Weird Free Body Paradox

[mezarashi]

I was doing a little thought experiment myself and thought maybe I could use some help. The rod here is an ideally rigid rod, and the tracks are of course frictionless and fixed. How does this configuration suspend itself up while gravity is pulling on it and there are no vertical forces to counter it?

 

P.S. excuse my horrible paint skills again

 

Additional Explanation: The two large circles are fixed tracks. The rod in the middle has two wheel rollers attached to it. Each roller is in one of the tracks. Because the rod is rigid, it won't allow the wheels to be separated anymore than they are. The red lines are just force vectors showing the tendency for movement of the rod. Intuitively we know that such a configuration will be more or less static, but given the ideal conditions stated above, what is the analysis behind it.

[Severian]

You are almost correct.

 

Imagine first a not quite rigid body. The system cannot sit in this completely up-down symmetric position. What will happen is that the body will move down very slightly, so that the rail pushes the wheels outwards. but since we are slightly off the central point (ie. the height where the distance between the rails is smallest) the rails are at a very slight angle. So part of this force is pushing upwards and it will balance out the force of gravity. Where this point is depends on how rigid the body is.

 

Now for a rigid body. If the body is entirely rigid then the wheels cannot turn (since they are rigid) and the body cannot move. If you allow the wheels and rod to be separately rigid, then you need to say that the wheel is running frictionlessly, but as the rod moves down it will exert an increasing sideways force on the wheel and any definition of 'frictionless' will at some point break down (otherwise the rod would pass throught the wheel!).

 

In other words, the premises of 'rigid' and 'frictionless' don't work in the case you demonstrate because they would require the object to fall apart.

[calbiterol]

Hmmm. Mezarashi, or Severian, could you explain the diagram to me? I'm rather interested, but I don't know what's what in the diagram. Alternatively, is there anything I could search for about this setup?

[mezarashi]

I put some additional explanations, hope that helps. Feel free to post what you think. I think Severian has already posted an interesting conclusion? It may serve to say that that sometimes the physical systems we know in the real world cannot be modeled ideally.

[calbiterol]

Ahah. Thanks, the extra explanation makes much more sense to me. One question, though, are the wheels on the inside of the tracks spherical or cylindrical?

[mezarashi]

Honestly I never knew you could get spherical wheels, but as this is a 2 dimensional problem of course, we can just assume that they are cylindrical.

[calbiterol]

I've never seen them either, but... I'm sure there's a way. Somehow. That could make for an interesting problem right there.

[Severian]

I have seen a prototype of a car with spherical wheels. It allowed the car to move sideways (well, at any angle I suppose) which looked really handy for parking. I have no idea how they worked though.

 

(Didn't the Audi in 'I Robot' also have spherical wheels?)

[calbiterol]

Yep, at least that how I saw it. I actually liked that movie, it was an interesting way of interpreting and synthesizing Asimov's stories.

[ydoaPs]

i am not sure about the cars, but the big trucks that carried the robots did.

[calbiterol]

Do you think you could use magnets to both push them away and pull them in at the same time? Like with this? Only problem is, no way to rotate thewheel. Hmm...

[Edit: I figured I should explain this more. The spherical halbach array (the "wheel") is attracted by the hemisherical, fixed halbach array in the vehicle. The other magnet repels the "wheel" when it gets close to it - it would have to have very dense field lines, though. Alternatively, the outer magnet could also be a halbach array, but still repelling the "wheel." Perhaps a halbach array made out of superconductor magnets?]

[J.C.MacSwell]

I was doing a little thought experiment myself and thought maybe I could use some help. The rod here is an ideally rigid rod' date=' and the tracks are of course frictionless and fixed. How does this configuration suspend itself up while gravity is pulling on it and there are no vertical forces to counter it?

 

[img']http://img233.echo.cx/img233/1931/12410yi.gif[/img]

P.S. excuse my horrible paint skills again

 

Additional Explanation: The two large circles are fixed tracks. The rod in the middle has two wheel rollers attached to it. Each roller is in one of the tracks. Because the rod is rigid, it won't allow the wheels to be separated anymore than they are. The red lines are just force vectors showing the tendency for movement of the rod. Intuitively we know that such a configuration will be more or less static, but given the ideal conditions stated above, what is the analysis behind it.

 

If it is perfectly rigid and frictionless an infinite force is produced horizontally.