# Thought experiment on potential energy

## Main Question or Discussion Point

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.

Related Special and General Relativity News on Phys.org
Doc Al
Mentor
The potential energy is not a property of the object, but of the "object + earth" system.

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

Doc Al
Mentor
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!

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.

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
Mentor
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.)

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

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.

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

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.)

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

Last edited:
Doc Al
Mentor
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 seperate 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.

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

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

Doc Al
Mentor
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.)

I took, "..and it wont have more weight when it is set..." to be about the mousetrap question. Was this a misinterpretation?

Last edited:
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 wont have more weight when it is set..." to be about the mousetrap question. Was this a misinterpretation?
But lets 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?

Last edited:
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.)

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 $\simeq$9.8N. 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.

Last edited: