Thought experiment on potential energy

[snoopies622]

If I lift an object, does its mass increase due to my investing it with potential energy? If I then drop it, does its mass increase further as it falls due to the acceleration? I don't see a contradiction here, but my intuition tells me that something is amiss.

 

[Doc Al]

The potential energy is not a property of the object, but of the "object + earth" system.

 

[snoopies622]

So does the mass of both the Earth and the object increase?

 

[Doc Al]

I'd say yes, but at the expense of whatever lifted the object. Since you lifted the object, your mass would decrease. (Some of your chemical potential energy has been transferred to the "object + Earth" system.)

Interesting question!

 

[snoopies622]

And after I drop it, does its mass again increase as it falls?

 

[LURCH]

No; as the object falls, kinetic energy increases while potential energy decreases. Total energy (and therefore total mass) of the system remains constant. As the object reaches the ground, potential energy goes to zero kinetic energy has reached maximum and the total energy input into the object by the work done by the person who originally lifted it is entirely converted to kinetic energy. This energy is released in the form of sound waves, shot ways in the ground, and some heat at the moment of impact. Total energy of the Earth-object system never changes.

 

[snoopies622]

Thank you both.

Here's a follow-up, if you're interested: after I lift the object and its mass increases, if I then elevate myself by the same amount, does the mass of the object change (from my perspective) back to what it was? I am imagining that if both the object and I then fall together back to the ground, for me it will appear that we are both at rest as the ground approaches us, and so our masses shouldn't be changing as we go, and a moment before we hit the ground the ratio of our masses should be the same as it was before I lifted the object in the first place.

 

[Doc Al]

after I lift the object and its mass increases...

Don't attribute the increased mass (when you lift the object) as being a property of the object; it's a property of the system (object + earth). Think of it as being stored in the gravitational field. Nothing happens to the object's mass, considered as a separate entity.

(Perhaps someone can give you a more technically accurate description from the viewpoint of general relativity.)

 

[snoopies622]

I guess what I don't understand is how mass-energy equivalence manifests itself. If an object is heated, if a spring is stretched, if a sphere is made to rotate faster, is there an increase in mass (equal to the invested energy divided by c squared) or does the equivalence relation only appear when mass is converted into energy and vice versa?

 

[LURCH]

Wait, clarrification, please:

Don't attribute the increased mass (when you lift the object) as being a property of the object; it's a property of the system (object + earth). Think of it as being stored in the gravitational field. Nothing happens to the object's mass, considered as a separate entity.

(Perhaps someone can give you a more technically accurate description from the viewpoint of general relativity.)

I think that, while the mass of the total system remains constant, the mass of the object does increase, while the work done by the person or thing that does the lifting causes them to lose the same amount of mass, keeping the overall mass of the object+person+Earth remains constant.

Somebody please verify.

 

[Doc Al]

Why would the mass of the object increase? Not sure what you mean.

 

[snoopies622]

Maybe this is the essence of my question: does potential energy effect an object's inertia? Will a mousetrap in the 'set' position weight more than one that is not set? (I realize that in a case like this the difference would not register on a household scale.)

 

[Schrodinger's Dog]

Maybe this is the essence of my question: does potential energy effect an object's inertia? Will a mousetrap in the 'set' position weight more than one that is not set? (I realize that in a case like this the difference would not register on a household scale.)

I think from a classical view point I've always looked at potential energy whether gravitational or stored in some way as a potential to do work, thus say an object that is moved from the bottom of a shelf to the top has no more energy or mass in it as such, but because of the way we describe energy it has more potential energy.

I don't think that in general relativity either, the object has more mass, but the person does in a naive sense. Nor do I think the object has more energy in the same way but the system does. I could be completely mistaken, but I think it's better to consider the whole system, and not to focus on the parts too closely, as that will lead you to make faulty assumptions. The system isn't an object and it isn't a person, it is object + person=.

The object won't weigh more or less unless it is moved spatially, and it won't have more weight when it is set necessarily either, although it may given certain conditions. But let's not confuse weight with mass. In physics particularly they are two very different measures.

 

[Doc Al]

Maybe this is the essence of my question: does potential energy effect an object's inertia? Will a mousetrap in the 'set' position weight more than one that is not set? (I realize that in a case like this the difference would not register on a household scale.)

I'd say yes.

 

[Ich]

Will a mousetrap in the 'set' position weight more than one that is not set?

The mousetrap itself, if you can separate it, yes.
If you include the one that set the mousetrap: The mass of this system will not change. He expended energy, put it into the mousetrap. So, if you can separate both afterwards, he will weigh less the same amount that the mousetrap weighs more. It's quite the same for a planet and a particle, but it's more difficult to separate the two.

 

[snoopies622]

I guess there's a difference of opinion out there.

 

[Doc Al]

I guess there's a difference of opinion out there.

About what?

 

[snoopies622]

About whether or not setting the mousetrap will change its weight. (See the last paragraph of Schrodinger's Dog's entry.)

 

[Doc Al]

About whether or not setting the mousetrap will change its weight. (See the last paragraph of Schrodinger's Dog's entry.)

That entry doesn't even mention the mousetrap example.

The difference between the raised object example and the mousetrap is that the mousetrap is self-contained: All the elastic potential energy added to the mousetrap is part of the mousetrap. With the raised object, the added potential energy belongs to the system of object + earth. (So asking about the increased mass of the object doesn't make sense to me.)

 

[snoopies622]

I took, "..and it won't have more weight when it is set..." to be about the mousetrap question. Was this a misinterpretation?

 

[Schrodinger's Dog]

That entry doesn't even mention the mousetrap example.

The difference between the raised object example and the mousetrap is that the mousetrap is self-contained: All the elastic potential energy added to the mousetrap is part of the mousetrap. With the raised object, the added potential energy belongs to the system of object + earth. (So asking about the increased mass of the object doesn't make sense to me.)

That was the idea I was trying to convey, but to extrapolate on the second bit rather more analy than is strictly necessary:

I took, "..and it won't have more weight when it is set..." to be about the mousetrap question. Was this a misinterpretation?

But let's say I was talking about a mouse trap and not an object as such: it will have a piece of cheese on it when set as well.

Does the mouse trap in the mouse trap game have more weight when set? What about if we only consider the basket and the stick at the end? I wonder? What about one way traps that have no moving parts as such or spring traps, or at least they are set only by the mouses motion, they work on the principle that they'll let a mouse in but not out.

Also there's a fine distinction here between mass and weight. Which are two very different concepts. In physics they are absolutely separate qualities. if something weighs 10kg it doesn't mean its mass is 10kg one is a measure of gravitational attraction on an object, the other is an absolute amount of matter. Unless you put a proviso or two on it they are not the same. On Earth they are roughly proportional, but how much mass does a Kg of iron have on the moon? How much does it weigh?

 

[snoopies622]

Well, I guess I'll take all that to mean that I was in error and that there is in fact no disagreement about the mousetrap matter. I'm still not perfectly clear about the lifted object, though. How can a change in mass be stored in a gravitational field (entry #8)? Does a gravitational field have weight/inertia? If so, how could such a thing be measured? (I understand that mass and weight are not the same thing.)

 

[Schrodinger's Dog]

Well, I guess I'll take all that to mean that I was in error and that there is in fact no disagreement about the mousetrap matter. I'm still not perfectly clear about the lifted object, though. How can a change in mass be stored in a gravitational field (entry #8)? Does a gravitational field have weight/inertia? If so, how could such a thing be measured? (I understand that mass and weight are not the same thing.)

Does an electromagnetic field have weight/inertia? Not sure what you are asking here?

In GR the weight is proportional to the mass of an object and it's attraction to another mass object, on Earth this would be denoted by f=ma, where weight/force(N)=mass(kg) \times\simeq9.8ms^{-2} so 1Kg in weight \simeq9.8N. What attracts two bodies is a curvature in space time. According to GR, with this in mind what are you asking? I don't think if you lift an object it is "stored" in the sense you mean, but I'm not sure what you mean.

 

[snoopies622]

Read the first eight entries of this thread.

 

[LURCH]

Why would the mass of the object increase? Not sure what you mean.

Becuase the mass of the person who lifted it decreased. If the person lifting the object expends energy and looses mass, and the overall mass of the total system remains the same, doesn't the Earth+person part of the system have to gain mass? I should think that the energy lost from the person has been converted to potential energy in the object, which means energy has been transferred from the person to the object.

Though, now that I hear mysaelf say it, I must admit that it would be a more accurate statement is to say that the potential energy between the Earth and the object has increased. After all, when the persons he ended his lifting the object away from the Earth, is equally true that the persons feet and pushing the Earth away from the object. So, I suppose the mass of both the Earth and the object must increase when the mass of the person decreases. Otherwise, how does the overall mass of the system remain constant?

 

[MeJennifer]

Otherwise, how does the overall mass of the system remain constant?

It does not in GR.

If you have a bunch of masses then the combined mass depends on the individual distances between them.

You could make a whole "guess the mass" quiz by stacking 50 Lego blocks in various ways.

 

[Ich]

If you have a bunch of masses then the combined mass depends on the individual distances between them.

No, the total mass is conserved. Only if you change (here:reduce) the distances and let the gained kinetic/thermal energy get away, the mass will decrease. Not from the change in potential energy, which is always balanced by an increased kinetic energy, but from the real energy loss you admitted.

 

[Chrisc]

Snoopies662
It can be confusing because mass is a term used to describe the energy of matter.
If you keep in mind that gravitational mass is a measure of the energy exerted by the Earth and the body you lift on each other and that it is decreased as the distance between them increases, then it is easier to understand that the gravitational mass (both) decreases after you lift the body. But as the gravitational attraction of the Earth will still bring it back down, it has greater potential energy, that if released with become greater kinetic energy during its fall.

The inertial mass of the body is a measure of the energy outside of a gravitational context. It is the total energy required to set the body in motion from rest or to bring it to rest from motion. So if after you lift it you bring it to rest in its new position with respect to yourself, its inertial mass does not change.

The second part of your question is answered by the first. While the body is falling its inertial mass is increased with respect to you or anything not falling with it.

Will a set mouse trap have greater mass than an unset trap?
That's a great question. It definitely will. It is much more difficult to explain the increase of mass in this situation, but its the essence of the question Einstein asked that resulted in his famous equation E=mc2. Einstein reasoned that a radiating body loses mass. In other words the more energy a body possesses the greater its mass. The more it loses, the less its mass.

 

[snoopies622]

Is not one of the premises of GR that gravitational mass and inertial mass are the same thing?

 

[JesseM]

I guess what I don't understand is how mass-energy equivalence manifests itself. If an object is heated, if a spring is stretched, if a sphere is made to rotate faster, is there an increase in mass (equal to the invested energy divided by c squared) or does the equivalence relation only appear when mass is converted into energy and vice versa?

For any bound system made of multiple particles (like a collection of atoms in a box), the total inertial mass of the system in the rest frame of the total system (the rest frame of the box), which tells you how much force is needed to accelerate it at a given rate from a state of rest, will be the sum of the rest masses, kinetic energies, and potential energies of all the particles (including the particles that make up the wall of the box). So for example a heated brick is harder to accelerate than a cold one because of the increased kinetic energy of its particles (which means it will also weigh more on a balance scale), and the inertial mass of an atom is actually less than the inertial masses of the protons, neutrons and electrons that make it up if they are measured when free, because their potential energy is smaller when all brought together in an atom.

 

[snoopies622]

By "potential energies of all the particles", are you including gravitational potential energy?

 

[JesseM]

By "potential energies of all the particles", are you including gravitational potential energy?

That's a good question. I'd guess it would contribute to the "inertial mass" of a bound system in the usual way for an asymptotically flat spacetime (where all the mass is in the bound system, so as you get far from the mass spacetime gets arbitrarily close to the flat spacetime of special relativity), and if you look at pervect's comment #13 and pmb_phy's comment #14 on this thread, they seem to confirm this. But as pervect's link points out, talking about gravitational energy in more general circumstances (not confined to asymptotically flat spacetime) can be tricky.

 

[snoopies622]

Thanks, JesseM; that's quite a thread. It looks like there's more to this than I thought. Maybe after another year of studying GR (I haven't learned killing vectors yet) I'll finally understand the answer to my question.

Thanks to everyone who participated in this thread!

 

[Schrodinger's Dog]

Read the first eight entries of this thread.

Still doesn't make it any clearer.

But Jesse's post appears to answer your question anyway. I was just wondering exactly what you meant, because it seemed there were a few things flying around, that might confuse the issue.

 

[snoopies622]

Well, I guess the questions were: does energy always have a mass-like manifestation, and if so, where is it in the case of gravitational potential energy? I thought it might be in the mass of the objects experiencing gravitational "pull", but entry #8 suggested that it was in the field itself, which I didn't understand, since I don't know how such mass/inertia could be perceived or measured.