# Work done by balanced force?

Thanks for your answer to my kinetic energy question.I still would like to know how the ball initially at rest can "jump" to some velocity,it is a similar problem to Zeno's paradox but I still cannot see how the ball can go from no velocity to some velocity.PS.is a "free particle" one that is in equilibrium? if so the ball in question is not a free particle and it's it's kinetic energy that I was referring to.

Let me answer this by not actually answering your question.

I have an issue that bugs me, that is connected with it...

Suppose I lift a box. Let us consider that the box is rigid, totally.

I lift the box. I am separating it from the earth. When one separates attracting bodies, the one can think of it as giving the system some amount of potential energy.

But there is a subtle issue here. Note that I cannot give a system potential energy unless I HAD PREVIOUSLY given it some kinetic energy: you cannot separate two bodies without giving them a certain speed! (at least not in the world of pots and pans). You simply cannot....

This is the same as to say that the Force that is lifting the box up (separating it from the earth), the same force that is equalized by the so called force of gravity, does NO WORK unless there HAD been another UNBALANCED force that has given this body a small kick of an acceleration. Now the body has a constant speed and the force that is lifting the box, the same force which is being balanced by the gravity, SUDDENLY is able to do some work!

I find it kooky, bizarre. One might say that there is nothing strange about it...bla bla.. but it sort of is strange...

It seems to give rise to the idea, that one needs to give a system an extra kick to let other forces do some work...
Like in order to light a match, one has to invest a tiny amount of energy to get this process going...
Like it is with chemical reactions (which also involve physical work).
Like it is with proton fusion in stars: a slight amount of energy is needed in order to get a process going that releases MORE energy..
There are some similarities between those example, except that in the case of lifting the box, there is no energy actually released.

This idea also fits with the equation describing physical POWER:

POWER = FORCE x SPEED

If the speed of the body is 0, there is no power, no work done.

So if I wanted to hide behind the math, I would say: shut up and don't complain: you see the equation: it does give 0 work since it is zero power. So everything is ok, mathematics explains everything.. huh...

But there is something bizarre about it. A body's speed somehow seems to contribute to a balanced force's ability to do work.
On the other hand it makes sort of sense if we consider that one cannot separate the two bodies unless you give them at least a little speed.

## Answers and Replies

DaveC426913
Gold Member
Let us consider that the box is rigid, totally.
Note that perfectly rigid objects are forbidden by relativity. The moment you break physics and allow them, you can get all sorts of paradoxen, such as superluminal communication using a very long stick.

Note that perfectly rigid objects are forbidden by relativity. The moment you break physics and allow them, you can get all sorts of paradoxen, such as superluminal communication using a very long stick.

Perfectly rigid bodies are "forbidden" even by classical mechanics! What is your point?

PeterDonis
Mentor
2020 Award
Perfectly rigid bodies are "forbidden" even by classical mechanics! What is your point?

His point is that, in classical mechanics, there is no limit in principle to how rigid an object can be. In relativity, there is: one way of stating the limit is that the sound speed inside the object, which is the speed at which forces exerted on one part of the object are propagated to other parts, cannot exceed the speed of light. That makes a big difference.

His point is that, in classical mechanics, there is no limit in principle to how rigid an object can be. In relativity, there is: one way of stating the limit is that the sound speed inside the object, which is the speed at which forces exerted on one part of the object are propagated to other parts, cannot exceed the speed of light. That makes a big difference.

I still fail to understand how it is relevant to the work and power issue I brought up.

PeterDonis
Mentor
2020 Award
I still fail to understand how it is relevant to the work and power issue I brought up.

What role does your assumption that the body is "rigid, totally" play in your scenario? Do you need that assumption? Would it be sufficient for the body to just be rigid within the limits allowed by relativity?

PeterDonis
Mentor
2020 Award
the Force that is lifting the box up (separating it from the earth), the same force that is equalized by the so called force of gravity, does NO WORK unless there HAD been another UNBALANCED force that has given this body a small kick of an acceleration. Now the body has a constant speed and the force that is lifting the box, the same force which is being balanced by the gravity, SUDDENLY is able to do some work!

No. The work isn't being done by the force that just balances gravity; it's being done by the extra force that isn't balanced by gravity. Take away that extra force and the body stops moving, and no work is done.

What role does your assumption that the body is "rigid, totally" play in your scenario? Do you need that assumption? Would it be sufficient for the body to just be rigid within the limits allowed by relativity?

If the body is not rigid, then extra work is done by the stretching it or compressing it while lifting.

No. The work isn't being done by the force that just balances gravity; it's being done by the extra force that isn't balanced by gravity. Take away that extra force and the body stops moving, and no work is done.

That is not true. Unbalanced forces cause acceleration and acceleration only. If one pushes a box at a constant speed, then the net force is ZERO!
But the box gets hotter: that means that it gains energy and that means that work is done.
And that work is heating up the box by pushing!
The force that is heating it up, is most certainly the one that is balanced by the frictional force.
The frictional force is converting all of the energy given to the box into heat form BUT the conversion from one type of energy into another is not work.
But at the same time it gives energy to the floor, which also heats up. That means that the frictional force heats up the floor at the expense of the box. That means that the frictional force does negative work on the box.

Consider an uniform electric field pushing a charged box so that the box moves at the constant speed. There is a definite amount of energy in that field. The electric force is balanced by the frictional force! But the box heats up. You can be sure that all of the energy is directly coming from the electric field! The force mediating the energy to the box is most certainly the electric force which is balanced by the friction.

The fact that a force is balanced by another force does not mean that it cannot do work!

PeterDonis
Mentor
2020 Award
If the body is not rigid, then extra work is done by the stretching it or compressing it while lifting.

Yes, in principle this is true. But is it possible to have a box that is rigid enough, within the limits imposed by relativity, that this extra work is negligible in comparison to the work you are interested in? If so, why not adopt that as your assumption?

If one pushes a box at a constant speed, then the net force is ZERO!

If one has to push the box for it to move at a constant speed, then one is not in an inertial frame. Newtonian mechanics is somewhat inconsistent on this, because gravity is considered to be a force, yet a body moving solely under gravity is in free fall, feeling no force. GR fixes this issue by not considering gravity a force (I note that in your first post you put "force" in quotes when referring to gravity, indicating that you are aware of this issue). In a non-inertial frame like the one you are using, yes, a force can do work even though it is not causing acceleration in that frame. (I apologize for mis-stating this in my previous post.) But this concept of acceleration is more properly called "coordinate acceleration" in GR, because you can change it just by changing the coordinates you use.

Also, relativity is more consistent than Newtonian mechanics about recognizing that energy is coordinate-dependent; an object can have constant energy in one coordinate chart while gaining energy in another. So, for example, if your box is just sitting on the Earth's surface, not changing height, then its energy is constant according to the coordinates you are using (fixed with respect to the Earth), but not according to the coordinates of a local inertial frame centered on the box at a given instant of time (since in that frame, the box is accelerating upward at 9.8 m/s^2, and therefore is gaining kinetic energy).

If the box gains height (relative to the Earth), then in the local inertial frame I just described, it simply gains a little more kinetic energy than it would if it were not gaining height. In the frame fixed to the Earth, it gains potential energy--the small kinetic energy it gains while it moves upward is lost again when it stops moving. Since the latter frame is a non-inertial frame, the box can gain energy even though, on net, the forces on it are balanced (there is a small imbalance when the box starts moving, and a small imbalance in the opposite direction when it stops moving, but those effects basically cancel out--I was incorrect in my previous post in attributing the work done to those imbalances).

the box gets hotter: that means that it gains energy and that means that work is done.

The energy the box gains (in coordinates fixed with respect to the Earth) is gravitational potential energy. The box's height can change without its temperature changing at all.

The force that is heating it up, is most certainly the one that is balanced by the frictional force.

What frictional force? If you mean air resistance, then do the experiment in a vacuum. If you mean gravity, gravity is not a frictional force; in fact, in GR it is not a force at all. See above. I think the rest of your post is all based on a similar misconception, so I won't comment further on it.

PeterDonis
Mentor
2020 Award
This is the same as to say that the Force that is lifting the box up (separating it from the earth), the same force that is equalized by the so called force of gravity, does NO WORK unless there HAD been another UNBALANCED force that has given this body a small kick of an acceleration. Now the body has a constant speed and the force that is lifting the box, the same force which is being balanced by the gravity, SUDDENLY is able to do some work!

Following on from my previous post, since energy is coordinate-dependent, so is work. In coordinates fixed to the Earth, the force only does work when the box changes height; this is a non-inertial frame, so that is ok. In coordinates of a local inertial frame, the force always does work (how much work it does changes slightly when the box changes height, compared to when it is at constant height).

DaveC426913
Gold Member
I still fail to understand how it is relevant to the work and power issue I brought up.
You may find that the whole puzzle you describe might vanish. You can impart kinetic energy on only part of a system, without actually moving the whole system.

I'm not stating it outright, I'm simply suggesting that you remove the unrealistic 'perfectly rigid' constraint, and try your thought experiment again.

Newtonian mechanics is somewhat inconsistent on this, because gravity is considered to be a force, yet a body moving solely under gravity is in free fall, feeling no force.

Is that true? It seems to me that in Newtonian mechanics, a body in free fall above the earth feels a force of gravity, which is equal and opposite to the force felt by the earth. They are an action reaction force pair.

GR fixes this issue by not considering gravity a force

I don’t think that was what needed fixing. Einstein’s motivation was to extend non-inertial frames to the special relativity inertial frames idea that the laws of physics should be the same in any frame of reference. Choosing the earth to be our frame of reference makes gravity a fictitious force in the same way that choosing a merry-go-round as a frame of reference creates fictitious forces. It that case there is a situation that can be fixed, if we choose to view it that way.

PeterDonis
Mentor
2020 Award
It seems to me that in Newtonian mechanics, a body in free fall above the earth feels a force of gravity, which is equal and opposite to the force felt by the earth. They are an action reaction force pair.

The force of gravity acts on the body, according to Newtonian mechanics; but the body does not feel that force, in the sense that if you attach a strain gauge or an accelerometer to the body, it will read zero.

For the action-reaction part, see further comments below.

Einstein’s motivation was to extend non-inertial frames to the special relativity inertial frames idea that the laws of physics should be the same in any frame of reference.

Yes.

Choosing the earth to be our frame of reference makes gravity a fictitious force in the same way that choosing a merry-go-round as a frame of reference creates fictitious forces.

Yes. And in both cases we can also choose an inertial frame in which the fictitious forces vanish. In the case of the Earth, the inertial frame will be one in which an object free-falling downward with respect to the Earth is at rest. (Note that this will only be a local inertial frame, covering a small patch of spacetime.) In the case of a real force, once that is felt (and can be measured with a strain gauge or an accelerometer), you can't make it vanish by choosing a different frame.

But the definition of "force" I used in the above paragraph is the GR definition, not the Newtonian definition. Under the Newtonian definition, centrifugal force (the force observed in the merry-go-round frame) is a fictitious force, but gravity is not. So Newtonian mechanics considers a frame centered on the Earth to be an inertial frame (whereas in GR it isn't). Newtonian mechanics has to do this, because, as you noted, it applies the Third Law to gravity (whereas it doesn't apply the Third Law to fictitious forces like centrifugal force--the merry-go-round frame is a non-inertial frame in Newtonian mechanics).

What all this does in Newtonian mechanics is to leave the concept of "force" without a really clear physical meaning. A "real" force isn't picked out by the fact that it can be directly measured (with a strain gauge or an accelerometer), because gravity can't be. A "fictitious" force isn't picked out by the fact that it can't be directly measured, because gravity isn't considered a fictitious force.

Thanks Peter. Just as an aside, I think that the case of the Newtonian view of the merry-go-round frame having fictitious forces is a bit strange as well, because the way that we do that is to simply discard (and forget about) one part of what was previously part of an action reaction pair.

It seems we could come up with a better method for doing that (changing frames without messing up our accounting of forces). Or is that what GR is all about?

Following on from my previous post, since energy is coordinate-dependent, so is work. In coordinates fixed to the Earth, the force only does work when the box changes height; this is a non-inertial frame, so that is ok. In coordinates of a local inertial frame, the force always does work (how much work it does changes slightly when the box changes height, compared to when it is at constant height).

When I was talking about a balanced force doing work, I was bringing more examples than just lifting a box. If you read carefully, you will note that I am also talking about PUSHING a box at a constant speed and also, letting the box being pushed by an uniform electric field.

• ulianjay
You may find that the whole puzzle you describe might vanish. You can impart kinetic energy on only part of a system, without actually moving the whole system.

I'm not stating it outright, I'm simply suggesting that you remove the unrealistic 'perfectly rigid' constraint, and try your thought experiment again.

But it still doesn't help, because sooner or later the whole box catches up with it... having a speed

The energy the box gains (in coordinates fixed with respect to the Earth) is gravitational potential energy. The box's height can change without its temperature changing at all.

What frictional force? If you mean air resistance, then do the experiment in a vacuum. If you mean gravity, gravity is not a frictional force; in fact, in GR it is not a force at all. See above. I think the rest of your post is all based on a similar misconception, so I won't comment further on it.

Read carefully, what I posted. I was bringing an example of PUSHING a box on the ground! I even made the electric field push the box.

But since we are at it, I am still not sure what you are saying: how does general relativity solve this issue?

Consider now a CHARGED box in space, being against the platform that is oppositely charged.
Now someone is starting to separate the box from the platform. Gravity is negligible!
And still we get the same problem: you cannot separate the box and the platform unless you give the box a little speed!
You can lift the box so that it is moving uniformly and it order for that balanced force (which is now balanced by the electric force!) to do work, a small kick of speed has to be given to that box! So seemingly out of nothing this balanced force, again can suddenly do work by giving the system electric potential energy.

PeterDonis
Mentor
2020 Award
I think that the case of the Newtonian view of the merry-go-round frame having fictitious forces is a bit strange as well, because the way that we do that is to simply discard (and forget about) one part of what was previously part of an action reaction pair.

I'm not sure I understand. Fictitious forces only appear in non-inertial frames, and the Third Law is never applied to them. Real forces that existed in an inertial frame still exist in a non-inertial frame, so what is being discarded?

It seems we could come up with a better method for doing that (changing frames without messing up our accounting of forces). Or is that what GR is all about?

GR formulates the laws of physics in such a way that they are valid in any frame, inertial or not. This includes laws that have forces appearing in them.

PeterDonis
Mentor
2020 Award
I was bringing an example of PUSHING a box on the ground! I even made the electric field push the box.

You didn't make it very clear that you were changing examples in the middle of the discussion. That's generally not a good idea, particularly when we haven't even gotten clear about the original example you gave (lifting a box in a gravitational field). Also, one of your other examples adds a new element, a dissipative force (friction), which complicates the analysis.

That said, I'm still not sure what exactly the problem is. You say:

you cannot separate the box and the platform unless you give the box a little speed!

This is true, but I don't see why it's a problem. Then you say:

You can lift the box so that it is moving uniformly

Yes, it can be moving uniformly, apart from a brief initial acceleration. The brief initial acceleration doesn't have to do any appreciable work by itself; but it is still necessary for there to be relative motion, which means a brief period of unbalanced force is necessary. See below.

seemingly out of nothing this balanced force, again can suddenly do work by giving the system electric potential energy.

This is just because you can't do work at all unless there is relative motion. And to get relative motion, there must be an unbalanced force for at least a brief period.

Now to go back to this question:

how does general relativity solve this issue?

I'm no longer sure your issue has anything to do with relativity. I thought it did originally because your original example involved gravity, and it seemed like you were confused about how gravitational potential energy works. But now it just seems like you're confused about how work in general works. You could just as easily ask your question in the framework of Newtonian mechanics, and the answer would be the same (it's what I gave above).

You didn't make it very clear that you were changing examples in the middle of the discussion. That's generally not a good idea, particularly when we haven't even gotten clear about the original example you gave (lifting a box in a gravitational field). Also, one of your other examples adds a new element, a dissipative force (friction), which complicates the analysis.

That said, I'm still not sure what exactly the problem is. You say:

This is true, but I don't see why it's a problem. Then you say:

Yes, it can be moving uniformly, apart from a brief initial acceleration. The brief initial acceleration doesn't have to do any appreciable work by itself; but it is still necessary for there to be relative motion, which means a brief period of unbalanced force is necessary. See below.

This is just because you can't do work at all unless there is relative motion. And to get relative motion, there must be an unbalanced force for at least a brief period.

Now to go back to this question:

I'm no longer sure your issue has anything to do with relativity. I thought it did originally because your original example involved gravity, and it seemed like you were confused about how gravitational potential energy works. But now it just seems like you're confused about how work in general works. You could just as easily ask your question in the framework of Newtonian mechanics, and the answer would be the same (it's what I gave above).

I brought the friction in because I wanted to demonstrate that a balanced force most certainly can do work. Friction is a force that very often balances other forces...