# Is holding something in a gravitational field doing work ?

1. Aug 22, 2008

### bunburryist

Is holding something in a gravitational field "doing work"?

My son and I are on opposite sides of this question - if I am holding something in my hand in a gravitational field, am I doing work? My position is this - if I constantly accelerate a ball in space I am doing work. Since acceleration is equivalent to being in a gravitational field, and since holding a ball on earth is in a gravitational field, I am doing work when I hold it. My son's position is that since the thing I am holding is not moving (there is a net acceleration of zero) I am doing no work. Is it simply that we are disagreeing about what is the relevant frame of reference - mine being the earths gravitational field, his being my body?

2. Aug 22, 2008

### Razzor7

Re: Is holding something in a gravitational field "doing work"?

You're not doing any work on it.

3. Aug 22, 2008

### uart

Re: Is holding something in a gravitational field "doing work"?

Bunburryist, what about if you place the object on a table and allow it to resist the gravity instead of your hand? Is the table doing work on the object? If so then where is it's energy coming from.

4. Aug 22, 2008

### hdunham

Re: Is holding something in a gravitational field "doing work"?

Have your son hold a 5 lb weight straight out at arms length for 5 minutes, then ask him if he felt like he did any work.

Yes, it's a frame of reference difference. As your son holds the mass, have him imagine he is on the moon. The displacement is obvious.

The NET work is zero if you are just holding the ball still. Gravity is doing work to accelerate the ball toward Earth and you are doing equal and opposite work in holding it still, balancing out gravity. That can be seen by taking your son's frame of reference at which the displacement of the ball is zero, thus zero work.

One must not be confused between the work one does against gravity, the work gravity does, and the net work.

5. Aug 22, 2008

### Hootenanny

Staff Emeritus
Re: Is holding something in a gravitational field "doing work"?

Could you please quantify the work done by gravity on the 5 lb weight, assuming that the weight is held stationary by the boy?

6. Aug 22, 2008

### Topher925

Re: Is holding something in a gravitational field "doing work"?

hdunham, I disagree. No work is being done. Work is a force applied over a displacement. In equation form:

W = F x d

You are providing the force, but where is the displacement? There isn't any. If you dropped the weight, or picked it up, then work would be done. I understand your moon analogy, but the force is not being applied in the same direction as the displacement.

Last edited: Aug 22, 2008
7. Aug 22, 2008

### atyy

Re: Is holding something in a gravitational field "doing work"?

When you accelerate the ball in empty space, you are acting like a gravitational field for the ball - you are doing the work that the gravitational field would be doing if the ball was free falling in a gravitational field. So the analogy should be you in empty space equals gravitational field, but you the way you told it is you in empty space equals you in a gravitational field. If you pushed the ball hard in empty space, and some one pushed equally hard the other way, no one would be doing any work. So when you oppose the pull of gravity on the ball, neither you nor the field is doing any work either (that's my guess).

8. Aug 22, 2008

### hdunham

Re: Is holding something in a gravitational field "doing work"?

yeah, ok, i'm going to have to eat crow. it's still early. what i meant to say was forces, not work. net FORCE is zero, so as with displacement its zero.

any displacement due to rotation is tangential to the force so the dot product is zero.

sorry!

9. Aug 22, 2008

### atyy

Re: Is holding something in a gravitational field "doing work"?

But in general, work and energy are frame dependent.

10. Aug 22, 2008

### Renge Ishyo

Re: Is holding something in a gravitational field "doing work"?

As many have said you are doing no work *on the ball* and your son is right. What confuses people is that they "feel" as if they are doing work. Indeed, there IS work being done in this example, but it is work done *inside your body* to contract your muscles so that you can hold the ball (the muscles are constantly being tugged into position with the expense of ATP so that you can support the ball). But you are not transferring any of that work to the ball, because you have not moved the ball against gravity through a distance. In effect, all that work done by your muscles is just wasting away as heat, and your body "feels" this expense whether you move the ball against gravity (doing work on the ball) or keep the ball at the same height above the ground (doing no work on the ball).

11. Aug 22, 2008

### bunburryist

Re: Is holding something in a gravitational field "doing work"?

A balls natural state in a gravitational field would be to fall (follow a geodesic). That is, when it is in freefall relative to the earth it is "standing still" (following a geodesic) in spacetime. So anything we do to force the ball to deviate from it's "standing still" in spacetime would be an acceleration. So when we hold the ball from falling we are constantly accelerating it against it's natural state of falling. Does this make sense?

We can, I think, agree that work is being done as rockets on my feet accelerate me and the ball in my hand. Let's have me holding a scale with the ball on the scale, so that, as I accelerate, it registers "weight." Imagine that there is another person next to me in space who is accelerating along with me. This person might, mistakenly, believe that since the ball is "standing still" in his frame of reference, that there is no work being done. He might even conclude that we were merely in a gravitational field, that that is why the ball is "pressing" on the scale, and that no work was being done. (We could replace my body on earth with a rocket that pushes just hard enough to keep the ball suspended.) Of course, he is mistaken. I am doing work (work is being done) as the ball is accelerating constantly. So whether it is my legs or rockets on my feet, if the ball registers weight, it is undergoing an acceleration. If it is undergoing an acceleration, then there must be work being done.

Someone next to me on the earth is analogous to my fellow rocketeer. He is, along with the ball, me, etc. constantly accelerating relative to the earths gravitational field.

Does this make sense?

12. Aug 22, 2008

### Renge Ishyo

Re: Is holding something in a gravitational field "doing work"?

That's simply not the correct way to apply the concept of work. Ignoring complicated relativistic considerations you can look at the issue this way, doing work on a something raises the energy of that something regardless of what frame of reference you look at it in. It is clear that by holding the ball static at a certain height for 20 seconds and then dropping it a certain distance and by holding the ball at that same certain height for 5 minutes and then dropping it a certain distance that in both cases the ball would have the same kinetic energy as it hits the ground (even if some measured values differ based on the frame of reference, the *energy change* between start point and end point would be measured the same regardless).

If you believed that you were doing work on the ball by holding the ball in place you would predict that the ball held for 5 minutes would have much more kinetic energy when it was dropped through the same height as the ball held for 20 seconds. Experiment shows this not to be the case. In fact, the ball has the same final kinetic energy in both examples (and this is why, by definition, no "work" is being done on the ball while you hold it in place).

Last edited: Aug 22, 2008
13. Aug 22, 2008

### Staff: Mentor

Re: Is holding something in a gravitational field "doing work"?

I just wanted to briefly expand on the idea of energy and reference frames. Work is defined by f.d, but since d is frame-dependent f.d is naturally also frame dependent. Let's consider three cases:

a) Ball on a table in a uniform gravity field. As the ball sits on the table the real upwards normal force exerted by the table is counteracted by the real downward gravity force exerted by the earth. The ball does not accelerate, the displacement is 0 in this reference frame, so the work is zero and the KE is also 0.

b) Ball on a table in a uniformly accelerating rocket as considered from an inertial reference frame. As the ball sits on the table the real upwards normal force exerted by the table is not counteracted by any force. The ball does accelerate, the displacement is non-zero in this reference frame, so the work is non-zero and the KE increases.

c) Ball on a table in a uniformly accelerating rocket as considered from the rocket's accelerated reference frame. As the ball sits on the table the real upwards normal force exerted by the table is counteracted by the ficticious inertial force. The ball does not accelerate, the displacement is 0 in this reference frame, so the work done is 0 and the KE is also 0.

The usual convention on the surface of the earth is to use the reference frame as shown in a), but it is just a convention. There are no absolute answers to typical energy questions, it is all relative to the reference frame chosen.

14. Aug 22, 2008

### Staff: Mentor

Re: Is holding something in a gravitational field "doing work"?

No. That's not how work is defined. Work is force times distance. Where's the distance here?

What you are describing is analagous to a circular orbit. There is a constant acceleration in an orbit, yet no work being done because the force is perpendicular to the direction of motion.

In your example of equivalence between gravitational force and acceleration, you are right that you can look at the force from different frames. But in no frame is there motion, so in no frame is any work being done.

15. Aug 22, 2008

### Renge Ishyo

Re: Is holding something in a gravitational field "doing work"?

The *change* in energy is independent of the reference frame chosen (you just have to be careful that you are measuring the initial and final parameters from the same reference frame). For example, neglecting friction if I push a box measured initially to be at rest on earth with 5N through a distance of 1M, the kinetic energy it would gain would be 5J. I would say the initial kinetic energy here is 0J and the final is 5J. The change in kinetic energy is 5J-0J = 5J.

Now say you watch the same situation while you look down at me in space. Due to the spinning earth you see the box as moving with an initial kinetic energy of 250J in some direction. If I then on earth exert a force on the box of 5N through a distance of 1M along this same direction, you would measure the final kinetic energy of the box in space as 255J, not 5J. Yet the *difference* in energy that we calculate would be the same in both frames of reference. In space you would calculate, 255J-250J=5J.

But wait wait wait Renge you say (as Dalespam said above in his example c), let us suppose that I start measuring the box at rest at 0J in a reference frame and then my reference frame accelerates along with the ball so that as you add your 5J to the ball it always appears that the ball is at rest. I would calculate the change in energy to be 0J-0J=0J! Energy changes are relative!

No, they are not. The flaw in this case is that the observer didn't measure the final parameter and the initial parameter in the *same* chosen reference frame (the laws of physics require that you chose one reference frame and make both the final and initial measurements relative to that). The number calculated was off, because the observer didn't take into account that his reference frame had changed between the time he made his initial measurement and final measurement. His result was a calculation error brought about by an error in measurement, the box still gained the same amount of energy in all three examples. The laws of conservation of energy survived relativity in this way, and so did the son's explanation of work.

Last edited: Aug 22, 2008
16. Aug 22, 2008

### Andy Resnick

Re: Is holding something in a gravitational field "doing work"?

As Renge points out, there is no mechanical work being performed on the ball. However, your body (or whoever is holding the ball) is performing chemical work to maintain their arm in a nonequilibrium configuration. If you like, this is simply non-PV work.

17. Aug 22, 2008

### cesiumfrog

Re: Is holding something in a gravitational field "doing work"?

No.. in every *inertial* frame (that is, relative to any free-fall observer) work is being done to lift the ball's trajectory. (The energy is supplied from the work of the floor on the person, not from the person's metabolism. The effect originates from the curved space-time of GR: no single flat inertial frame can be applied to the entire system, and as has already been noted, energy is always frame-dependent.)

Last edited: Aug 22, 2008
18. Aug 22, 2008

### Staff: Mentor

Re: Is holding something in a gravitational field "doing work"?

I understand that, but in this example, there is no free-fall observer, nor any easy/convincing way to create one.

19. Aug 22, 2008

### LukeJD

Re: Is holding something in a gravitational field "doing work"?

I would agree that you are exerting a force, because you are accelerating the ball at 9.8m/s^2 in the opposite direction from the center of the gravitational field. However mathematically, Work is defined by Force x Displacement, there is no displacement. So no work is being done.

20. Aug 22, 2008

### atyy

Re: Is holding something in a gravitational field "doing work"?

Well, perhaps your fellow accelerating rocketeer should be equivalent to a person free falling in a gravitational field. So he is not analogous to someone standing still next to you on earth. While you are standing on earth, a free falling observer will see 'mistakenly' think that you are doing work on the ball. (I'm not sure the 'mistakenly' is really meaningful, just using it to parallel your description of your fellow accelerating rocketeer 'mistakenly' thinking no work is done.)

21. Aug 22, 2008

### Gear300

Re: Is holding something in a gravitational field "doing work"?

You're not doing work on the object, but your body is doing work on itself. Your muscles are constantly contracting, which is why you'll start to eventually feel strain.

22. Aug 22, 2008

### Staff: Mentor

Re: Is holding something in a gravitational field "doing work"?

Actually, even the change in energy is frame dependent. For example, consider an object of mass 1 at rest. The four-momentum p0 = (1,0,0,0) (in units where c=1), if you exert a force on it in the x direction accelerating it to .6 c the four-momentum is p1 = (1.25,.75,0,0) for a change in energy of .25. Now, if you boost p0 and p1 by .5 c you get p0' = (1.15,0.58,0,0) and p1' = (1.87,1.59,0,0) for a change in energy of .72. Just like time is relative so is energy.

I believe that the only time the change in energy is independent of the reference frame is when you are considering only inertial frames in Newtonian physics.

23. Aug 23, 2008

### bunburryist

Re: Is holding something in a gravitational field "doing work"?

Imagine that a rocket takes off, and inside the rocket is a ball on a scale. As the rocket accelerates, the ball “pushes down” on the scale – it has “weight.” The whole idea of equivalence, in this context, is that people in the rocket wouldn’t know (just from experiments they could do in the rocket) whether they were undergoing an acceleration or whether they were feeling the effects of a gravitational field.

So the rocket accelerates for a certain length of time, all of which the ball exhibits weight on the scale – evidence of the acceleration of the rocket. Suppose the rocket turns off its engine. The ball would no longer have “weight” – it would not push on the scale any more. No more work is being performed by the rocket.

Relative to the rocket, the ball has no more energy than before it took off. Someone on the rocket would conclude (mistakenly, as would someone holding a ball and believing he didn’t do any work) that “since the ball has no more energy now than when we started, we must not have done any work.” That is mistaken because it is using the wrong frame of reference relative to which it must be determined if the ball has more energy. It is the frame of reference from whence the rocket/ball started that the ball has more energy relative to. To people in that frame of reference, the ball and the rocket both have more energy than before it took off.

When we hold a ball, we are like the person on the rocket. We can hold it all day long, and even though it has weight (evidence of our doing work), it doesn’t seem to have any more energy than it did an hour or a day ago – within our reference frame. So we conclude that we have done no work. So who/what in our ball holding example is equivalent to the person where the rocket took off from, relative to which the ball does have more energy? It would be someone in free fall.

The longer you hold a ball in a gravitational field, the more energy it has relative to someone in freefall in the gravitational field. That is, the longer you hold the ball, the faster it is accelerating away from the free-falling person. But who is doing the accelerating – you, the earth, and the ball, or the free-faller? Although we, on the earth, learn to think of ourselves as being at rest, and falling things as being those which are accelerating, it isn’t the person in free-fall who is undergoing an acceleration. It is the ball – and you and the earth - that is accelerating. Remember, feeling the force of a gravitational field is equivalent to the acceleration. As you hold the ball, it is moving away from the free-faller faster and faster.

Imagine someone standing on a trap door next to you as you hold the ball. You can hold the ball all the live-long-day, and it will not have any more energy from your friends’ perspective, just as it wouldn’t from yours. Suppose now that the trap door opened and your friend was to start a free fall. The ball you are holding, you, and the earth would move away from/past him faster and faster – it would be accelerating. We tend to think that it is the falling person who is accelerating, but that is mistaken. It is we, the balls we hold, and the ground we are standing on that is accelerating. Remember – if the person falling had a ball on a scale it would not have weight – he would not be undergoing an acceleration. That is, no experiment he could do would show that he was either accelerating or was under the effect of a gravitational field. He might as well be very far removed from any massive body – he wouldn’t know the difference.

If the ball you are holding is undergoing an acceleration, then it should have more and more energy as time goes by, right? How can this energy be used by the free-falling person, showing that work has been done and that the ball (and you and the earth) have more energy? How can he measure the fact that you, the ball, and the earth have more energy with each passing moment? Suppose that your falling friend was connected to a very long comb shaped device with paddles sticking out, so that one passes near the ball every ten seconds. If someone on the comb device wants to measure the kinetic energy of the ball as it passes by, he adjusts the next paddle slightly so it hits the ball. As the comb device falls faster and faster, the ball is moving faster and faster in the other direction, so that with each passing paddle (if it was to be made to hit the ball) there would be a greater ability to “do work” (impose a force on the paddle).

So who is undergoing an acceleration? Is it the ball you are holding (along with you, the earth,etc.), or the comb-device with your free-falling friend? If there was a ball on a scale on the comb device there would be no evidence of acceleration or of feeling the force of a gravitational field (the ball would not “weigh” anything on their scale, but would, like the rest of the comb device, be in free-fall). But the ball you are holding on your scale continuously has weight – evidence that it is undergoing an acceleration.

There is a tendency to think that the gravitation field is imparting more and more energy to your falling friend (doing work) as he accelerates away from you. But it is you who is doing work. Actually, it is you, the ground you’re standing on – the whole earth – everything that stops you from following a geodesic – doing work – gaining more and more energy in your free-falling friends frame of reference. He is “standing still” in space-time – following a geodesic. So it’s not merely the ball that gets more and more energy – it is the whole structure of you, earth, etc. that is moving faster and faster relative to the free-falling friend, as all of it has more energy relative to the free-falling friend, and so is capable of doing more and more work in his frame of reference.

You and your friend were originally in the same frame of reference. When the trap door opened it wasn’t him that accelerated. Rather, it was him who stopped accelerating as he went into free-fall – his scale would stop registering weight of his ball on his scale showing that he was neither accelerating nor under the effect of a gravitational field. It would be equivalent to the rocket stopping it's acceleration. Imagine two rockets side by side with balls on scales. If one rocket stopped accelerating, it would appear to the one who was accelerating that he (the one who stopped accelerating) was falling. But it is really the one on the rocket that is still firing that is accelerating.

I know I’m going on and on about this, but I really am trying to make sense of it. I understand that within the context of the people holding the ball that the ball does not gain energy as it is being held. On the other hand, feeling the effects of a gravitational field is equivalent to acceleration, and it takes work to perform an acceleration. I think there is some subtle aspect of GR that I either don’t understand completely or I understand incorrectly.

Maybe I should have posted this in the general relativity section!

24. Aug 23, 2008

### Landru

Re: Is holding something in a gravitational field "doing work"?

Do the chemical reactions which occur in the arms muscles which work to resist the effect of gravity count as 'work'? Do the chemical reactions themselves constitute force * distance?

25. Aug 23, 2008

### Staff: Mentor

Re: Is holding something in a gravitational field "doing work"?

Yes, the myosin fibers burn ATP which pulls the thin filament. Therefore on a molecular level f.d>0 so there is work being done internally. The external work done may be 0, implying 0% efficiency. In my examples I used a table precisely to avoid this confusion.