I Is it accurate to say work is motion against an opposing force?

[A.T.]

Isn't "work" equal to the integral

$W =\int \vec F \cdot \, d\vec S$

Where's the "resistance"?

He probably meant reaction(?).

OK, where's the "reaction"?

 

[sophiecentaur]

OK, where's the "reaction"?

The friction force and any force due to acceleration. Your equation contains an F; that's the one.

 

[A.T.]

Your equation contains an F; that's the one.

F is the force doing the work. Where is the "reaction" to F in the equation?

 

[sophiecentaur]

F is the force doing the work. Where is the "reaction" to F in the equation?

I have already asked you for a reactionless force (a concrete practical example). Nothing so far.

 

[A.T.]

I have already asked you for a reactionless force (a concrete practical example). Nothing so far.

See post #47

 

[sophiecentaur]

????

For a practical example, consider a rocket ship with uniform acceleration. There is a "gravitational" force which can do work on falling objects.

Sorry but I just don't get that.
1. A gravitational force is experienced by an object on the floor of the ship and the object gains KE.
2. Alternatively, the work done in expelling the ejecta is the force on ship and/or ejecta times distance moved.
3. Or you could be saying that the work done with the engine is changing the relative velocity of a free object before it hits the floor. But work is still being done on the ship in a conventional way as it accelerates to 'overtake' the free object.
How are those three changes in energy not examples of N3?

There is KE 'in abeyance' as the object and ship have relative motion change but that applies to any pair of objects in the Universe that are under different forces. So everything is involved in some sort of work translation with respect to everything else? I see how that's a possible view but is it of any use? You may be able to convince me more. Try.

 

[jbriggs444]

????

Sorry but I just don't get that.
1. A gravitational force is experienced by an object on the floor of the ship and the object gains KE.

No. In the frame of the ship, the object on the floor is just sitting there. It gains no KE.

It is subject to an interaction force from the floor of the ship. But that force is applied over zero displacement, so it does zero work. It is subject to the fictitious force from the acceleration of the ship. But that force is also applied over zero displacement so it also does zero work. KE is unchanged. Which is correct.

2. Alternatively, the work done in expelling the ejecta is the force on ship and/or ejecta times distance moved.

The [nearly] instantaneous action of expelling a particular bit of ejecta does the same amount of work in the accelerating frame as it does in the inertial frame. There is no time for acceleration to make a significant difference.

However, a bit of ejecta seen by the accelerating rocket will be accelerating away, gaining kinetic energy as it does so. This is explained by the fictious force from the acceleration of the frame.

3. Or you could be saying that the work done with the engine is changing the relative velocity of a free object before it hits the floor. But work is still being done on the ship in a conventional way as it accelerates to 'overtake' the free object.

This one.

A falling object is accelerating as viewed by an observer standing on the rocket floor. It is gaining kinetic energy. This is explained by the work done by the fictious force arising from the acceleration of the reference frame.

How are those three changes in energy not examples of N3?

The first one (object on the floor) did not involve a change in kinetic energy at all. If you think it did, you are frame jumping.

The second one (expelling the ejecta) is an ordinary interaction force. Of course it involves N3.

The second one (watching the ejecta after ejection) does not involve a N3 interaction. The gain in KE is from a fictitious force. Fictitious forces lack a third law partner.

The third one (watching a falling object) does not involve a N3 interaction. The gain in KE is from a fictitious force. Fictitious forces lack a third law partner.

There is KE 'in abeyance' as the object and ship have relative motion change but that applies to any pair of objects in the Universe that are under different forces. So everything is involved in some sort of work translation with respect to everything else? I see how that's a possible view but is it of any use? You may be able to convince me more. Try.

The only notion of "KE in abeyance" that I know of is potential energy. Some accelerating frames allow a potential to be defined. A uniformly accelerating frame is one example. A rotating frame is another.

In a uniformly accelerating frame, the potential is the familiar $\text{PE} = mgh$.

In a rotating frame, I believe that the centrifugal potential is $\text{PE} = -\frac{1}{2}m\omega^2r^2$

 

[sophiecentaur]

No. In the frame of the ship, the object on the floor is just sitting there. It gains no KE.

I like the description 'frame jumping'. It needs good discipline to avoid that unconsciously. it's as if the quantity 'Work' should only be restricted to inertial frames or else the KE in abeyance needs to be taken into account. Work is done to accelerate the ejecta of the ship and there you have an N3 pair and those GJ contained in the propellant are responsible for the Work.

Here's a thing. The object is suspended from the ceiling of the ship and 'weighs' ma. You let go of the object and the ship will (?) accelerate a bit faster. (in the observer frame) the object will gain KE and lose PE on the way to the floor. Can that energy be identified as Work? An accelerometer on the ship will record an 'upwards' acceleration as the crew notice the object accelerating 'downwards'. Doesn't that represent a third law pair in the same way that gravitational force between the Earth and a dropped object works on both bodies? Don't dismiss that as ravings until you've positively destroyed the idea because it gives work a chance to be the same in all situations. I would be much happier if it could be.

I have a feeling that there's a similar situation in a rotating frame where, as radius changes the centrifugal force changes for both the object and the central attractor (as you vary the string length).

 

[jbriggs444]

You let go of the object and the ship will (?) accelerate a bit faster.

Since it is specified that the rocket accelerates uniformly, this declaration is false.

If one discards the uniformly accelerating frame and uses the ship's rest frame instead then it is true that the object accelerates a bit faster. The same way that a fat man accelerates a bit faster toward the gravitating earth [in the Earth's rest frame] than a thin man. Yet we do not normally discard the notion of a fixed value for $g$.

You proceed to focus on this approximation error and miss the point entirely.

Since a not-uniformly-accelerating frame has a time-varying acceleration, the resulting potential field will be time varying. It would not be proper to define a single static potential for such a field. The notion of potential energy could not be usefully defined except, perhaps, as an approximation.

 

[hmmm27]

Since a not-uniformly-accelerating frame has a time-varying acceleration, the resulting potential field will be time varying. It would not be proper to define a single static potential for such a field. The notion of potential energy could not be usefully defined except, perhaps, as an approximation.

IMO, assuming $a$ is otherwise constant, and not including the impulses(right term?) at "string-cut" and "thud", the "in-air" segment of the journey can be seen as totally discontinuous from the rest, and PE&KE (object) and $a$(ship) are constant during that time.

(unless of course I skipped a post or ten upstream and that's not what you're talking about)

 

[jbriggs444]

IMO, assuming $a$ is otherwise constant, and not including the impulses(right term?) at "string-cut" and "thud", the "in-air" segment of the journey can be seen as totally discontinuous from the rest, and PE&KE (object) and $a$(ship) are constant during that time.

(unless of course I skipped a post or ten upstream and that's not what you're talking about)

I am not sure what you are talking about here. Presumably you wish to consider the case where the mass of the falling object is a large fraction of the mass of the rocket so that the acceleration rate changes when the string is cut and again when the object strikes the floor.

I was making the simplifying assumption that the falling object has negligible mass compared to the rocket. Or that the rocket motors are controlled to deliver constant acceleration rather than constant thrust. But we can consider your scenario where they deliver constant thrust.

There is no impulse at the string cut. No instantaneous transfer of momentum. Only an instantaneous change of acceleration. And a corresponding instantaneous change in PE as the entire PE field is replaced.

[Do not try to conserve PE + KE across a change in the potential field. That is one reason that it is convenient to assume constant acceleration. This would result in an unchanging potential field]

There is an impulse at impact on the floor. An instantaneous transfer of momentum. An instantaneous change in rocket velocity. An instantaneous change in the rocket's acceleration as it resumes its prior steady state. This means an instantaneous change in the falling object's KE both due to the impulse on the object and due to the discontinuous change in the velocity of the frame of reference. There is also an instantaneous change in PE as the entire PE field is replaced again.

Things are much nicer if we consider the fall period only. Now the PE and KE change smoothly and in lock step so that their sum remains constant. Just as if we were standing on the Earth watching an apple as it falls from tree to ground.

It should be no surprise that the Earth case and the rocket case are similar. That's the equivalence principle at work.

 

[A.T.]

The object is suspended from the ceiling of the ship and 'weighs' ma. You let go of the object and the ship will (?) accelerate a bit faster.

You are distracted by details of one particular example, that are not generally required. Just strip it down to the core:

A mass m that is floating in space inertially (its accelerometer reads zero) is analyzed from a non-inertial reference frame that accelerates with a constant acceleration a, in which the object is initially at rest. In that frame the mass is subject to just one force, the inertial force -ma. That force has no Newton 3rd reaction but it does work on the mass to increase its KE.

 

[hmmm27]

you wish to consider the case where the mass of the falling object is a large fraction of the mass of the rocket

yes, that was it.

But we can consider your scenario where they deliver constant thrust.

in reference to the above, that's a better term.

There is no impulse at the string cut.

ah good - that was bothering me - in which case . . .

There is an impulse at impact on the floor.

. . ."inelastic collision" might work better for the scenario (for me).Thankyou.

 

[sophiecentaur]

A mass m that is floating in space inertially (its accelerometer reads zero) is analyzed from a non-inertial reference frame that accelerates with a constant acceleration a, in which the object is initially at rest. In that frame the mass is subject to just one force, the inertial force -ma. That force has no Newton 3rd reaction but it does work on the mass to increase its KE.

That's a consistent piece of Maths. Fine. But what is the Physics of the situation? The reference frame is totally abstract so you might question where the model applies to real world.
Work is a concept that involves Energy and energy exchange. In your basic scenario you are ignoring how those conditions can be achieved. This is in the same way as when a post arrives saying "What happens when the Sun is taken away?" We all jump on it like a ton of bricks because the suggested situation is not achievable and we rationalise it by saying that too much energy would be involved.

My problem was that I have been insisting that Work should apply everywhere. I could come to terms with this 'other thing', involving non inertial frames if Work were only defined under an inertial frame. It's the only case where there are third law pairs at work and where there is a real force involved with the force times distance. Perhaps the change in energy when non inertial frames are involved, would best be described as just that? When non inertial frames are involved there are implied other energy changes - like how the reference frame is maintained.

Final point: are there any non-inertial situations in which the concept of Work is used? e.g. orbital mechanics or Cosmology.

 

[A.T.]

The reference frame is totally abstract

All reference frames are abstract.

Work is a concept that involves Energy and energy exchange.

Work and Energy are frame dependent, and thus just as abstract as reference frames.

 

[jbriggs444]

That's a consistent piece of Maths. Fine. But what is the Physics of the situation? The reference frame is totally abstract so you might question where the model applies to real world.

An apple falling from a tree. You do realize that gravity is a fictitious force, right?

Work is a concept that involves Energy and energy exchange. In your basic scenario you are ignoring how those conditions can be achieved.

Not at all. Energy is not a real thing. It is a mathematical abstraction that works.

Final point: are there any non-inertial situations in which the concept of Work is used? e.g. orbital mechanics or Cosmology.

An apple falling from a tree. The rest frame of the Earth's surface is non-inertial.

 

[sophiecentaur]

An apple falling from a tree. You do realize that gravity is a fictitious force, right?

Why is it ficticious? (Any more than any other force.)
This conversation is on another plane now.

 

[jbriggs444]

Why is it ficticious? (Any more than any other force.)
This conversation is on another plane now.

General relativity.

Or, to be more mundane, the equivalence principle. It does not matter whether it is a fictitious force or the interaction force from Newtonian gravity. The two are locally indistinguishable. General relativity adopts the interpretation that "gravity" is a fictitious force.

That is to say, in a small lab, an able experimenter will be unable to determine whether he is working in a uniformly accelerating 1 g rocket in deep space or in a room on the surface of the Earth. [Or, for that matter, in a rocket with 1 g of proper acceleration hovering near the surface of the Earth]

If you consider that gravity can do work then you must agree that fictitious forces can do work.

Note that we are talking about "local" experiments. For instance, the able experimenter is not allowed to try to detect the tidal effects of a $\frac{1}{r^2}$ central gravitational force law. And he is for sure not allowed to send satellites into orbit.

 

[sophiecentaur]

General relativity.

Your reasoning makes good sense and I am inclined to accept it, thank you; I appreciate your flexible approach to this. However, does the same thing apply to electromagnetic forces?
A strong magnet on the floor of a very light spaceship (ignore its mass) will attract a lump of iron, mass m with a force $F_B$. That same force could be produced with no magnet but a rocket causing acceleration a=$F_B/m$
Could you tell the difference? GR wouldn't come into the magnetic version. I'd like to be able to close that loop and then I'd be happy.

 

[jbriggs444]

Your reasoning makes good sense and I am inclined to accept it, thank you; I appreciate your flexible approach to this. However, does the same thing apply to electromagnetic forces?
A strong magnet on the floor of a very light spaceship (ignore its mass) will attract a lump of iron, mass m with a force $F_B$. That same force could be produced with no magnet but a rocket causing acceleration a=$F_B/m$
Could you tell the difference? GR wouldn't come into the magnetic version. I'd like to be able to close that loop and then I'd be happy.

There are two questions lurking here.

1. Can one tell the difference between an electromagnetic force and an inertial/gravitational force?

The answer to question this one is an obvious yes. You can tell the difference. A lump of iron is affected. A lump of plastic is not. As you point out, the electromagnetic force cannot be transformed away by a change in frame of reference. [At least not easily. Kaluza-Klein theory makes the attempt, though I know little more about that beyond the name].

2. Can the familiar notion of "work" apply to electromagnetism, gravity and other fictitious forces at the same time?

The answer to question this one is also yes. You can define "work" so that both fictitious forces and interaction forces are considered to do work. For @A.T. and myself, this is the default posture. One force is as good as the next for this purpose. As you have suggested previously, it is important to be consistent in using the accelerating frame without waffling and inadvertently shifting to an inertial frame where the fictitious force you are considering does not exist.

 

[A.T.]

2. Can the familiar notion of "work" apply to electromagnetism, gravity and other fictitious forces at the same time?

Going back to the OP and the statement in question:

"work is done to achieve motion against an opposing force"

I really don't see how the above would apply to two charges repelling/attracting each other. There is only one force on either of the charges that accelerates it, and does work on it. Nothing is "opposing" that force. So even for interaction forces, that do have a Newton 3rd reaction (the force on the other charge), the statement makes no sense.

 

[sophiecentaur]

A lump of iron is affected. A lump of plastic is not.

I chose magnetic force as an example because it would be larger and work over a distance. I could have suggested a Coulomb force .

You can define "work" so that both fictitious forces and interaction forces are considered to do work. For @A.T. and myself, this is the default posture.

I would need a lot of effort to construct a world where there would be a difference (two different classes of Work) I think I am convinced.

There is only one force on either of the charges that accelerates it, and does work on it. Nothing is "opposing" that force.

Unfortunately, that would have to apply to all the electrical forces even between two 'solid' objects. At a microscopic level, molecules are held in place by non-contact, electric forces; one object would just be pushed by a field and the other would be pushed back by a field there would be no N3 pairs at all. Take away on object and the other object would have no force. This, of course is where I have trouble with gravitational interaction. An object in a gravitational field is creating a force on the object creating the field.

Only when you bring in the business of frames does the Maths say something different.

 

[jbriggs444]

Unfortunately, that would have to apply to all the electrical forces even between two 'solid' objects. At a microscopic level, molecules are held in place by non-contact, electric forces; one object would just be pushed by a field and the other would be pushed back by a field there would be no N3 pairs at all. Take away on object and the other object would have no force.

Whoah there. In the simplistic picture of Newtonian gravity and the Coulomb force, there is no field as a tangible entity. There are just forces-at-a-distance. Real honest N3 pairs.

Untold numbers ($\frac{n(n-1)}{2}$ for a system with $n$ charged or massive particles) of force pairs, certainly. But that is no problem. We have the continuum approximation and Newton's shell theorem to dial things back to a managable number of interacting bodies.

To be sure, if one looks closely at the details, there is a time lag from action to effect and the notion of a field with properties (e.g. momentum, energy, electromagnetic waves and gravitational waves) becomes necessary. But from a first year physics point of view, action at a distance and N3 are adequate.

 

[A.T.]

one object would just be pushed by a field and the other would be pushed back by a field there would be no N3 pairs at all.

Whatever pushes the objects back, it does work on them even if there is no "opposing force", so they can separate freely. Thus the statement in question doesn't apply in general.

 

[sophiecentaur]

even if there is no "opposing force",

With Coulomb force repulsion there are two forces; one on each charge. When there is 'clear air' between them it's just the same as when two solid objects push on each other. If this is not true then where would you draw the line?
I now appreciate that "the statement" doesn't apply universally but you can't chuck out all instances of 'non-contact' interaction.

BTW I was just searching my memory for frame dependent forces where there is no possibility of finding a reaction force somewhere. Coriolis Force comes to mind. My experiences on fairground rides suggest that, to overcome the Coriolis force, you still have to apply a real force (to move your arm you have to restrain your body). That's a N3 pair but, as has been pointed out, that pair doesn't actually involve the Coriolis force.

 

[A.T.]

I now appreciate that "the statement" doesn't apply universally ...

Which was my sole point. One counter example should be enough, and I have given one for interaction forces and one for inertial forces.