Potential Energy and Conservation of Energy principle

[Thomas2054]

In light of the Conservation of Energy principle, what would happen to the potential energy of the Earth's moon, or any satellite of Earth, if the mass of the Earth were instantaneously reduced by half? That is, is potential energy conserved, is it transferred to another form of energy or is it simply "lost" (not destroyed, but removed from consideration) by virtue of the fact that the potential energy reference body, i.e., the Earth, has been changed? In this ridiculous, hypothetical situation I would expect that the kinetic energy of the moon would cause it to move off is some curved path since the gravitational influence of the Earth would have been reduced but not eliminated.

Thomas

 

[jtbell]

In light of the Conservation of Energy principle, what would happen to the potential energy of the Earth's moon, or any satellite of Earth, if the mass of the Earth were instantaneously reduced by half?

That depends on how the mass of the Earth is reduced by half. You can't simply say the mass magically disappears, without something else appearing in its place or elsewhere. That would be an unphysical assumption, and you wouldn't be able to make sensible conclusions from it in the context of the physics that we know about.

 

[Brin]

I agree with jtbell, and I don't want to even assume that people could start doing something like that - however, isn't this how Einstein came about theorizing about the "speed of gravity?" With the thought experiment: "What would happen if the sun just magically disappeared? How long until the Earth felt the effect? Would we still see the light from the sun, even though we're no longer in an orbit?"

I don't actually know if that's true, but I heard it from this show called http://www.pbs.org/wgbh/nova/elegant/"

 

[Thomas2054]

jtbell,

I know it is ridiculous, but I was doing the "thought experiment", Brin suggests, as to what would happen if...

So, what happens to the potential energy of the moon or satellite if the Earth mass is reduced in half suddenly. It seems to me that the question can be addressed purely for fun.

Thomas

 

[Brin]

I assume that in order for the PE to be halved, that the moon must either move a certain distance closer to the planet (smaller orbit), or its mass is affected somehow, or heck, Earth might instantly shrink itself.

Should the moon shrink instantly - its effect on the planet would definitely be less (in terms of tides, that's the only effect I am aware that the moon would have) - also it would appear smaller (I am assuming that mass and size in this case are related).

If it moved closer, we'd notice an even greater effect on on tides - and I am really curious as to whether or not the two gravitational fields would pull each other together at this point (if the distance was halved for example).

The other possibility that the PE of the moon could be halved by is that the Earth's gravity is made lesser. In this situation I'd assume we'd have even greater problems :P

 

[wysard]

In this wonderfully fanciful concept I'll take a stab.

1) Earth suddenly looses half it's mass.. *poof*

2) The attraction between the moon and the Earth halves.
Since F = k * (M1 * M2 / d) and you just halved on of the Masses.

3) Magically halving the binding force between them will result in the moon suddenly having way to much kinetic energy for it's position and will go from object in orbit to object doing a fly by. The moon will move tangenetally away from the Earth until either it is recaptured at roughly twice it's old orbital distance, or if there is to much energy becoming a free body.

4) The Earth, now suddenly 1/3 the mass of the moon will definitely be forced into a much greater co-rotation with the moon.

5) At suddenly half it's mass, but with the same velocity relative to the sun would the Earth not try to change it's orbit too? Perhaps out by Mars? Or just get shredded in the tidal forces of adjusting to it's new status as this seasons winner of The Biggest Loser - Solar System Edition?

 

[Dale]

I think there are two ways you could approach this. First would be to say the outer half of the mass is somehow converted into radiant energy all going outward in one big sphere. You would probably need to address this with GR where potential energy is notoriously problematic anyway.

The second way would be to take the Newtonian approach and just haul half of the Earth away to infinity. This would require a considerable amount of energy, but if you did the analysis I am sure that you would find that the total energy was conserved. Since the PE of the moon would be less, my initial guess is that would imply that some of it went into moving the chunk. But I would have to work the problem to be sure.

this seasons winner of The Biggest Loser - Solar System Edition?

Hehe!

 

[jtbell]

So, what happens to the potential energy of the moon or satellite if the Earth mass is reduced in half suddenly.

Well, if you have no problem with half the Earth's mass simply disappearing, you should have no problem with half the potential energy of the system simply disappearing.

As a first approximation, ignore the effect of the Sun, and assume the moon simply orbits around the center of the Earth. (Actually, the Earth and Moon both orbit around their center of mass.)

The moon will still have the same kinetic energy immediately after this event. If the total of kinetic plus potential energy remains less than zero, then the moon remains gravitationally bound to the earth, but is now in an elliptical orbit, with its perigee at the location where it was when the mass-disappearance took place. If the total energy becomes greater than zero, the moon flies off to infinity on a hyperbolic path. I haven't worked out the numbers myself, but if you know the formulas for kinetic energy and gravitational potential energy, it should be easy enough for you to calculate the result.

Taking the Sun into account should make things more interesting, because as I recall the Moon is more tightly bound to the Sun than to the Earth, so that the Moon doesn't orbit the Earth in the same way that one of our artificial satellites does. It's more like the Moon and Earth are dancing together as partners around the Sun. So the question is whether they remain partners, or start dancing independently of each other (although still in the same general orbit around the Sun).