Mach versus Einstein

[sol invictus]

Generally a fictitious force is referred to as an inertial force but we may want to extrapolate on that and say that a fictitious force results in no overall net exchange of energy or momentum between particles.

Hmm... but standard fictitious forces can certainly change the energy of something. For example, if you are rotating while holding a weight in your outstretched hand, it will take some effort to bring the weight in close to your body. So you did some work against the centrifugal force.

Now obviously total energy was conserved there - but energy is always conserved by all forces, fictitious or not, so I don't see how that can be used to distinguish them.

 

[The Man]

Hmm... but standard fictitious forces can certainly change the energy of something. For example, if you are rotating while holding a weight in your outstretched hand, it will take some effort to bring the weight in close to your body. So you did some work against the centrifugal force.

Now obviously total energy was conserved there - but energy is always conserved by all forces, fictitious or not, so I don't see how that can be used to distinguish them.

No not necessarily, your example is synonymous with a bound state going from a higher energy level to a lower energy level. The weight has a higher tangential velocity when your arm is extended the when it is retracted, provided the RPM has not changed. Likewise, the weight also has a higher kinetic energy when your arm is extended. The work you do in retracting your arm only seems like it is countering the fictitious centrifugal force but actually you are slowing the tangential velocity of the weight and reducing its kinetic energy.


ETA: If we allow the tangential velocity of the weights to remain constant and the RPMs to change, then the work you have done has gone into increasing the angular momentum and energy of both you and the weights.

 

[sol invictus]

The work you do in retracting your arm only seems like it is countering the fictitious centrifugal force but actually you are slowing the tangential velocity of the weight and reducing its kinetic energy.

You changed the energy of the block by doing work on it. In the unrotating frame you slowed it down and reduced its kinetic energy. In the rotating frame, the block started and ended at rest, and the work was due to pulling it in against the centrifugal force. You changed its "centrifugal potential energy" by moving it closer to the center.

As far as I can see the fictitious force acts just like any other force, at least in this regard. So I don't see any way to distinguish between ordinary forces and fictitious forces that way.

 

[shadron]

It appears I have to correct myself here. I am inappropriately equating "frame of reference" with "inertial frame of reference" (which is usually an unstated assumption in discussions of SR but not in GR). I apologize for the error.

I'm sure there are a fair number of us who are straddled across a good understanding of SR and a rather less-good understanding of GR. To us (why, just a mere engineer *blush*), this sort of thread is a wonderful mind-expansion exercise.

Lay on, y'all.

 

[ponderingturtle]

On the contrary. The laws are invariant under any coordinate transformation; that is the underlying principle of general relativity.

But they only hold for inertial reference frames, and a rotating reference frame is not an IRF.

 

[The Man]

You changed the energy of the block by doing work on it. In the unrotating frame you slowed it down and reduced its kinetic energy. In the rotating frame, the block started and ended at rest, and the work was due to pulling it in against the centrifugal force. You changed its "centrifugal potential energy" by moving it closer to the center.

As far as I can see the fictitious force acts just like any other force, at least in this regard. So I don't see any way to distinguish between ordinary forces and fictitious forces that way.

There is no such thing as “centrifugal potential energy” that I am aware of. If you are commoving with the blocks in an inertial reference frame then they only have the potential energy of their rest mass. If you are not co-moving with the blocks in different inertial reference frames then they will have some kinetic energy based on relative motion between those inertial reference frames. There is no such thing as potential energy based entirely on relative motion. Objects either have relative motion and kinetic energy in relation to each other or they do not. In a gravitational field and object has potential energy based on its distance from the center of the gravitation field. The motion of that object is a stable orbit can maintain that distance and the gravitational potential energy. In the co-moving or rotating reference frame the block starts and ends at rest but its position changes, the work is done by accelerating the block then decelerating it until it comes to rest at the new position. There is no centrifugal anything because the only motion of the block in that reference frame is the motion to the new position which has no different potential then any other position in that reference frame.

 

[sol invictus]

But they only hold for inertial reference frames, and a rotating reference frame is not an IRF.

No - they hold in all reference frames. Read the quote from Einstein in post #5 above. They just look different when you write them in non-inertial coordinates.

If you prefer, we can say it like this: we have a set of physical laws that apply in all inertial frames. We also have (because Einstein wrote them down) a set of rules that tell us how those laws change when we transform to non-inertial frames. Since a coordinate transformation is not a physical operation, those rules had better leave the physical predictions of the theory invariant (if there is a tension in the rope in inertial coordinates, there had better be a tension in the rope in non-inertial coordinates). GR satisfies that criterion.

 

[sol invictus]

There is no such thing as “centrifugal potential energy” that I am aware of.

It's a bit of an abuse of terminology, which is why I put it in quotes. However one could perfectly well define it - the potential in this case would be [latex]\tiny $V(r) = -{m \omega^2 \over 2} r^2$[/latex], r being the distance from the origin in cylindrical coordinates. The gradient of that potential gives you the centrifugal force.

If you are commoving with the blocks in an inertial reference frame then they only have the potential energy of their rest mass.
...

You seem to have forgotten the problem in question. This has nothing to do with gravity in the ordinary sense; forget about that. In the inertial frame there are no relevant forces other than the tension in the rope. But the blocks are rotating, which means they are at rest in a non-inertial frame - and that's the whole point. In a non-inertial frame there are "fictitious" forces; in this case, a centrifugal force that increases linearly with radius. From that point of view the configuration is completely static, with the tension in the rope is balancing the centrifugal force. If you shorten the rope, you do work against the centrifugal force (you may have to be careful about Coriolis forces too).

 

[The Man]

Well I suppose that could be a way of looking at it, as long as we remember that it is a fictitious “potential energy”. It would be similar to the simulated gravity of a rotating space station (sorry to bring gravity back into it). If in a room on large enough space station (with a small enough Coriolis force) we might have no way of knowing that we were on a rotating space station and not a rotating planet. If we can develop a consistent and definitive distinction between a fictitious and non-fictitious force, we may find a new way to look at things.

 

[69dodge]

Fictitious forces don't obey Newton's third law.

For example, in a rotating frame of reference, an object is subject to centrifugal force. The object ought to apply an equal and opposite force to the source of this centrifugal force. But it has no source. It just magically is there.

 

[sol invictus]

Fictitious forces don't obey Newton's third law.

For example, in a rotating frame of reference, an object is subject to centrifugal force. The object ought to apply an equal and opposite force to the source of this centrifugal force. But it has no source. It just magically is there.

Interesting point.

However Newton's third is just conservation of momentum. That's a law of nature, and as such it does actually hold in the rotating reference frame, just in a different form.

Still, maybe there's a way to use that somehow...

 

[Schneibster]

Fictitious forces don't obey Newton's third law.

For example, in a rotating frame of reference, an object is subject to centrifugal force. The object ought to apply an equal and opposite force to the source of this centrifugal force. But it has no source. It just magically is there.

Nope. An object that experiences a centrifugal force experiences it because of a centripetal force. That's the force applied by whatever holds it to the rotating course instead of it pursuing the zero-force inertial path. In the case of the rotating object, that would be the molecular and atomic bonds that hold the object together.

Fictitious forces are subject to Newton's Third Law. You're hiding the equal and opposite reaction inside the object, but it's still there.

 

[sol invictus]

Fictitious forces are subject to Newton's Third Law. You're hiding the equal and opposite reaction inside the object, but it's still there.

You're missing his point. Go to the rotating frame and let the object (which was at rest in that frame) break apart. Now the only force acting is the centrifugal force, and the pieces of the object will fly out from the origin, accelerating as they go. In those coordinates, mass times velocity is NOT conserved. Neither is mass times velocity squared, incidentally.

However there is a more complicated quantity that is conserved - simply the total linear momentum (and total energy) written in the rotating coordinates.

This is a good example of how the laws of physics still hold in non-inertial frames, but look very different written in those coordinates.

 

[The Man]

Perhaps the description of a fictitious force as an inertial force is the best one after all, if we just give it clarification. If we consider a large cylinder out in space and at the center we have two boxes attached with and ideal compressed spring with a person floating free in the center of each box. If we release the spring the boxes are accelerated towards the inner wall of the cylinder until they impact that wall. In the cylinders reference frame or in a reference frame of one of the boxes each person would be pushed against the bottom of the box until the box impacted the wall of the cylinder at which time the person would be propelled into the top of the box (real force).


If we now repeat this experiment with the cylinder and boxes rotating about the cylinders axis but now the boxes are attached to each other with a rope. Then we cut the rope. In the rotating reference frame of the cylinder the boxes will accelerate towards the cylinder wall just as they did in the other example, but now come the differences. In the reference frame of one of the boxes, before the rope was cut, the person would be pinned against the top of the box. When the rope is cut the person floats free and experiences no force. At the same time in the cylinders reference frame the boxes are accelerating towards the cylinder walls under the perceived centrifugal force. When the boxes hit the wall in both of those reference frames the persons are again slammed against the top of the boxes.


So in the rotating reference frame of the cylinder when the rope is cut the box appears to be non-inertial or undergoing a (non-rotating in this case) acceleration due to a force. However in the reference frame of the box when the rope is cut, that frame is completely inertial and the occupant feels no force, until the impact.

So here perhaps is a new definition of a fictitious force, a force that when non-rotationally accelerating a reference frame is not perceivable in that reference frame.

 

[69dodge]

An object that experiences a centrifugal force experiences it because of a centripetal force.

I swing a ball on a rope around me in a circle. I exert on the ball, via the rope, a centripetal force, a force directed toward me. And in response to this force, in accordance with F = ma, the ball accelerates toward me. End of story. Where is the centrifugal force? There is none.

Centrifugal force is a fictitious force that we postulate when we want to refer all motion to a rotating frame of reference rather than an inertial one.

In a frame of reference rotating about me together with the ball, the ball is motionless. In particular, it is not accelerating. How can this be, when I am still exerting a force on it? We postulate a fictitious centrifugal force on the ball, equal and opposite to the force I'm exerting on it. That way, the net force on it is zero, which is consistent with its lack of acceleration.

 

[Schneibster]

You're missing his point. Go to the rotating frame and let the object (which was at rest in that frame) break apart. Now the only force acting is the centrifugal force, and the pieces of the object will fly out from the origin, accelerating as they go. In those coordinates, mass times velocity is NOT conserved. Neither is mass times velocity squared, incidentally.

But if that happens, there will no longer be a rotating object. There will be two objects, which depending on the details of the breakup might or might not be rotating themselves, leaving the location of the breakup of the original rotating object along geodesics.

Furthermore, they will not be accelerating (except as they are rotating- and they may not be). They will (except for any rotation) be inertial- and their centers of mass will move along geodesics, whether they are rotating or not. The amount of linear momentum they have will depend on whether they are rotating or not; if they are not, then the original angular momentum, converted into linear momentum, will be at a maximum value, conserving the converted quantity; if they are, then it will be some lesser value, and their new angular momentum plus the converted quantity will add up to the original value.

Now, what he said was,

Fictitious forces don't obey Newton's third law.

For example, in a rotating frame of reference, an object is subject to centrifugal force. The object ought to apply an equal and opposite force to the source of this centrifugal force. But it has no source. It just magically is there.

Newton's Third Law is Action and Reaction. Mathematically, it's conservation of momentum. What you've said is, momentum is conserved; what I've said is, momentum is conserved. But by claiming, "It just magically is there," 69dodge has asserted that momentum is NOT conserved. What you said showed the breakup of the object, but is not directed at the original object, which 69dodge asserted had no equal and opposite force to the centrifugal force, a different subject since after the breakup, the two (or more) pieces do not maintain physical contact, do not therefore exert centripetal force, and stop moving rotationally and start moving linearly (except for any residual rotation they experience as a result of, for example, an uneven break). The centripetal force is gone, and the centrifugal force disappears with it. Instead of these forces, we now have inertial movement by the two objects (again, except for any residual rotation they retain).

However there is a more complicated quantity that is conserved - simply the total linear momentum (and total energy) written in the rotating coordinates.

This is a good example of how the laws of physics still hold in non-inertial frames, but look very different written in those coordinates.

Sure, but it's not a demonstration of anything "magically appearing."

 

[Schneibster]

I swing a ball on a rope around me in a circle. I exert on the ball, via the rope, a centripetal force, a force directed toward me. And in response to this force, in accordance with F = ma, the ball accelerates toward me. End of story. Where is the centrifugal force? There is none.

The centrifugal force acts to pull the object away from you, keeping the rope taut, and is equal and opposite to the centripetal force that makes it move in a circle instead of a straight line. If there were none, the ball would simply move toward you and smack you in the face; there'd be nothing to hold it out at the end of the rope, or hold the rope taut.

Centrifugal force is a fictitious force that we postulate when we want to refer all motion to a rotating frame of reference rather than an inertial one.

I just demonstrated both centrifugal and centripetal force in an inertial frame. Tell me how the rope being taut, or the ball being accelerated in rotation rather than inertial in linear motion, can be explained without them.

In a frame of reference rotating about me together with the ball, the ball is motionless. In particular, it is not accelerating. How can this be, when I am still exerting a force on it? We postulate a fictitious centrifugal force on the ball, equal and opposite to the force I'm exerting on it. That way, the net force on it is zero, which is consistent with its lack of acceleration.

You've chosen a Euclidean frame to describe a curved spacetime; there are, therefore, discrepancies which must be explained by forces. Furthermore, you can measure the force of the ball on the rope, the rope on you, you on the rope, and the rope on the ball, and being measurable, these aren't "fictitious," they're factual and measurable. They have real physical consequences. If they act, there will be changes in momentum; since they are in equilibrium, they do not act.

 

[69dodge]

I'm talking about Newtonian mechanics here. No relativity or curved spacetime or anything like that.

I don't know what you mean by "centrifugal force". On what, according to you, does this force act? Me or the ball?

I have been talking only about forces on the ball. There are also forces on me, but they don't affect the motion of the ball, of course. An object responds to forces on it, not to forces on other objects.

Why would the ball move directly toward me, in the absence of whatever you're calling "centrifugal force"? Granted, I am pulling the ball directly toward me, but objects don't move in the direction of the force on them; they accelerate in that direction. That is Newton's second law: F = ma. And an object moving in a circle at constant speed is, in fact, accelerating directly toward the center of the circle.

 

[Schneibster]

I'm talking about Newtonian mechanics here. No relativity or curved spacetime or anything like that.

I'm not sure, then, how you expect your example to conform to reality when you apply rotating coordinates. This is a situation where we know that Newtonian mechanics and relativity give different answers, and the answers relativity gives conform more closely to reality. On the other hand, in this response, you don't appear to be talking about rotating coordinates any more, but only about flat spacetime, where Newtonian mechanics closely approximate relativity. I'll answer using that assumption.

I don't know what you mean by "centrifugal force". On what, according to you, does this force act? Me or the ball?

That depends on whether you are rotating yourself; if you are not, then only to the ball. If you are, then to some degree to you as well, though less than to the ball since your radius of curvature is smaller.

I have been talking only about forces on the ball. There are also forces on me, but they don't affect the motion of the ball, of course. An object responds to forces on it, not to forces on other objects.

You are touching the rope, and the rope is touching the ball. You, the rope, and the ball are a single system; you might be a single object, in some definitions of the word "object." You are able to transmit a force through the rope to the ball; without the rope, or without you, the ball does not move in a circle. The force you transmit is centripetal; the force the ball transmits back is centrifugal. These forces are equal and opposite at any instant, holding the ball in a circular orbit about you.

Why would the ball move directly toward me, in the absence of whatever you're calling "centrifugal force"?

Because you are exerting centripetal force on it through the rope. If that is not balanced by centrifugal force, then it will pull the ball directly toward you.

Granted, I am pulling the ball directly toward me, but objects don't move in the direction of the force on them; they accelerate in that direction. That is Newton's second law: F = ma.

The only way to remove the centrifugal force is to remove momentum; and without momentum, there is nothing to vector sum with the centripetal force. That force then acts to accelerate the ball directly toward you.

The ball exerts centrifugal force; you exert centripetal force. Each is transmitted by the rope, in opposite directions. You feel the centrifugal force; the ball feels the centripetal force.

And an object moving in a circle at constant speed is, in fact, accelerating directly toward the center of the circle.

Correct, under the influence of the centripetal force exerted by you on it through the rope.

 

[sol invictus]

I'm not exactly sure what the issue is here. We could try to define momentum in the rotating frame as mass times velocity, where velocity is the time derivative of the rotating frame coordinates. With that definition momentum (and kinetic energy) would not be conserved. For example an object that started at rest at a non-zero radius would accelerate away along a radial line. I thought that was 69dodge's point.

But this is simply a reflection of the fact that the laws of physics we're used to look different when you write them in non-inertial coordinates. There is still conservation of something in the rotating coordinates, but it's not mass times the time-derivative of the position.