In the column http://www.straightdope.com/classics/a1_165.html, ** the Man** says that one will eventually stop at the center, due to air resistance. But since air resistance is a function of (among other things) the speed squared, would that be true? The last bit of speed would never be removed, since resistance is a fraction of the energi. I’m pretty sure this isn’t another version of Zeno’s paradox, and it would in this case take an infinite amount of time before reaching a standstill.
The idealized numbers in math are infinitely divisible but nothing made of real matter is. No matter how you slice it, you eventually come to a point in which motion stops because the energy available to move an object - which is continually decreasing - is less than the frictional forces opposing it - which are constant.
There are no infinities in the real world, which is the point that early mathematicians kept getting hung up on. They assumed that math was a representation of reality.
I can see that at a point, there is going to be a molecule of air that is going to stop you, in reality. But I don’t like the phrase “is less than the frictional forces opposing it - which are constant”. The friction isn’t constant, it’s a function of speed. I guess that the amplitude is going to decrease exponentially, and we’ll have a dampened frequency, something like x = (1-e^(-kt))Acos(w*t), in theory, but in reality, the energi to push aside the final molecule of air won’t be there. In the column The Man writes that he couldn’t figure out how long it would take. That comforts me, it would be a pretty incredible feat.
Just to be safe, I have to say that my formular isn’t the real one. I have no idea what the real one is. But it would have similarities. In my formular, t is time, w is rad/sec, I can’t remember what that parameter is called in Eglish, something angle-speed?, A is amplitude, in this case the radius of the earth, and k is some constant, dependant on the air friction.
You also say that there are no infinities in the real world. In the case of Zeno, I would say that both time and distance are infinitely divisible, but I guess that is more a question of religion than science.
Fluid dynamics is full of assumptions, and just how well your assumptions work depend on what regime you’re operating in. There are a number of different effects that combine to produce drag, and under some conditions on or another effect will become dominant, so you can ignore the others.
Let’s assume that we’re dropping a sphere, rather than a human being, through the center of the earth. At relatively high speeds, the drag from air resistance is pretty close to proportional to the velocity squared, as you say. However, as the velocity of the sphere got small (an inch or so per second), the flow around the sphere becomes laminar, and the drag is more proportional to something like the velocity to the 1-1/2 power. As the velocity got smaller yet (say, in the range of a few in/hour), the sphere enters the creeping flow regime. Here, the drag force is directly proportional to the velocity to the first power.
Now, another assumption made in fluid dynamics is what Exapno Mapcase was referring to, and that’s the assumption that the fluid is one continuous medium. In reality, it’s not: it’s a collection of molecules. In the aggragate, at reasonable speeds and sizes, the assumption that air is a continuous smear of matter is pretty good. However, as speed decreases, the small amount of kinetic energy left in the sphere is going to approach the small amount of kinetic energy in groups of molecules hitting the sphere. Remember that air molecules are continually in motion in all directions.
As a quick f’rinstance of time scale, once a sphere slows to creeping flow, I calculate that a 150 pound, 2 ft diameter sphere will decrease its velocity by a factor of 10 every 230 seconds. Someone else can figure out how slow the sphere needs to go before the collisions of molecules are an important consideration.
Finally, since we’re talking about a human, and not a sphere, it’s important to remember that humans move. I’m not sure just how much acceleration is imparted by waving your hand or turning your head, but it can’t be negligible when we’re considering the effects of individual molecule collisions.
After thinking about it over the weekend, and discussing it with some friends, I think that no matter what was dropped, it would never stop completely. But as zut says, it would be a range of different areas we’d be passing through. To begin with, the movement would be somewhat as I described. Then the object would slow down and behave like zut describes it. Finally it would be pushed around by the brownian movements of the air. So as Exapno says there are no real infinities in the real world, there would never be anything that stands absolutely still. But it wouldn’t be the pendulum movement, it would just be some ultra slow, small vibration.
On the other hand, once the object (whatever it is) gets to the point that the dominant effects on it are Brownian motion, one might as well call it “at rest”. If we’re not allowed to make simplifications like that, then physics has a tendancy to grind to a halt.
Yes, but it was exapno who brought the movement down to a molecular/atomic level. If the simplification you talk about was used, then my first statement would be true. Either that, or someone ought to give me a number for the amount of energy required to move, so that I can figure out when this thing is going to stop.
Sorry if that was confusing, but I only meant that as long as you have real particles involved you have real physical limitations - which the world of math does not have.
Brownian motion does not have real world application to particles larger than atoms in any case. The human body that is part of the original example would come to a halt for all measureable purposes.
As Chronos says in this thread on pi:
The same limitations apply to motion, if you see the analogy.
Man, I’ve missed out on a some interesting discussions.
Anyway, I choose to believe that the object dropped will continue to move eternally, and have “eternally” mean “a really long time”, instead of saying that it stops, and having “stops” mean “barely moves at all”. I don’t work with things on the atomic level, I’m an engineer, so that definition is a bit more practical for my world view. Either way, I don’t think any of us could figure out when the object is going to stop, even if we could agree on what “stops” means…
I was just surprised that The Man tried to figure out how long it would take in the orignal column.
But I think we’ve gotten as far as possible on this thread.
Isn’t there an analogy with a pendulum’s motion? I think most would accept that friction will damp a pendulum’s swings to nothing in a reasonably short time.
Well, yes and no. If you look at frictional forces on a pendulum, you have to include the Coulomb friction at the pendulum pivot joint. Coulomb (or dry) friction is typically modelled as a force that is proportional to the normal force and oriented in the direction that is opposite the motion or impending motion. This force is not proportional to velocity, and, in fact, tends to be somewhat higher when velocity is zero.
The driving torque in a pendulum is the torque created by the hanging weight, when it is not directly below the pivot joint. As the friction slows down the pendulum, there will come a point where, after reaching the end of the swing (with velocity zero), this driving torque is smaller than the potential opposing frictional torque, and the pendulum will not move again.
However, the frictional force on an object falling through the center of the Earth is fluid friction. This frictional force is modelled as proportional to velocity (or velocity squared). Thus, when the falling object reaches the end of its “swing” (with velocity zero), there is no frictional force to oppose the object’s motion.
I was thinking in terms of a pendulum suspended by a thin strip of flexible metal. Would the friction model you describe apply to that?
Actually, that’s a darned good question. Umm…I don’t think so. What you’re thinking of here is where the “internal friction” in the thin strip removes energy from the system; this is called, properly, “material damping” or “structural damping”. As far as I know (and I’m not an expert at all), the mechanisms that cause material damping are not very well understood. However, there are a number of ways of modeling these effects that work fairly well in most circumstances (Here’s a page with a somewhat technical, reasonably thorough, but not overly intimidating, discussion of material damping).
The damping modeling, as far as I’ve seen, have assumed that the damping force is proportional to either velocity or amplitude. In the either case, the mathematical solutions predict that the object (pendulum, in this case) will oscillate forever, but with an ever-decreasing amplitude. For example, the response from the former model will match the response from a classic underdamped spring-mass-damper system.
However, these models are used because they are simple and give reasonable results in most circumstances, not because they actually model the physical damping. From the first link: “Damping models do not imply any detailed explanation of the underlying physics.” So what exactly happens at ultra-low amplitudes is, I think, not understood by anyone. Certainly the pendulum does not continue forever, but does it keep oscillating until the movement is so small that quantum effects take over? Or do the (not well understood) internal mechanisms that cause structural damping have some nonlinear feature that halts the motion? I don’t know. And maybe no one does.
In all this thread, and in the master’s original work, no-one has noted the effect of the rotation of the earth.
The master’s original discussion is only valid if the hole drills from pole to pole.
If you fall from a point on the equator to an opposite point you have the problem of starting off moving sideways at 1000 mph in one direction and ending up some 90 minutes later at the other side moving at a 1000 mph in the opposite direction.
The entire journey would involve bumping against the sides of the tunnel at an average accelleration of 2000 mph in 1.5 hrs or 1333 mph in 1 hr which works out at 0.3 mph per second
As a side note, assuming you could travel sideways in a hollow shell-world, you would end up in some sort of orbit around the centre. You would never be able to cross the dead centre
We’ve discussed it many times before, but I notice that he stipulates a frictionless interpolar tube–just to avoid those complications, I’m sure.
That’s only 2% of Earth gravity, right? Maybe up to 3% at the center.
Not sure what you’re assuming here. Are we still using a frictionless interpolar tube? One that empties out into a vast spherical airless void? If so, I’d think you would go through dead centre.
What do you mean by “orbit”?
If there is air in that tube, you’re going to stop. I really don’t feel like pulling out my aero books for drag equations, because it doesn’t really matter.
If the mathematical assumptions relating drag to velocity are absolutely correct and apply perfectly to the real world, then yes, speed approaces zero asymptotically over time. But damped harmonic motion is very real and very well understood, and anyone can tell you that it will eventually settle to zero because of real-world frictional effects. And my guess is that it would take a fairly short time to come to rest in that interpolar tube.
zut said:
I think that pretty much sums it up. I find that any argument can be effectively halted by the introduction of the word “nonlinear.”
As another point, the OP said:
Time and distance are not infinitely divisible. At least not distance, anyway. Once you get to the Planck length (about 1.616 * 10^-35 meters), space-time loses all smoothness, and the very concept of length measurement becomes meaningless.
I’ve refuted this claim dozens of times in the past, but at the moment, I’m too tired, so I’ll just say that this isn’t true, so far as we know.
I was alway taught that if you fell into a hole through the earth, whether it was through the centre or not, you would fall down, come out the other side, and fall through it again, to get back to you’re original starting point. My astronomy professor also made a point of saying that the trip would take ninety minutes.
I mean, eventually, with enough times going through, I’m sure you would stop, but I think the velocity you build falling through the Earth, would keep you from stopping when you reached the centre.
Chronos, could you point me in the direction of your previous arguments? What I posted was the SOTA of theoretical physics as far as I knew. I assumed that getting my information from Hawking was good enough.
When did Hawking say this? Sometimes his books do tend to create misconceptions. The Planck length represents the length that quantum gravity should predominate (though it was orginally conceived as part of a unitless system of measurement rather than to reperesent a fundamental constant), though there isn’t a working theory of this at the moment. IIRC measuremnts of interfernce from far off em sources show that any quanitzation of space must take place at a smaller scale than the Planck length.