A harmless riddle

bmanbs2 · July 17, 2006, 9:26pm

This is what sky, who’s theory I am now supporting as opposed to my prop one, said when I asked him to post here for all of you from the Engineering Majors thread:

bmanbs2,

As much as I would like to, I don’t think I’d have the energy to enter another debate on this topic.

All the debate in the MIT thread is meaningless. The simple fact of the matter, this problem can be simply solved using the first thing they teach you in high school physics… using a free body diagram. An airplane will have 4 forces acting on it, thrust, drag, lift, and weight (due to gravity). Lift can be ignored for this problem. Let’s assume a stationary airmass along the “runway.” Obviously if you have a headwind, you can take off with a lower ground speed, but assuming it is stationary will make things easy (and is a perfectly fine assumption. Zero wind is a great condition for pilots). Aerodynamic drag will not become a significant force during the initial roll, so that can be ignored at first. Thrust of the engine will be assumed constant as a function of velocity. The only remaining forces we are looking at are the thrust of the engine and the frictional forces due to the landing gear. Without even looking at the mechanics of the frictional forces, I think it is pretty clear that the forces due to friction will be much much less than the forces due to the propulsion system. The net force in the longitudinal direction can mean only one thing in Newtonian physics: an acceleration. The airplane will continue to accelerate until the forces balance (no net force = no acceleration). Aerodynamic drag will certainly increase with velocity (squared), but the frictional forces at the tire and in the wheel bearings are dependent primarily upon forces acting normal to them (based on the weight), which will decrease as the airplane starts producing lift that opposes the weight. These frictional forces generally are independent of velocity. “Startup” forces that get bearings spinning are much higher than any friction that is caused by spinning the bearings faster (as what would happen when the treadmill speed increases).

Remember how we assumed a stationary airmass? As the airplane velocity increases (since it is accelerating down the runway), there is now a airflow over the wings which produces lift. Even if you don’t assume a stationary airmass, then you will get another velocity component between the wing and the airmass, but regardless, the airplane’s velocity is generally much higher than the airmass.

And with respect to your puller prop statement… It could help… but only in a certain configuration. The prop wash in a tractor (ie. puller) configuration can increase the lift a wing generates if the wing is in the slipstream of the propeller. Most pilots will know that the “power-on” approach can result in a lower airspeed as compared to the “power-off” approach. This is because a propeller does increase the airflow over the wings, and therefore does produce a bit more lift. However, for this problem, this fact is pretty much irrelevant.

Well… dang! It looks like in the time that it took me to write this, I could have posted the same message in the MIT forum. Regardless, if you want, you can link them to this post, or cut and paste it yourself.

I’ll also add my credentials as a pilot and aerospace engineer if it helps add any weight to my argument.

lowofo · July 17, 2006, 9:44pm

bmanbs2:

What if that rocket-engine equipped cart was on a treadmill running backwards with an acceleration matching that produced by the rocket engine if it were on stationary ground? There we go again.

At this point, the question isn’t really whether it’ll fly or not, it’s whether it will move forward, because if it will move forward, the hard part’s done.

Sky:

“I think it is pretty clear that the forces due to friction will be much much less than the forces due to the propulsion system.”

Static friction between the landing gear tires and the “runway” causes the tires to spin, it does not cause a “drag” force on the aircraft, per se.

The way I think of it, the “moving runway” is equivalent to having the airplane start off rolling backwards on a “stationary runway”, as opposed to not moving at all. Sure, if it starts rolling backwards at a constant velocity, and the thrust, always produced forward acceleration, will overcome that negative velocity, achieve positive velocity, lift, etc.

But the thing is, the treadmill, as confirmed by the OP, accelerates backwards at exactly the same rate, at every instant, as the airplane’s thrust might be accelerating the craft forward; I say this for perhaps the fourth time, if that is the case, it is impossble for the airplane to accelerate anywhere from the frame of reference of some stationary point off of the treadmill. If you were on the treadmill, sure, the airplane would be accelerating forward (relative to your position on the treadmill), but its the former we are concerned with, as air does not move with the treadmill.

I am a high school graduate from one of the nation’s worst public school systems, if that helps add any weight to my argument.

Slorg · July 17, 2006, 9:50pm

Wow. You guys are stupid. Assuming the plane has wheels, it obviously flies.

Spartan_Pho3nix · July 17, 2006, 9:53pm

Hm yah. It’s pretty obvious when I actually thought about the question. It’s a plane. The wheels spin freely, the taking off depends on the air.

lowofo · July 17, 2006, 9:57pm

So then if the aircraft is placed on the moving treadmill, the wheels spin and the airplane stays still? I would like to think the wheels don’t spin, and the aircraft moves back with the treadmill.

I don’t even know what I’m talking about even more. All I can say is, with respect to any part of the treadmill surface, yes, the airplane will “accelerate,” with respect to the ground, and the still air, the airplane, I believe, stays still, no flight.

Slorg · July 17, 2006, 9:58pm

“If you push an airport cart the wrong way on a moving walkway at the same speed as the walkway, it won’t move anywhere with respect to the air.”

This example doesn’t support your side at all. Pretend you’re standing with the card 10 feet away from the walkway entrance. The walkway is moving at 5 mph. You run at 5 mph for 10 feet and release the cart. It will keep rolling down the walkway at 5 mph. It won’t magically freeze once the wheels touch the walkway. Instead, the wheels will just start spinning faster. And, depending on the efficiency of the bearings inside the wheel, the cart will gradually slow down.

sky · July 17, 2006, 10:02pm

lowofo,

You are confusing speed and forces here. Two totally different things. You are assuming that the force the runway exerts on the airplane is the same force that the propulsion system on the airplane is producing. (Equal in magnitude, opposite in line of action). That is simply not the case. The friction of the tires on the runway, and any other forces that would oppose forward motion (bearing friction, inertial effects, etc etc) are NOT dependent (only in higher orders) on the speed of the airplane moving down the runway (if you assume that lift is staying the same, which isnt a bad assumption since the airplane doesn’t really produce a lot of lift until it rotates and becomes airborne). Any forces due to the runway (whether stationary, or a treadmill as in this case) will be relatively constant no matter how fast the airplane (and therefore the treadmill is moving). Rolling friction, bearing friction, and other forces that oppose the forward motion of the plane are independent of the speed; they depend only on the frictional coefficients and the loads the tires and bearings are experiencing (which will remain constant since the plane doesn’t change weight as it accelerates (at least measurably… sorry Mr. Einstein)…

Slorg · July 17, 2006, 10:03pm

“So then if the aircraft is placed on the moving treadmill, the wheels spin and the airplane stays still? I would like to think the wheels don’t spin, and the aircraft moves back with the treadmill.”

Go find some roller skates and a treadmill. Get on the treadmill. Hold the bars on the treadmill. Turn the treadmill on. Remember what happens. Turn the treadmill off.

Now take the roller skates off. Lie down on the treadmill. Grab the bars. Turn the treadmill on.

While wearing roller skates, the force needed to hold you still is very low. Even if you turned the treadmill speed up to 60 mph, the force would be low.

This is not the case when you’re dragging yourself on the ground. The force required to hand on while the ground is moving at 60 mph is much larger.

lowofo · July 17, 2006, 10:15pm

Slorg:

Yes, I understand wheels were invented as a means of combating friction. How about this. Put on some skates. Go stand on a treadmill. Don’t hold on to the bar. Turn the treadmill on. Do you go back with the treadmill, or stay where you are, having only the wheels on your skates roll?

sky:

I’m gonna have to think on that one. What say you, if a plane is placed on a treadmill running backward; does the plane go back with the treadmill, no wheel spinning whatsoever, or does the plane stay stationary relative to the observer, wheels spinning at a rate as would be induced by the velocity of the treadmill? I think that’s the source of our discontent. Half of us think the plane goes back with the tread, the other half thinks it stays put. From that comes everything regarding flight and no flight. If you don’t agree with anything I’ve said before, at least agree with these last two sentences.

Slorg · July 17, 2006, 10:21pm

“Yes, I understand wheels were invented as a means of combating friction. How about this. Put on some skates. Go stand on a treadmill. Don’t hold on to the bar. Turn the treadmill on. Do you go back with the treadmill, or stay where you are, having only the wheels on your skates roll?”

I slowly move backward (because wheels aren’t perfectly efficient). But it’s much more slowly than the treadmill is moving. Similarly, the force that the treamill exerts on the plane is much, much less than if the plane didn’t have wheels.

How about this. Get in a car. Bring a basketball. Accelerate forward. Does the ball roll backward (relative to you)? Does it pretty much stay still relative to the Earth (except for spinning faster)?

lowofo · July 17, 2006, 10:25pm

Slorg:

You slowly move backward, or you move backward at exactly the same rate the treadmill is running backward?

As for your basketball analogy, the relative to me in the car, the basketball does not move. Relative to a stationary point on the ground, the basketball accelerates with the automobile.

Slorg · July 17, 2006, 10:25pm

Note that a car with wings wouldn’t take off. This is because cars use their wheels for thrust.

tech_fan · July 17, 2006, 10:26pm

I now buy what sky says. This riddle only gets people riled up because it tries very hard to make you think the treadmill operates to make the plane stand still (though of course it doesn’t say that). So, yes, I think it takes off.

I think, however, this requires the wheels to slip against the treadmill. If they don’t slip, there’s no way for the plane to move forward assuming the treadmill “rotates” at the same rate they do in the opposite direction.

Slorg · July 17, 2006, 10:27pm

"You slowly move backward, or you move backward at exactly the same rate the treadmill is running backward?

As for your basketball analogy, the relative to me in the car, the basketball does not move. Relative to a stationary point on the ground, the basketball accelerates with the automobile."

Much, much more slowly than the rate at which the treadmill is running backward.

Um. Ok, maybe your car has inertial dampeners. Get a wagon. Put a ball in it. When you pull the wagon, the ball appears to move backward. In fact, it starts spinning and moves a little forward relative to the Earth.

But the key point is that it doesn’t move forward as fast as the cart does. That energy is translated into rotation.

Similarly, the treadmill doesn’t move the plane backward at the same speed the plane is moving forward precisely because of the wheels. That force spins the wheels instead of moving the plane.

lowofo · July 17, 2006, 10:33pm

“Much, much more slowly than the rate at which the treadmill is running backward.”

How? Convince me this is true and your plane can take off.

As for the basketball, if friction was so great that the ball was as good as stuck to the car, what I said is perfectly fine. If it were less, what you said would probably be true, to varying degrees. No need to involve Star Trek.

EDIT:

Slorg, there it is! Ah, goddammit. Yes, so work done by static friction from treadmill is converted into KErotational and KEtranslational. Goddammit. So, I have been assuming this would be completely transformed into KEtranslational. If that were so, the plane would not take off.

You assume, and correctly so, that it will transform into some mixture of KErotational and KEtranslational. As such, the plane will eventually take off, but much only after much greater time. Under these circumstances, the treadmill would have to out-accelerate the potential acceleration of the thrust of the aircraft in order to keep it from moving forward (relative to something stationary).

Slorg · July 17, 2006, 10:36pm

If the efficiency of wheel bearings was 100%, treadmills wouldn’t affect wheeled objects.

For the treadmill force exerted on wheeled objects, the only thing you need to accept is that wheel bearings have greater than 0% efficiency.

(ignoring static friction)

Slorg · July 17, 2006, 10:39pm

Get two cans of food and a rug. Put one on its side and the other on its end. Pull the rug. The one that can roll will move less because it starts spinning.

lowofo · July 17, 2006, 10:39pm

Slorg, there it is! Ah, goddammit. Yes, you’re right. So work done by static friction from treadmill is converted into KE rotational and KE translational, on the wheel. Goddammit. So, I have been assuming this would be completely transformed into KEtranslational. If that were so, the plane would not take off.

You assume, and correctly so, that it will transform into some mixture of KE rotational and KE translational (like you said, dependent upon bearing efficiency). As such, the plane will eventually take off, but much only after much greater time. Under these circumstances, the treadmill would have to out-accelerate the potential acceleration of the thrust of the aircraft in order to keep it from moving forward (relative to something stationary).

Would you agree with that, Slorg?

River_Phoenix · July 17, 2006, 11:05pm

Ben: The statement of the riddle implies the treadmill keeping in speed to match to plane, that much is clear. If somehow the author of the riddle intended it as a scam and the treadmill has constant speed, then the plane obviously takes off and it’s not a very good riddle, so we should answer the other one.

Having said that, here’s another bad answer: Yes, the plane can take off, it just doesn’t in this case.

tech_fan · July 17, 2006, 11:40pm

The plane cannot move forward if the wheels don’t slip. This is not even a physical fact, just a mathematical one. Write down equations for the motion of the treadmill, the wheel (a point on it will suffice), and the plane, and see if you can make the plane move forward without the wheel slipping.

Since the wheels are assumed not to slip, it does turn out that the various forces exerted around the wheel can counteract the forces of jet engines and keep the plane from moving. One of the simplest facts you learn in early physics is that tiny objects can exert huge forces (third law). So apparently sky was wrong in his intuition – even engineers and pilots can be wrong.

In reality, any realistic wheels would slip and the plane would move forward, but that isn’t very interesting.

A harmless riddle

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