Airplane on a converter belt question.

[kbfc]

The primary reason a rolling wheel goes slower than a sliding one is because of energy and momentum conservation, not friction.

Do an energy balance and you will see this is clearly true. For the same potential energy starting conditions, a rolling wheel converts into both rotational and translational kinetic energy/momentum. Therefore the translational kinetic energy is necessarily smaller -> velocity is smaller.


Anyway, my whole point in this is that the problem is answerable without obfuscating everything with irrelevant concerns. With non-slipping wheels, there is no kinetic friction going on, as there is no relative motion between surfaces, which would be a variable one might actually care about. There are a whole host of small things going on that could affect matters. The problem can be shown to be a contradiction without even addressing this issues, though. If you were solving for something like how long it would take to fly or how fast the wheels would be turning at liftoff, you might be interested in dealing with rolling friction. That wasn't the concern of the problem.

[Sharkey]

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The primary reason a rolling wheel goes slower than a sliding one is because of energy and momentum conservation, not friction.

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The only force tending to increase the angular velocity of the wheels is tire-to-belt friction.

[kbfc]

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The primary reason a rolling wheel goes slower than a sliding one is because of energy and momentum conservation, not friction.

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The only force tending to increase the angular velocity of the wheels is tire-to-belt friction.

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I think we're getting frictions confused here....I meant that there wasn't any kinetic friction between the wheel and belt that was causing it to slow down. Whatever....not important. I gotta go. This whole sidetrack is irrelevant and I just proved how retarded I am. I initially posted on it to make the point that it was a waste of time.....and then I went and wasted a bunch of time.....

[Matt R.]

Sweet. We made it to 300 replies. [img]/images/graemlins/heart.gif[/img]

[ QUOTE ]
The primary reason a rolling wheel goes slower than a sliding one is because of energy and momentum conservation, not friction.

Do an energy balance and you will see this is clearly true. For the same potential energy starting conditions, a rolling wheel converts into both rotational and translational kinetic energy/momentum. Therefore the translational kinetic energy is necessarily smaller -> velocity is smaller.

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Yes, but the entire reason that the wheel is rotating is because the static friction imparts a torque on the wheel. Without the static friction, the wheel would slide and it would have zero rotational kinetic energy and momentum. Of course you can also look at the KE and momentum to determine the rotational dynamics of the wheel. Both ways are mathematically equivalent and equations which describe the motion using any of these concepts can be derived from one another. Just note that force = dp/dt (time rate change of momentum) and K2 - K1 = Work = Fd (change in kinetic energy is equal to force dot displacement). So force is an integral part of all of these concepts.

I agree with your last paragraph. The only reason I mentioned friction and complicated matters was to address those people questioning what force could possibly be opposing thrust in such a scenario. It's unreasonable to assume f could be large enough to stop an airline at full thrust though, which you've pointed out. My intention was simply to show that there *was* a frictional force, not that its magnitude could equal the thrust in the real world.

[Raydain]

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I've thought about this problem for who knows how long and I recently did a presentation which involved explaining the answer to this question. Let me attempt to give a clear answer and put this issue to rest.

The answer is: Yes.

Why? Because a plane generates speed (thrust) from its engines; it does not step on the gas so that the wheels turn. As a matter of fact, the wheels surve no purpose other than to reduce friction. First, imagine this scenario: The speed of the wheels is zero. In other words, the runway doens't move backwards (it moved at speed zero) and the wheels don't move either. The plane turns its engines on. If they can provide enough force, the plane will just begin to slide on the runway until it slides fast enough to take off.

From this, you can see where I'm going. If the wheel speed was, say 100 mph, the runway would move back at 100 mph. But this doens't mean that the plane itself couldn't move faster. If the plane travels at 250 mph, the wheels basically compensate for 100 mph of that and the rest is just from sliding.

So, basically, yes the plane can take off if its engines are powerful enough to overcome the friction of the "still" wheels.

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Didn't this question get completely answered on the first page of this thread? No matter how much the engines speed up the plane, the conveyor belt will match the speed of the wheels. But the plane will move faster than the speed of the wheels. Since the plane will push the plane forward, the wheels will not neccesarily move as fast as the plane moves, so the conveyor belt will not match the forward speed of the plane.

SUMMARY
The conveyor belt will match the speed of the wheels
The plane will move forward regardless of the fact that the speeds of the wheels and conveyor belt cancel each other out.
The plane will be skidding across the conveyor belt.


One last point:
If instead of a conveyor belt we had wheels that never spun. They were locked into place. The plane would still be able to move forward during a takeoff even though the wheels wouldn't spin. This is the same situation as the conveyor belt question.

[kbfc]

Ok....this long ridiculous thread seems to be coming to an end. hurrah!

[Sharkey]

[ QUOTE ]
SUMMARY
The conveyor belt will match the speed of the wheels
The plane will move forward regardless of the fact that the speeds of the wheels and conveyor belt cancel each other out.
The plane will be skidding across the conveyor belt.


One last point:
If instead of a conveyor belt we had wheels that never spun. They were locked into place. The plane would still be able to move forward during a takeoff even though the wheels wouldn't spin. This is the same situation as the conveyor belt question.

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“The conveyor belt will match the speed of the wheels”

Correct.

“The plane will move forward regardless of the fact that the speeds of the wheels and conveyor belt cancel each other out.”

Incorrect. If wheel speed = belt speed, plane speed = 0.

“The plane will be skidding across the conveyor belt.”

Says you. There’s nothing to that effect in the OP.

“If instead of a conveyor belt we had wheels that never spun. They were locked into place. The plane would still be able to move forward during a takeoff even though the wheels wouldn't spin. This is the same situation as the conveyor belt question.”

Not if the belt continues to accelerate as long as the plane is moving forward.

[suzzer99]

Please forgive if someone had already posted this link. I didn't read all the middle pages. I did search.

http://www.straightdope.com/columns/060303.html

[flatline]

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Please forgive if someone had already posted this link. I didn't read all the middle pages. I did search.

http://www.straightdope.com/columns/060303.html

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That article is about the question "what if the converter belt is moving at the opposite speed of the plane?" The answer to this is obviously that the plane will still take off.

The question under discussion is whether the plane will take off if the converter belt moves at a speed opposite the wheel speed, in which case the solution is undefined.

[Sharkey]

There two different setup conditions and two different solutions.

1. Belt speed = plane speed. The plane takes off with wheel speed = 2 * plane speed.

2. Belt speed = wheel speed (the OP). The plane does not take off. Belt speed increases until the backward force of friction on the plane equals maximum engine thrust, or the wheels disintegrate, whichever comes first.