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Here's a brain tickler (mathematical discussion)
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f1000
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Mar 3, 2007, 05:20 PM
 
If you aren't a professional physicist, mathematician, or aeronautical engineer, I'd prefer that you not post in this thread. I will ask the mods to delete posts as necessary to keep the discussion rigorous.


Imagine an airplane is sitting on a massive conveyor belt, as wide and as long as a runway. The conveyer belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Can the airplane take off?


Although the problem is ambiguously worded, it can be best interpreted by using the following free body diagrams.




If a wheel of radius r rolls along a stationary surface for a distance of s over time t, then

rθ(t) = s(t)
θ'(t) = s'(t)/r




If the aforementioned wheel instead rolls down a conveyor belt that slides a distance of d in the opposite direction over time t, then

rθ(t) = s(t) + d(t)
θ'(t) = [s'(t) + d'(t)]/r

If the belt's speed is also matched with the wheel's linear speed[1], then

d'(t) = s'(t)
θ'(t) = 2s'(t)/r

The wheel's rotational (angular) speed, θ'(t), is therefore doubled.


Since θ(0) = s(0) = 0,

rθ(t) = 2s(t)
rθ'(t) = 2s'(t)
rθ"(t) = 2s"(t)


<AT THIS POINT, ANYONE WITH A GOOD BACKGROUND IN PHYSICS SHOULD SEE THAT THE PLANE TAKES OFF. ADDING FORCE VECTORS WOULD BE PEDANTIC; HOWEVER, I INTEND TO DO SO TO COMPUTE HOW MUCH OF A DIFFERENCE A TREADMILL MAKES ON THE ACTUAL TAKEOFF TIME. THOSE WHO SAY THAT THE TREADMILL IS COMPLETELY IRRELEVANT ARE INCORRECT, BUT THEY ARE CORRECT IF THEY SAY THAT IT IS MERELY NEGLIGIBLE.

Fthrust(t)* - Fdrag(s') - Frollingfriction(θ',Flift) - Fbearingfriction(θ',Flift) - Ftorque(θ") = m(t)s"(t)
Flift(s') = Fgravity(t) (Takeoff condition)

*Will consider constant thrust (with brakes applied) and a ramp up of thrust from zero.





SOME NEEDED DATA

MASS OF AN AIRPLANE
THRUST OF SAID AIRPLANE
POUNDS OF FUEL CONSUMED TO ACHIEVE SAID THRUST BY SAID AIRPLANE
NUMBER OF WHEELS ON SAID AIRPLANE
MOMENT OF INERTIA FOR SAID AIRPLANE'S WHEELS<-
ROLLING FRICTION FOR SAID AIRPLANE'S WHEELS<-
BEARING FRICTION FOR SAID AIRPLANE'S WHEELS<-
COEFFICIENTS OF DRAG AND LIFT FOR SAID AIRPLANE?<-

ALL OF THIS DATA MUST COME FROM ONE AIRPLANE. DON'T QUOTE ME THE MASS OF A BOEING 747, WHICH I CAN LOOK UP MYSELF, IF YOU CAN'T TELL ME ALL OF THE OTHER RELEVANT DATA AS WELL.

SOME MORE POINTS FOR CONSIDERATION:

AS A PLANE TAXIS DOWN THE RUNWAY, LIFT REDUCES THE NORMAL COMPONENT OF FRICTION.>

  1. Some have suggested that the phrase "speed of the wheels" should be interpreted as meaning the magnitudes of their tangential velocities[2]. Since the tangential velocity vector at any fixed point on a rotating wheel is continually changing direction, how can a horizontally constrained belt be made to always move in a direction opposite to such a vector unless one is talking about using only the oscillating horizontal component of the latter? Similarly, if the belt's velocity were instead matched with the tangential velocity at some constant phase of a wheel's rotation, what phase should it be? If for the sake of argument one were to choose a phase of π (the apex of the wheel), then one would have to contend with the following tautology:

    d'(t) = s'(t) + rθ'(t)
    d'(t) = s'(t) + [s'(t) + d'(t)]
    s'(t) = 0

    In other words, the original word problem now becomes the following: if s'(t) = 0, can s'(t) ≠ 0? More subtly, if s'(t) = 0, then s'(t) = 0?. In plaintalk: if a wheel's linear speed is always zero, then is it always zero? The answer to this question is that of course it is, even if physically impossible, because one has simply defined it to be so.

    Ignoring the flaws associated with the above interpretations, is setting s'(t) = 0 even physically possible within the conditions of the original word problem?

    <TO BE CONTINUED>

  2. Interpreting tangential speed as being relative to the airplane or conveyor belt as opposed to the stationary air won't eliminate the tautology that s'(t) = 0.
( Last edited by f1000; Mar 6, 2007 at 08:38 PM. )
     
itistoday
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Mar 3, 2007, 05:37 PM
 
[neva mind]
     
brassplayersrock²
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Mar 3, 2007, 05:49 PM
 
forum search is your friend
     
analogika
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Mar 3, 2007, 05:57 PM
 
This aspect of that question has been bothering me:

Let's assume that the plane is on stationary tarmac. If the plane starts moving forward at X speed initially. This means that the wheels move at rotational speed x'.
Now X increases over time as the plane accelerates. Thus, so does rotational speed x'.

Now, the problem I have is the phrasing:
The conveyor belt matches the speed of the WHEELS, NOT of the PLANE.

So, in other words, if the conveyer belt speed is a function of rotational speed x', NOT of plane speed X, wouldn't that mean:

As X increases, x' increases. As rotational speed x' increases, conveyor belt speed f(x') increases. However, we now enter a feedback loop, since rotational speed is a direct function of speed over ground (conveyor belt), which is a direct function of rotational speed.

In other words: The way the question is phrased (probably not how it is intended), the instant the plane starts moving, wheels and belt start accelerating uncontrollably until either the conveyor belt mechanism gives or the wheel bearings overheat and lock up, dragging the whole plane backwards in a huge flaming fireball of glory.

So: no, it doesn't take off.
     
f1000  (op)
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Mar 3, 2007, 08:05 PM
 
Originally Posted by analogika View Post
So, in other words, if the conveyer belt speed is a function of rotational speed x', NOT of plane speed X, wouldn't that mean:
Rotational velocity is in units such as radians/second whereas treadmill velocity is in units such as meters/second. Meters and radians are not directly comparable.
     
analogika
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Mar 3, 2007, 08:08 PM
 
No, but in this case, treadmill velocity is a direct linear function of rotational velocity. (Since radians at a certain radius translate directly into circumference traversed.) As radians/second increase, so do meters/second.

I'm not comparing them, I'm saying they are dependent upon one another - AS THE PROBLEM IS WORDED (not in any sensible way).
     
f1000  (op)
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Mar 3, 2007, 08:11 PM
 
Originally Posted by analogika View Post
No, but in this case, treadmill velocity is a direct function of rotational velocity.

I'm not comparing them, I'm saying they are dependent upon one another - AS THE PROBLEM IS WORDED (not in any sensible way).
Sorry for the brevity of my response, but I had explained the difference between rotational and tangential velocities in the original thread and pointed out the existence of a tautology when using the latter.
     
analogika
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Mar 3, 2007, 08:13 PM
 
I confess I'm not terribly firm on the terminology involved.

That doesn't invalidate my post, though.
     
f1000  (op)
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Mar 3, 2007, 08:18 PM
 
Originally Posted by analogika View Post
I confess I'm not terribly firm on the terminology involved.

That doesn't invalidate my post, though.
I'll put up equations explaining the tautology later. Maybe.
     
Gossamer
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Mar 4, 2007, 02:05 AM
 
Originally Posted by brassplayersrock View Post
forum search is your friend
Yours too.

http://forums.macnn.com/89/macnn-lou...6/#post3316344

It wasn't a coincidence that he picked the exact same subject line...
     
brassplayersrock²
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Mar 4, 2007, 02:19 AM
 
that made no sense gossamer

edit:

or did you want me to include a link to what I was talking about to make things easier for people?
     
analogika
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Mar 4, 2007, 04:19 AM
 
Good idea, bpr.

I have no idea what you meant, either.
     
f1000  (op)
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Mar 4, 2007, 03:55 PM
 
Originally Posted by analogika View Post
I confess I'm not terribly firm on the terminology involved.

That doesn't invalidate my post, though.
I reread your posts and noticed that you had used the word speed, so you weren't wrong in trying to associate belt speed with rotational speed (tangential speed). Sorry, I was tired.
     
analogika
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Mar 4, 2007, 04:08 PM
 
So would you agree that the scenario described in the problem would result in a spectacular fireball?
     
f1000  (op)
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Mar 4, 2007, 04:22 PM
 
Originally Posted by analogika View Post
So would you agree that the scenario described in the problem would result in a spectacular fireball?
If s'(t) = 0, then given the physics of the problem the angular speed of the wheels seems to become directly proportional to time. This means that at some point any real wheel would disintegrate and the plane would fall onto the extremely fast moving belt below and be obliterated, ending up as a flaming ball of wreckage hurtling backwards down the runway. I'm still waiting for someone to supply me with some precise data before I come to that conclusion.

In the meanwhile, we could always load up the plane with pyrotechnics just to be sure.
     
christ
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Mar 4, 2007, 06:53 PM
 
The airplane can't take off, because lift is generated by the differential pressure caused by the differences in airspeed of the air travelling over the two surfaces of the aerofoil section of the wing - if there is no air flow over the wing, there is no lift.
Chris. T.

"... in 6 months if WMD are found, I hope all clear-thinking people who opposed the war will say "You're right, we were wrong -- good job". Similarly, if after 6 months no WMD are found, people who supported the war should say the same thing -- and move to impeach Mr. Bush." - moki, 04/16/03
     
Gossamer
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Mar 4, 2007, 07:24 PM
 
Originally Posted by brassplayersrock View Post
that made no sense gossamer

edit:

or did you want me to include a link to what I was talking about to make things easier for people?
Sorry if I'm mistaken. I was assuming you were telling him that there was already a thread about this and that he should have searched and found it before starting a new thread about the same thing. I was linking to his specific post in that other thread where he stated that he was going to start a new thread to discuss this issue purely in a mathematical sense.
     
Chuckit
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Mar 4, 2007, 07:26 PM
 
There's only one thing to do here: Somebody get a Boeing 747 and post a video of this experiment on YouTube.
Chuck
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analogika
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Mar 4, 2007, 08:37 PM
 
Originally Posted by christ View Post
The airplane can't take off, because lift is generated by the differential pressure caused by the differences in airspeed of the air travelling over the two surfaces of the aerofoil section of the wing - if there is no air flow over the wing, there is no lift.
     
brassplayersrock²
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Mar 4, 2007, 10:07 PM
 
Originally Posted by Gossamer View Post
Sorry if I'm mistaken. I was assuming you were telling him that there was already a thread about this and that he should have searched and found it before starting a new thread about the same thing. I was linking to his specific post in that other thread where he stated that he was going to start a new thread to discuss this issue purely in a mathematical sense.
good for you want a cookie?
     
   
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