one is correct?

Any assistance is very much appreciated!

--- End quote ---
If you do get a question asking you this, pay attention to the relative magnitude of the radius 'r'. As a general rule IF 'r' is relatively small so for example if the object ball or whatever is still inside the Earths atmosphere, use the formula U = mgh. Otherwise if it is orbiting it is INCORRECT to use the formula U = mgh.

I must stress that because any trial paper or textbook that uses U = mgh and gives you a scenario where the satellite is orbiting is giving you a flawed question. If you get it in your trial paper and your teacher deducts a mark you can argue for that mark because you are correct. In the HSC if they give you a satellite in orbit and say it's radius doubles, you will have to use the latter.

louisaaa01:

--- Quote from: DrDusk on November 08, 2019, 07:25:51 pm ---If you do get a question asking you this, pay attention to the relative magnitude of the radius 'r'. As a general rule IF 'r' is relatively small so for example if the object ball or whatever is still inside the Earths atmosphere, use the formula U = mgh. Otherwise if it is orbiting it is INCORRECT to use the formula U = mgh.

I must stress that because any trial paper or textbook that uses U = mgh and gives you a scenario where the satellite is orbiting is giving you a flawed question. If you get it in your trial paper and your teacher deducts a mark you can argue for that mark because you are correct. In the HSC if they give you a satellite in orbit and say it's radius doubles, you will have to use the latter.

--- End quote ---

This is very helpful, thank you!

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