Below I've tried to put together a reasonable definition of gravitational potential energy.
No, you haven't. You derived the formula for gravitational potential from the formula for gravitational force, without understanding what it is your final formula means, or how it is used.
And, as kellog already pointed out, it is dependent on your reference point.
The scientifically accepted definition of
Gravitational potential energy (GPE from now on) is the
potential energy an object possesses because of its
position in a gravitational field. The calculation of GPE for a
static point can be interpreted as the potential energy
lost as an object moves from infinity to its current position (this is why GPE is always negative). But when
moving masses in a gravitational field, changes in GPE represent energy released or consumed in that movement.
So where you are making your mistake is in how gravitational potential energy can be used to determine the energy released/consumed when
moving and object from one position in a gravitational field to another - and since we're talking about moving objects this value is critical.
In the case of a moving object - which is exactly what your two merging planets are - the
difference in GPE from the start/end points represents the amount of energy (not potential energy) released (if moving closer to the reference point), or consumed (if moving away from the reference point).
Gravitational potential energy = U = -G*m1*m2/r, where r is the distance separating the object from the reference point.
And now where you've gone wrong - the reference point and how potential energy changes based on changes in its position relative to a reference point.
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We have your two masses - lets call them 'A' (with a mass of mA) and 'B' (with a mass of mB), whose centers of mass are separated by distance 'r'. The gravitational potential energy of 'B',
relative to 'A' is equal to: U = -[G*mA*mB]/r
[G*mA*mB] will be a constant in our system - G and the mass of our planets do not change. For the sake of simplicity, lets pretend that [G*mA*mB] = 1, and r is 1m. This means we have a potential energy of: U = -1/1m = -1J.
Keep in mind that negative sign - its important, and where you are going wrong.
Lets pretend we now move B
closer to A such that they are half as far apart as they were before:
U = -1/0.5m
U = -2J
We've had a change in potential energy of:
deltaU = U(initial) - U(final)
deltaU = [-1J] - [-2J] = 1J
So in moving B 0.5m closer to A we
release 1J worth of potential energy, which we can use to preform work. In the case of gravity, this work will be in the form of gravitational acceleration, meaning that B will now have a kinetic energy of 1J, relative to A. In other words, when you move two objects
towards each other in a gravitational field the gravitational potential energy gets converted
into work - as in you can then use that work to do stuff - say destroy a planet.
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Now, what if we do the opposite - move B
away from A, to a distance of 2m?
U = -1/2m
U = -0.5J
deltaU = [-1J] - [-0.5J] = -0.5J
Note the negative sign. This means we've had to
add potential energy to move B farther away from A, which means we've had to use work to move B.
Now, what about your case of separating things to infinity:
U = -1/infinity
U = 0J
deltaU = -1J-0J = -1J
So to separate our two planets to infinity, we would have to do 1J worth of work. In other words, when you move two objects
away from each other in a gravitational field, you must
preform work to separate two masses.
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The above shows where you went wrong in this, and the other, thread. In looking back I realize it was partly my fault - I assumed as a self-professed mathematician you would understand the impact of inverse relationships and how GPE translates into energy, and thus skipped the (obvious to me) deltaU calculation.
For that I apologize - but it doesn't change the fact you are wrong, and the numbers I calculated were correct.
In fact, I was overly generous and gave you a huge amount of wiggle room, in terms of the amount of movement - and thus the change in GPE (and thus the amount of energy released).
Going back to the original thread, where we had two massive planets immediately juxtaposed to each other. The gravitational potential energy of that initial state was -5.95 X 10^31 J. Since gravity is acting on the system, those two planets are drawn together. As you can see from the above examples, that means that after the merger the GPE will be even more negative, and the difference in GPE will represent the work done on the planet. Since the relationship is an inverse one...well, we'll get back to that.
The exact amount of work that gets conducted depends on just how far the two centers of mass can move before equilibrium is achieved (i.e. they merge to form a new center of mass). As I stated (and you never disagreed) they would move in ~50% of the way, simply because they are 2 ~equal spheres moving, and thus their merged center of mass should be at a point roughly where they first meet.
Keep in mind, GPE is essentially -a/r, where a is a constant determined by the (unchanging) mass in the system and the (unchanging) gravitational constant. So a movement of 50% the radius would mean the change in GPE is deltaU = -a - [-a/0.5] = -a - -2a = a.
Which is the value I used in my explanation. I assumed a self-professed mathematician would be aware of the impact of inverse relationships - apparently I was wrong.
So to go back to the point I made in the last thread - in your system, starting with the two planets touching and NO OTHER ENERGY, and assuming the center of mass of the final planet is located roughly where the two planets first meet, you will release 5.95 X 10^31 J of energy due to the change in gravitational potential energy.
Whining that if you were to apply 5.95 X 10^31 J of energy you would separate the masses to infinity is pointless - we're not pushing these things apart; gravity is moving them together.
Bryan
