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Conservation of energy says total change in energy is $0$.

Let's assume a small object is in outer space (maybe like a football).

It moves towards a planet (assume planet has no atmosphere for simplicity) with a certain velocity. The object is out of the planet's gravitational field. Thus, the total energy is just the objects kinetic energy.

After a while, the object is in the planet's gravitational field. Thus, the total energy should be kinetic energy of object + gravitational potential energy of the object.

Here's my question, where did the extra energy come from?

I mean the planet is not gaining or losing any energy (I think). And, if I take the object have a gravitational field into account even though it is very weak, the planet should be accelerating slightly towards the object. This means the planet is also gaining energy.

What is wrong with the following example above?

Please explain this in simple terms.

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  • $\begingroup$ Have you considered the negative potential energy that the object "gains" as it moves from the zero point at infinite separation?? $\endgroup$ – DJohnM Aug 15 at 17:48
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Let's assume that the planet, mass $M$, and the object, mass $m$ are the only objects under consideration and there are no external forces acting.

The force that each object exerts on the other is the gravitational force $\dfrac{GMm}{R^2}$.
You will note that the forces gets smaller and smaller as $R$ gets bigger and bigger.

If both objects start at rest with a separation of $R_{\rm initial}$ then the gravitational potential energy of this system of 2 objects is $-\dfrac{GMm}{R_{\rm initial}}$ which assumes their gravitational potential energy to be zero when they are "infinitely" far apart.

If the objects are free to move they will accelerate towards one another under the influence of the gravitational force which is acting on each of them.
This means that both objects will gain kinetic energy and momentum.

Because there are no external forces acting the total momentum of the two objects must be zero because the objects started at rest.
Applying the conservation of momentum $Mv_{\rm M} + m v _{\rm m} =0$ where the $v'$s are the final velocities shows that both objects gain kinetic energy as they come closer together.
They gain this kinetic energy because the gravitational potential energy of the system (the two objects - not just one of the objects) has has changed from $-\dfrac{GMm}{R_{\rm initial }}$ to $-\dfrac{GMm}{R_{\rm final}}$.

Applying the law of conservation of energy $0+0-\dfrac{GMm}{R_{\rm initial }} = \frac 12 M v^2_{\rm M} + \frac 12 mv_{\rm m}^2-\dfrac{GMm}{R_{\rm final}}$

Now if one object is more massive than the other, ie $M\gg m$, then $v_{\rm M}\ll v_{\rm m}$ and to a good approximation the more massive object gains much,much less kinetic energy than the less massive object.

The conservation of energy equation can now be written as $\dfrac{GMm}{R_{\rm final}}-\dfrac{GMm}{R_{\rm initial }} = \frac 12 mv_{\rm m}^2$

The loss of gravitational potential energy is equal to the gain in kinetic energy of object mass $m$.

So there is no gain or loss of energy rather there is a change of energy from gravitational potential to kinetic.

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  • $\begingroup$ But that assumes the object in question already is in the gravitational field. So the gravitational force is acting on said object already, thus there is a gravitational potential. But if the object is outside the gravitational field, it first experiences no gravitational force.My question involves the object first being outside the gravitational field , then goes into a gravitational field $\endgroup$ – Bryan Foong Zhi Chuan Aug 15 at 16:10
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    $\begingroup$ @BryanFoongZhiChuan Looking at the formula for the force acting on two mutually attracting objects please explain how the gravitational field can be zero. $\endgroup$ – Farcher Aug 15 at 16:14
  • $\begingroup$ Ok, If the formula applies at all distances. Regardless of the distance between the two objects, then the concept of gravitational field is wrong. Because the definition of gravitational field is "the region of space surrounding a body in which another body experiences a force of gravitational attraction." $\endgroup$ – Bryan Foong Zhi Chuan Aug 15 at 16:22
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    $\begingroup$ @BryanFoongZhiChuan And that according to Newton's law of gravitation is everywhere. It might be that the force becomes so small that it cannot be measured or other forces are also acting which also makes it difficult to detect but it is still there. $\endgroup$ – Farcher Aug 15 at 16:29
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Potential energy doesn't really "come from" anywhere. One way to see is is to recognize that we can add any constant to our potential energy, but the physics doesn't change. So, I guess one could say potential energy "comes from" just being acted upon by a conservative force. However, what is more important are changes in potential energy.

A change in potential energy "comes from" work done by a conservative force. The familiar relation between the work $W$ done by a conservative force as the change in potential energy $\Delta U$ is given by $$W=-\Delta U$$ But keep in mind, in Newtonian mechanics these things are actually synonymous. In other words, the change in potential energy is essentially just a very useful tool in talking about the work done by a conservative force. You can either just talk about the work done by the conservative force the same as you would talk about the work done by a non-conservative force (as in the change in kinetic energy of an object is just the net work done on that object). Or, you can think about the change in potential energy, which I sure you know is a very useful concept.

I mean the planet is not gaining or losing any energy (I think). And, if I take the object have a gravitational field into account even though it is very weak, the planet should be accelerating slightly towards the object. This means the planet is also gaining energy.

Technically, the potential energy is a property of the object-planet system. For Newtonian gravity, this is given by $$U=-\frac{GmM}{r}$$ where $m$ and $M$ are the masses of the object and the planet respectively, and $r$ is the distance between them. You are right that both objects will be accelerating towards each other, and hence the potential energy of the system will be decreasing (as the kinetic energy of the system increases).

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In Newtonian physics, "Potential energy" in that case is the work of antiparallel and equal gravity forces $ \vec F = G \frac{mM}{r^2}\hat r$ that act on both the planet and the ball, which as Aaron Stevens mentioned should be: $$ U = - \frac{GmM}{r} $$

As Newtonian physics is formulated and since this work is applied with an opposite sign on both objects, the total energy of the system is zero. So conservation of energy still carries on. For the case of the ball, it's energy has just changed form as it also has for the planet.

Don't be tempted to think that the ball got extra energy. It's gravity field interacted with the planet's gravity field and this system-oriented energy only exists because of that.

To be honest, this is not a very complete answer and I don't even like writing it down. A complete answer would be given by field theory and how fields interact with each other - how the graviton carries energy through this interaction - but sadly I'm not the guy who can explain this. In short, you must have in mind that "fields", in any sense you want to think of them, carry energy on their own.

If you want a general relativity explanation, there is no energy involved. The ball has the exact same energy, $E = m_{ball} c^2$ and it's acceleration towards the planet is only because the curved geometry the planet creates around it with it's huge (relative to the ball) mass.

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  • $\begingroup$ Ehh, idk what general relativity is soo... I just need a intuitive way of understanding what is going on. $\endgroup$ – Bryan Foong Zhi Chuan Aug 15 at 15:52
  • $\begingroup$ If you need the Newtonian physics explanation, the extra energy as you're thinking of it, is stored in the system and has no meaning without thinking both of the objects. But as I said, this is not a complete answer. $\endgroup$ – Stamatis Tzanos Aug 15 at 15:54
  • $\begingroup$ General relativity in short says that masses bend space itself so the geometry objects move in, changes. This video is very popular and it may help: youtube.com/watch?v=MTY1Kje0yLg $\endgroup$ – Stamatis Tzanos Aug 15 at 15:55
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    $\begingroup$ Please note that the tags are for Newtonian mechanics, not GR. $\endgroup$ – Aaron Stevens Aug 15 at 16:22

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