http://pogue.blogs.nytimes.com/2006/12/11/...mill-conundrum/

An interesting conundrum. Planes need lift. Lift is generated by forward motion. No /forward/ motion no lift.

Well, not quite. Air passing over a wing will lead to lift, as in a wind tunnel where the wing/aeroplane is stationary. Incidentally, the Bernoulli principle (shape of the wing thing) does not explain lift, or aeroplanes could not fly upside-down, which some can of course.

Where in the hell is Rick?

The answer is simple. The plane would not fly.

Culture is correct; planes need forward motion to obtain lift.

If the treadmill negated all forward motion, the plane would not generate thrust. So, unless the winds exceed the threshold, the plan will not fly.


Some airplanes can alter their airfoils so as to fly upside down without stalling. Wings are not stationary.

After all, you wouldn't see a Boeing 747 flying upside down.

The airplane will fly. Forward motion is provided by the acceleration due to engine thrust. Without the brakes on, the wheels spin freely, so the treadmill has no effect.

By the way, I'll bet that an unloaded (empty) 747 can fly upside down (assuming the engines can run upside down).

Ok, I must admit, my response was somewhat hasty.

If you sit down and think about it, the question is actually quite complex.

In the end, here's the question that puzzles me:
If the plane moves forward, can the treadmill and the wheels possibly be moving at equal and opposite speeds without skidding?

Many people cite the child on a treadmill example:
A child with rollerskates is on a treadmill.
Even though the treadmill is moving backwards, if you push on the child's back (thrust), the child will move forward.
This is a true statement.

This is where most people stop thinking about the problem. Clearly the plane could move forward even if the treadmill is moving backwards. Right?

Of course, there is a problem with this logic. The wheels on the child's skates will still be moving faster in the forward direction than the treadmill's motion in the opposite direction! Does anyone disagree with this statement?

The problem requires the wheel speed and the treadmill speed to be the same.

Are you so certain it could accelerate the plane? Or would it just increase the treadmill's speed accordingly?

I'm still leaning towards no, but I'm no longer certain of my answer.

Very certain. With free spinning wheels, the only forces that can be transmitted from the treadmill to the plane are side-to-side and vertical. As long as the plane is steered straight ahead, there will be no side force (or in the real case of a side wind, wheel reactions need to be controlled to keep the plane on the treadmill by using nose wheel control), and as long as the plane is not yet airborne, the vertical force will support the plane on the treadmill. This true for both propeller planes and jets.

Maybe. Either way, I wouldn't want to be the co-pilot!!! The angle of the airfoils may permit it, but the wing design would likely work against it. I wouldn't take my chances in something that big.

Ok, then I have a question for you:

Your hypothetical plane is sitting on the treadmill runnway. Just for kicks, we release the brakes and set the treadmill in motion. Would you argue that the plane will stay in the same place because of it's freely rotating wheels?

If it moves, wouldn't there have to be a force acting on it?

Unless the pilot compensated by running the engines faster for more thrust, the airplane would move slowly backwards due to wheel friction. They may spin freely, but they are not entirely without friction.

On flying upside down, did you know that a B52 can do a barrel roll? If a fully loaded 747 is safe in air turbulance, you can bet the wings will stay on in calm air, unloaded, and upside down.

When I worked on the 777 design build team in the early 90's they took the first prototype which was locked in a tool Jig called the "Iron bird." They bent the wings until they snapped and they almost bent 90 degrees before they gave.

The wings are extremely strong, not only do they support the frame in flight but they contain the fuel cells and support the weight of the fuel and the engines on the ground.

When Tex Johnson flew the first 707 over lake washington he did an impromtu barrel roll and suprised the shit out of everyone. Any airplane should fly upside down.

So you've always had ambitions on heaven!
But not due to the Bernoulli Principle ( http://www.seed.slb.com/en/scictr/watch/sa...EU_sail_02.html ). Otherwise, a fixed wing, upside-down in an airflow, would descend. Generation of lift by an airfoil can be simply explained by the downward deflection and acceleration of air, and is derived from the angle of attack, flow velocity and air density.

ps Are all aeroplane engines able to operate upside-down? Is it a requirement?

http://www.darlugo.com/?id=24&p=22#

Let me try to explain this one more time. I'll try to be very thorough. If you disagree, please tell me which point you disagree with so I can better understand your point of view.

When I ask "do you agree?" please take your time to think about the given information before advancing if you do not agree.

Assumptions (not given in the problem)

• The wheels do not skid
• The wheels rotate freely.
• The treadmill INSTANTANEOUSLY matches the plane's wheel speed.
• The engines generate negligible vertical thrust.

These should approximate the given problem. Do you agree?

Note: I'm ignoring friction at the wheel bearings because it does not directly affect the problem. You can add it back in if you'd like.

Is the plane's motion connected to the wheel's motion? - YES
Ignore the plane and treadmill for a second, and think of any wheel resting on the ground. If the wheel rolls forward, it's center of mass must move forward. Do you agree?

Alright, now let's add the plane back in. The plane is directly connected to the wheel's center. If the wheel's center advances, the plane must advance. Do you agree?

Here's the hard part. Think carefully about this. If the plane advances, it's wheels will rotate forward (or it would skid). The reverse is also true. This has to do with the wheel's center of mass.

Clearly, we can see that the rotational motion of the wheel is directly translated into linear movement in the plane so long as the wheel does not skid. (Wheel rotation forces the center of mass to move. This forces the plane to move.)


Can the plane advance on the treadmill? - NO
Here's the problem: in order for a wheel to move forward without skidding it must move faster than the surface it's sitting on.

Here's how it works:
The engines will provide thrust. The thrust will apply a linear (forward) force on the bearings of the wheels. This force will translate into rotational motion. THEN (instantaneously) the treadmill will provide an equal and opposite force. This force will oppose the rotational movement and will try to move the wheel's center of mass backwards. This will in turn oppose the force on the bearings. The plane will not be able to move.

In essence, the treadmill will directly counter the force generated by the engines. So, if the pilot increases the throttle, the treadmill will speed up.

Can the plane take off? - NO
Without forward motion, the plane will likely remain on the ground.

Can the treadmill generate winds which allow takeoff? - NO
Don't forget that the pilot can directly control the speed of the treadmill. This treadmill is very large and will likely generage wind. If this wind was able to generate lift, the plane may be able to break contact with the ground. This doesn't mean it could take off. If the plane lifted off the treadmill, all winds would cease VERY quickly and the plane would likely fall back down and repeat the cycle.

Can a plane take off on a treadmill runway? - YES
If the treadmill is moving backwards, the wheels of the plane will move forward at a speed greater than that off the opposite treadmill speed. More work will be required for takeoff.
If the treadmill is moving forward, the wheels on the plane will still move forward at a speed greater than that of the treadmill. Less work will be required for takeoff. (Maybe none if the treadmill is going fast enough.)
Does this make sense? But if the forces are equal and opposite, the wheels can never advance. Thus, takeoff cannot be achieved.

The forces transfer like this:
Engines (linear) -> Plane (linear) -> Tires (linear) -> Tires (rotational) -> Treadmill (rotational)
The key is knowing that a rotating tire exerts a linear force as well.


Wow! That's really impressive.

The curvature of the wing can be compensated for by altering the position of the flaps.
This is how it banks in a turn changing the airflow on the surface of the wing.

Airplane engine designs are engineered to operate under extreme conditions, not limiting the movement of the airplane due to its need to maintain an upright position.

Thrust is not relative to the ground it is relative to air pressure. If you are at 30,000 ft and the wheels are up and the engines are operating it is not the air rushing against the wing on its own that the plane flies.
The treadmill moving would be like the plane in the air. Thrust creates forward movement in the air as well as on the ground. The engines do not create vertical thrust they create forward thrust. They would create vertical thrust if the plane was pointed upward in which case the treadmill theory would have to include the treadmill being vertical rather than horizontal.

Whether there are wheels or skids it doesn't matter if the thrust can compensate for the drag created by the friction. Wheels create little drag even if they spin at a faster rate.
Unless the wheels are bigger than the airplane the center of mass is not at the axle of the wheel.
If the wheels were bigger than the airplane and the center of mass was at the axle the plane wouldn't fly even if your dropped it from 30,000 feet with a rocket attached to it because the airflow over a sphere would not allow a controlled flight, unless you call falling flying.


As long as were talking a normal airplane rather than a hypothetical airplane with giant cartoon wheels it's gonna fly.

Picture another scenario.

If the treadmill is big enough the drag that is created on the surface of the treadmill is going to create airpressure and drag air across and under the wing lifting the airplane but the treadmill would have to be huge.

If the treadmill is your basic health store treadmill and the airplane a 747 the wheel is going to be larger than the tread, the weight of the plane too heavy to allow the treadmill to turn and the plane will roll forward dropping off of the treadmill before continuing to taxi to take off.

You misunderstood me.


I was not referring to the plane's center of mass. That is why I said "Ignore the plane" for a second. I agree, a plane would be absurd if it's center of mass was located at the wheel.

I'm merely trying to illustrate how the wheel's center of mass is a point where a force is applied to the plane.


That's an interesting assumption. What makes you say that?

The forward thrust is created by the engines creating pressure against the atmosphere. This forces the airplane forward creating surface tension as air crosses the wing creating lift.
The treadmill doesn't have an affect that can counter the thrust unless the treadmill is so huge and turning so fast that the drag created from the surface of the belt is enough to force air against the plane, to lift the plane and throw it backwards. The wind pressure would have to be at a high hurricane force to throw it back and greater than the thrust.
But if it creates enough drag to create wind it isn't going to hinder the plane from creating lift its going to aid it.

Unless the experiment is done in a vacuum the airplane will fly.

In response to the ignoring of the plane then you would also have to ignore the question because the wheels are not going to slow or impact the thrust unless you make the wheels bigger.

We are talking about a normal airplane right?

How big is the treadmill and how fast is it turning?

No, the wheels will not create thrust. But it will transfer the force generated at the engines.


Here's the problem directly taken from the link:

“Imagine a plane 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 plane take off?"

Just for kicks, I asked a few of my engineer friends and a physicist. They all agree that the plane will not take off. I have another friend who is an aerospace engineer. I suppose I could ask her for her opinion. (She's very bright)

More than the problem itself, I'm facinated by everyone's thought process when attempting to solve this problem.
How are you guys arriving at your answers?

• Formulas?
• Real world observations?
• Gut Instincts?
• Other People's Opinions?

The physicist solved it using formulas and "ideal/perfect" approximations.
The engineers took a practical approach and focused on friction and forces.

And most people (like me) seem to quickly arrive at an answer and then rethink it once someone brings up the free-spinning wheel argument.

All of the force generated by thrust is transfered to the wheels....



But none of them actually took an airplane and put it on a runway sized treadmill and took their ideas beyond theory.

Did the physicist include the air flow coming from the drag created by the belt moving at the speed of the wheels? And how fast were the wheels moving?

Lets say the airplane travels at 600 mph and the belt is travelling at 600 mph. What is the wind turbulence created at the level of the wings in terms of air speed from the drag created by the surface tension of the belt at that speed?

Just curious how the physicist or engineer really approached this problem?

I think there might be some confusion on how airplane engines work. They don't drive the wheels as a car's engine does. If the same thought experiment is done using a car instead of an airplane, the car could never get any airspeed because the treadmill would match the wheel speed. If you add a propeller to the car, it can then move forward through the air if the driver puts the wheel transmission in neutral.

Try thinking in terms of (row) vectors.

The variables and functions:

x = time
P(x) = plane's thrust
T(x) = treadmill's speed
L(x) = plane's lift

Plane's velocity vector = [ P(x) L(x) ]

Treadmill's velocity vector = [ - T(x) 0 ]

(1) Note that P and T(x) are dependent upon x, time, which means these quantities may change.
(2) Also note that P, T, and L are positive, which means that
> The plane is going this way --->
> The treadmill is going this way <--- (look for the minus sign in the treadmill's velocity vector).
> The plane is going up or not. (In which case L(x) = 0 for all x ...)

Think about what is necessary for the plane to take "flight." It needs to come off the ground, which means it needs lift. But not enough lift will not get it into the air.

What gives rise to the quantity of lift? Air needs to flow up and under the wings, which is what the wings are designed for. Denote the necessary speed at which the air speed relative to the wings must be at in order for the plane to take off as S. The only reason the air would "move past" the plane this way is if the treadmill dragged enough to make enough wind. So, make the premises that this does not create enough wind to lift the plane, and that there is no extraneous wind.

In which case the only other possible way air could be passing the the plane is if the plane is moving forward relative to the earth (and wind). This means that there exists time interval I such that

ʃ_I ( P(x) - T(x) ) dx > S,

so that the plane accumulates enough speed relative to the wind to go up and under the wings, giving it enough lift to make to take flight.

[Note this is integral calculus, and that this integral gives the total positional displacement relative to the earth over the time interval, and therefore relative to the wind]

If you think into the mathematics a little more, you can understand the possible cases and whether or not the plane would fly.

[Sorry for the long, possibly incoherent rant ]

Actually, P and L are forces, and the velocity at any time is the integral of the acceleration with respect to time, evaluated for the period in question.

Furthermore, the net vertical force is the lift minus weight of the airplane (W): L(x) - W.

To find the acceleration, apply the mass of the airplane (W / g (where g is gravitational acceleration)) to Newton's Second Law of motion, acceleration equals force divided by mass.