"Motion is impossible" claims modern Zeno

[darpan]

Let me set up a situation:

A piano rests on a frictionless surface. I am standing next to the piano (on a frictional surface) and I claim that the following two statements prove it is impossible for me to move the piano:

(1) ... The kinetic energy of the piano is equal to the work I've done on it.
(2) ... I can't do work on the piano unless it is moving. (But, because of (1), I can't get it moving unless I do work on it. But I can't do work on it unless it is moving... ad infinitum)

"Therefore the piano is immovable," I claim.

(1) is equivalent to the work-kinetic-energy theorem $W=\Delta E$
(2) is a special case (where $\frac{d\vec s}{dt}=0$) of the definition of work $dW=\vec F\cdot d\vec s=\vec F\cdot \frac{d\vec s}{dt}dt$

Please explain where and why my logic is flawed (assuming I don't know Newton's laws).

A pen is at rest on a table. I am standing next to the table. I claim that the following two statements will prove that it is impossible for me to pick up the pen:

(1) ... I will try and pick up the pen.
(2) ... I can ONLY try to pick up the pen. (But, because of (1), I will keep trying to pick up the pen. But I cannot ACTUALLY pick up the pen... ad infinitum)

"Therefore the pen can never be picked," I claim.

 

[darpan]

What if the piano rests on a frictional surface and you stand on a frictionless one.

 

[davenn]

A pen is at rest on a table. I am standing next to the table. I claim that the following two statements will prove that it is impossible for me to pick up the pen:

(1) ... I will try and pick up the pen.
(2) ... I can ONLY try to pick up the pen. (But, because of (1), I will keep trying to pick up the pen. But I cannot ACTUALLY pick up the pen... ad infinitum)

"Therefore the pen can never be picked," I claim.

But the fact, by practical demonstration, that you can pick up the pen negates your argument

 

[darpan]

But the fact, by practical demonstration, that you can pick up the pen negates your argument

Thank you Davenn. You just helped me prove, that Zeno's Paradox is practically unviable. Though its mathematical maze is an adventure to explore.

 

[votingmachine]

I think this is exactly a case parallel to Zeno's paradox. The problem is trying to define 0/0. In zeno's paradox, the infinite number of infinitely small intervals all are actually small dX's and they are still each covered in smaller and smaller dT's, allowing a constant V.

Consider that F=MA also implies that M=F/A. With no Force, and no Acceleration, we don't assume the piano has lost its mass since we last measured it.

Before a Force is applied, there is no dX. As soon as the Force is applied, there is a dX. You are asking how long does it take the piano to move no distance? dX/dT when dX=0.

The kinetic energy IS equal to the work you have done on it. The kinetic energy starts from 0, where you have done no work, also 0. You are taking the extreme of Zeno's paradox, and dividing the world into distances of 0 length. But there is no meaningful way to talk of a zero length motion.

Say that instead of the piano being at rest, it was sliding towards you and you were anchoring an ideal spring. Now it slows and slows and then STOPS, and then reverses. It only makes sense to talk of the acceleration as acting across very small distances. dx/dT/dT. The time intervals are also very small. There is no point where it is meaningful to accurately speak of what about when the dX is 0. That is a mathematical point. You need two points to have dX.

You either have something finite ... a distance between two points, no matter how small, or you have a meaningless singular point. And the problem is when you declare that the arbitrarily small intervals have to be a single point. And then the math breaks down, since it is based on differences (differentials).

I'm not sure that helps any. It is clear in my head, but seems very paradoxical as soon as you try to type it. I would recommend thinking about the spring variation, where the piano stops and reverses. How long is it at absolute rest? I think the answer is that it is at absolute rest for no time at all, even though before that moment it was in motion one direction, and then after that moment, in the other direction, it is at rest for no time at all!

 

[votingmachine]

I had another thought on this later. When you ask the amount of work for when you push during the interval from 8 inches to 9 inches, you can calculate that. But what about the work in the interval from 8 inches to 8 inches? Strictly speaking, there is no interval. I think the same problem is what you are running into when you start at 0, and define that point as an interval. You need some kind of elapsed distance or time interval.

 

[Chandra Prayaga]

Statement 2 by Nathaniel is wrong. "I can't do work on the piano unless it is moving". You can start with the piano at rest. Now push it. During the time when you are in contact with the piano, the force that you exert on the piano does work, which increases the kinetic energy from zero to some value. That is the energy point of view. You can also say that during the time you are in contact with the piano, you have delivered an impulse (force multiplied by the time of contact), which changes the momentum from zero to some value. This is the impulse-momentum point of view. You can also say, during the time that you are in contact, you are exerting a force on the piano, which, by F = ma, gives an acceleration to the piano, and over the time of contact, acceleration multiplied by time gives the final velocity. All three points of view are equivalent and give the same result.

 

[Drakkith]

Statement 2 by Nathaniel is wrong. "I can't do work on the piano unless it is moving". You can start with the piano at rest. Now push it. During the time when you are in contact with the piano, the force that you exert on the piano does work, which increases the kinetic energy from zero to some value.

Of course, but I think he was wanting an explanation that didn't involve Newton's laws or something.

 

[Chandra Prayaga]

Nathanael's statement 2 is wrong.
Start with the piano at rest. Push on it, exert a force.
Since there is no friction, you will be in contact only for a short time. You cannot keep pushing on it if there is no friction between your feet and the ground.
But during that short time:
F = ma, so there is an acceleration for a short time, and the acceleration multiplied by the tome of contact gives final velocity (view point 1)
The force multiplied by the time of contact is called the impulse, and that results in a change in momentum from zero to some final value (view point 2)
The force multiplied by the distance moved while in contact is the work done, which results in a change in kinetic energy from zero to some final value (view point 2)
All three points of view are equivalent

 

[Chandra Prayaga]

Of course, but I think he was wanting an explanation that didn't involve Newton's laws or something.

There is no such thing as an explanation that does not involve Newton's laws right? Not when you are pushing a piano.

 

[votingmachine]

Nathanael's statement 2 is wrong.
Start with the piano at rest. Push on it, exert a force.
Since there is no friction, you will be in contact only for a short time. You cannot keep pushing on it if there is no friction between your feet and the ground.
But during that short time:
F = ma, so there is an acceleration for a short time, and the acceleration multiplied by the tome of contact gives final velocity (view point 1)
The force multiplied by the time of contact is called the impulse, and that results in a change in momentum from zero to some final value (view point 2)
The force multiplied by the distance moved while in contact is the work done, which results in a change in kinetic energy from zero to some final value (view point 2)
All three points of view are equivalent

What you say is perfectly clear. I think the paradox he was building was equivalent to a force that really would not be a force ... so if you must exert a force over a distance to do work, what if you hit at the piano with a hammer which does not move beyond the starting point.

And to me the answer is that you can't hit with a hammer at point 0, and not beyond. If you did, you would not impart energy, but merely stop the hammer at the same place the piano starts. So yes, you have to actually have an interval ... even an infinitely small one to impart the force you posit. Think if you had a cam that was your force impeller. If the cam just exactly does contact the piano, there is no work. And as soon as you have more than that contact, you have an interval.

You don't have to "cross" the gap at zero. There is no gap between zero and the smallest number (as someone else pointed out). Either you exactly do not touch, or you exactly touch, or exactly do more than touch, and then do work.

The answer is the same as with Zeno. There is no gap between the most infinitely small difference and zero.

So yes, it is not in motion until work is done. And an infinitely small amount of work done in an infinitely small interval puts it into motion. I see it as asking how you cross the gap from position 0 to the first non-0 position. And there is no gap.

EDIT: I suppose the other way of saying it is to agree that there is no work until there is motion and there is no motion until there is work, but that they both begin to exist at the same infinitely small amount of time. No bit has to come before the other, but both exist as soon as the piano moves from position-zero, to the next point in space.

And as has been pointed out, that is no distance at all.

 

[Dadface]

I think the original statements should be clarified as follows:

1. The increase of kinetic energy is equal to the work done.
2. There can be no increase of kinetic energy unless work is done.

Aren't the above two statements and the original two statements different ways of saying the same thing? I can't see where there's a problem. Perhaps I'm not understanding it.

 

[A.T.]

I can't see where there's a problem. Perhaps I'm not understanding it.

See post #6.

 

[thinkandmull]

Calculus is used to say what a limit is. I struggle to understand how a limit can exist for something in space like a line. If from A to B I must always first go half a step, when will there ever be a final step. If the step can be divided in 2, its not the final step. If it cannot be divided, than I am already at my location. Again, there would be no final step.

 

[A.T.]

I must always first go half a step

The time you need to go those half steps goes towards zero.

 

[thinkandmull]

Yes but when does it reach zero? There is no final step of the series? A line seems to have the qualities of the finite and the infinite at the same time. Its seems almost like a round square to me

 

[A.T.]

A line seems to have the qualities of the finite and the infinite at the same time.

It has nothing to do with the qualities of a line, just with numbers which we use to quantify a line's length, a square's area or whatever. Positive numbers can be divided into infinitely many positive non-zero parts.

 

[Dale]

I struggle to understand how a limit can exist for something in space like a line.

Do you understand how a limit can exist for an infinite series of real numbers?

 

[thinkandmull]

Since I am using a line's quality as the bases for my understanding this, no I am not sure what a limit would be for infinite numbers or steps

 

[votingmachine]

Think of the series:
0.3 + 0.03 + 0.003 +0.0003 + 0.00003 + ...

The series goes on for an infinite amount of terms, but the limit is 1/3.

 

[jbriggs444]

Since I am using a line's quality as the bases for my understanding this, no I am not sure what a limit would be for infinite numbers or steps

A "limit" is, roughly speaking, the single, exact real point that is approached more and more closely by an infinite sequence -- if that sequence converges at all. The point is that the sequence does not need to have a last step in order to have a limit that it approaches.

In mathematics, the real number line are constructed in such a way that convergent sequences always have limits. If you have a sequence of numbers that get closer and closer to one another in a particular way (https://en.wikipedia.org/wiki/Cauchy_sequence) then there will always be a real number that is the limit of the sequence.

A course in Real Analysis would often take you through the definitions and theorems that make clear exactly how this works.

 

[thinkandmull]

If someone is always taking half-steps, he will never get to the destination. This is so counter-intuitive. If you are ever getting closer to something, how could you never reach it? The eternity of future's time seems to be bigger than any infinity of points

 

[jbriggs444]

If someone is always taking half-steps, he will never get to the destination. This is so counter-intuitive. If you are ever getting closer to something, how could you never reach it? The eternity of future's time seems to be bigger than any infinity of points

That's Zeno's paradox. If someone is always taking steps in half times, that sequence of steps will not cover all times out to eternity.

 

[votingmachine]

That's Zeno's paradox. If someone is always taking steps in half times, that sequence of steps will not cover all times out to eternity.

If you run a marathon in 2 hours, you also run two half-marathons in 1 hour each. Or 4 quarter-marathons in 1/2 hour each. Or 100 one-hundredth-marathons in 0.02 hours each. You can apply any division you want and you will always apply the same division to the time and to the distance.

It seems confusing because Zeno phrases it in terms of a sequence that is ahead of you, and therefore there are an infinite amount of steps. But you can ALWAYS divide the distance and the time by any arbitrary number, including the infinite halving sequence (Run a half-marathon, then a quarter-marathon, then an eighth-marathon, then ...).

The answer is that there is always a corresponding division of the time. Pick ANY number of divisions for that marathon. Divide the distance by that number. Divide the time by that number. Then multiply them together and get the original answer. In my example you will always get 1-marathon-per-2-hours.

 

[thinkandmull]

So we must say that continuous motion has a special quality that is a "sum greater than its parts", so to speak. Because the infinite half steps need to be passed over before one gets from A to B, but I think the continuous motion is greater than the motion that is taking half steps. But greater how?

 

[A.T.]

So we must say that continuous motion has a special quality that is a "sum greater than its parts", so to speak.

No, it has nothing to with motion being special. You can divide any number like that. An it's not greater than the sum of its parts.

 

[thinkandmull]

Than we are faced with the same paradox as Zeno's Arrow paradox. I think motion is "one step ahead" of matter so to speak. I don't know how else to say it. When you cross a finite segment, you pass over infinite points, and there is no final step. There are no discrete "one two three" steps to take and then wow! you are at your destination. I'm thinking that motion must be more mysterious, something about it that we are not getting straight from math, but can understand better with physics

 

[PeroK]

Than we are faced with the same paradox as Zeno's Arrow paradox. I think motion is "one step ahead" of matter so to speak. I don't know how else to say it. When you cross a finite segment, you pass over infinite points, and there is no final step. There are no discrete "one two three" steps to take and then wow! you are at your destination. I'm thinking that motion must be more mysterious, something about it that we are not getting straight from math, but can understand better with physics

My steps are about 1m, so if I have to travel 3m, then it is "one, two, three" and I'm at my destination.

I never thought much of Zeno's paradox, I must admit. Here's my version:

The hare starts 1m behind the tortoise.
The hare starts 1m behind the tortoise.
The hare starts 1m behind the tortoise ...

Ergo, the hare never even moves let alone catches the tortoise.

 

[Dale]

Since I am using a line's quality as the bases for my understanding this, no I am not sure what a limit would be for infinite numbers or steps

This is the thing that you need to learn then. Once you have understood the limit of an infinite sum of real numbers then the mapping from the real numbers to a line is trivial.

As far as the infinite sum goes, this is actually straightforward. If you have $\sum _{n=1}^k \frac{1}{2^n}$ then you can fairly easily convince your self (simply by writing it out for a few small values of $k$) that the answer is $1-2^{-k}$. As $k$ becomes larger and larger the sum gets closer and closer to 1. In fact, you can specify any arbitrary $0<\epsilon<<0.5$ and calculate the $k$ necessary to make the sum closer to 1 than $\epsilon$. This is what it means to have a limit. The limit of any finite series is less than 1, but as the finite series gets arbitrarily large the sum gets arbitrarily close to 1. So the infinite sum is equal to 1.

Please start with this website and think and mull it over a bit:
http://www.math.utah.edu/~carlson/teaching/calculus/series.html

 

[A.T.]

...so to speak. I don't know how else to say it.

Use math.