Will a plane on a treadmill take off?

Factual Questions
121

I don’t disagree with that – simply saying that in a “realistic” scenario, the initial force would be applied by the conveyor, with the engines producing a small amount of thrust to make the forces sum to zero. Actually, now that I’ve thought about it more, it seems like I’m seeing things from about the same point of view as you are.

I’ll use an example of a 737 sitting on this conveyor runway. From on each wing hangs a CFM56-7 turbofan, capable of delivering 100kN of takeoff thrust. The captain sets the brakes, and moves the power levers to the takeoff position. 200kN of force is now directed forward, through the engine pylons, through the wing, through the landing gear struts, through the brakes, to the tires. Friction between the tires and runway produces the 200kN reaction that keeps the airplane stationary.

The captain now releases the brakes. The runway’s logic controller knows it must keep the airplane stationary to ensure v[sub]t[/sub = v[sub]conv[/sub]. Therefore, it must somehow apply a 200kN through the airplane structure to the engines. It can only accomplish this by applying a force to the tires.

If the wheel bearings are frictionless, this will indeed cause a runaway angular acceleration of the wheels (F = I*alpha/r). A “realistic” wheel will exhibit rolling resistance, which is usually considered independent of speed at low speeds. In this problem, however, I’d expect bearing friction to increase as wheel speed increases beyond the normal range, and ultimately place an upper limit on the conveyor speed. Once the system reaches steady state, a 200kN load will be transmitted through the very quickly rotating wheel bearings and the aircraft structure to the engines, preventing the plane from accelerating.

122

I think that this explanation is flawed. With regard to the airplane on the treadmill, there is no rope attached to the front; no guy pushing from behind. Only the thrust of the engines.

In our example, there are magical frictionless bearings in aircraft axles and treadmill.

Let’s say the aircraft exerts 10,000 pounds forward thrust. On a normal runway, under normal conditions, this translates into about 100mph of forward speed.

On the treadmill, this translates into 100 mph of forward speed that is offset by the treadmill spinning 100 mph in the opposite direction. Effective forward speed of airplane = 0.

Airplane adds another 10,000 pounds forward thrust. It now has 20,000 pounds thrust and 200 mph potential forward speed. The treadmill adjusts to 200 mph in the opposite direction. Effective forward speed of airplane remains at 0.

Continue adding thrust. Continue countering potential forward speed with the treadmill spinning backward. Aircraft remains stationary.

No forward movement – no airflow. Airplane does not fly.

Now, buddy comes up from behind the airplane and firmly plants his feet on the ground and pushes on the airplane, generating a force that is equal to 100 pounds of force, then yes, the airplane will move forward at let’s say 5mph. But now you have introduced a force that is not related to the fact that the airplane is under its own power at all. In fact, shut down the airplane’s engines and have buddy continue to exert 100 pounds of force with his feet planted on the ground behind the treadmill, that plane will continue to move forward at an effective forward speed of 5 mph. If buddy can find a way of bracing himself off the treadmill and pushing or pulling the airplane to 100 mph, then the plane will be able to take off.

The point is that buddy must stand on solid ground. If he happens to get onto the treadmill and apply his 100 pounds of force, adding 5 mph, then the treadmill increases its backward speed by 5 mph and the airplane (and buddy) remains stationary.

Airplanes rely entirely on their own thrust to generate forward movement. There are no outside influences initiating forward momentum. If the treadmill can react immediately in a 1:1 backward movement, the airplane will not move forward and will not take off.

123

Unless the treadmill and drivetrain of the vehicle are frictionless, you certainly do need to apply a torque to the wheels. Put a car in neutral, and see how long you remain at a constant speed.

124

How did this thread becoming so long? Is this like the 0.999… = 1 thing, which some people just don’t seem to get?

Yes, the plane will take off. Although friction is quite complex, I believe that most simple models of rolling and kinetic friction have them proportional to the normal force, not the speed at which the two surfaces move (drag is another matter). Running the treadmill faster does not increase the force on the airplane.

125

Sheesh. I made a slip (ha ha) in defining my terms and it’s getting late enough that I’m having trouble correcting (should be trivial, I’ll look at it tomorrow).

That being said, I still maintain that the speed of the conveyor belt doesn’t have any real impact on the speed of the airplane. Let’s look at it backwards:

Get in a plane and fly it at 100mph over the conveyor belt.

Conveyor belt is initially turned off. Lower your landing gear so that the nose wheel just touches the belt. The wheel starts spinning.

Now turn the conveyor belt on. Start slow, say 1mph backwards. The wheel spins a little bit faster. Run the belt up to 100mph, matching the forward velocity of the plane. The nosewheel is spinning like crazy now (as far as the wheel is concerned, it’s touching solid ground at 200mph). Get nutty - go back down to 0mph for the belt, then run it up to 100mph going in the SAME direction as the plane. At that point the wheel will stop spinning altogether - from the wheel’s perspective you are hovering like a helicopter.

Tell me, at any point during this does the wheel skip or skid? No, it simply accelerates and decellerates smoothly.

At any point does the speed of the plane change? No, it is unaffected.

At any point does the plane drop out of the sky due to lack of lift? No, since the airspeed doesn’t change.

126

You are right that the speed of the conveyor belt doesn’t have any impact on the airplane. What we are given, though, is that the speed of the conveyor belt is equal to the speed of the wheels. This could be happening just by coincidence (seriously). The question is silent about how the two speeds happen to be equal. We could design some kind of Rube Goldberg contraption that could make them equal, or it could be happening by magic, or any number of things. The point, though, is that those two speeds are equal, and that means that the plane must be standing still.

127

But the thrust of the engines is not linked to the treadmill at all. It’s totally independent of the treadmill, like a rope pulling the plane would be. The thrust of an engine is not affected by how fast I’m spinning the wheels of the landing gear!

Not it’s not. All the treadmill does is spin the wheels backwards, but since the wheels do not drive the plane you can spin them at 1mph, 100mph or 10 jillion mph, whatever makes you happy. The center of mass of the plane will be unaffected - it’ll be moving 100mph forward, relative to the fixed earth. It’s just that the wheels will be spinning like mad.

Nope. Again, the engine thrust makes the center of mass of the plane move forward, it’s basic action/reaction stuff - conservation of momentum or center of mass of the system, however you want to do the math. The wheels do not enter into the equation since they are not propelling the plane!

128

OK, what is meant by “the speed of the wheels”? Is it the velocity of the center of mass of the wheel (which must be equal to the forward speed of the plane, since the wheel is rigidly connected at the bearing or axle or whathaveyou)? Or is it something else?

129

If it was the speed of the centre of mass the problem becomes a trivial, boring, non-problem because it would then be obvious that the plane could go at any speed we want. I took it to mean the speed at which the wheel is turning around its axis, i.e. the speed you would be going relative to the ground if the plane were rolling along. In this case the “ground”, i.e. the treadmill, is moving at the same speed in the opposite direction, so the two speeds cancel and the plane is motionless w.r.t. the air.

130

If the treadmill is stopped when your 100 MPH aircraft touches down what makes
the wheels start turning?
Let’s go back to the beginning. You have an aircraft sitting on the treadmill, it’s engine
is at idle and the brakes are on. The pilot releases the breaks and increases the
throttle. The plane begins to move forward, but the tread mill starts moving in the
opposite direction equal to the thrust moving the plane. As the pilot increases the
throttle, the treadmill increases speed to equal the increased thrust, the plane remains
stationary relative to the ground and the surrounding air. The energy expended by the
engine is dissipated in trying to overcome the treadmill. The energy required to
operate the treadmill will approximate the energy of the planes engine. No forward
movement over the surrounding land or air, no airflow over the wings, no lift, no
flight.
Don’t worry about what controls the speed of the treadmill, for the purpose of the
problem just assume that it can operate as described. The wheels will turn at the
same rate as the treadmill, but that’s irrelevant also. Just concentrate on what
principal allows a plane to fly, air flowing over the airfoil causing lower pressure on
the top and creating lift. No airflow, no lift. The plane must be moving through the air
before it can achieve flight.

131

Criminy, didn’t anyone take physics here?

The rotational intertia of the wheels and bearing resistance is going to be quite small in proportion to the thrust of the engines. If you try to forestall the forward motion of the plane by operating the treadmill in the opposite direction it will only work for very, very low thrust settings where bearing friction is equal to thrust.

Turn the problem around and turn the engines off. Starting the treadmill will move the airplane. At very slow accelerations the drag on the wheel bearings and tire drag will be greater than the inertial of the plane and they won’t rotate at all. Speed up the treadmill and you’ll have wheel rotation plus airplane movement as a result of the frictional forces of the wheels acting on the mass of the plane.

They have airplane treadmills, they’re called aircraft carriers. Unfortunately we don’t operate any piston engine planes - which have much slower stall speeds than jets - off carriers anymore but I’ve seen the surreal sight of a non-catapult deck launch of a C1 Trader on the Connie. With a 30+ knot ship speed and any appreciable headwind the stall speed of the C1 is quickly reached. On takeoff the pilot pushed the throttles to full power and releases brakes. The deck speed is barely above a walking pace but the airspeed is the only thing that matters so the plane climbs out after it reaches the end of the deck instead of plummeting into the sea.

132

Yes, it can take right off.

The assumption is that the treadmill will counteract whatever gain in speed the plane can generate. The problem with that assumption is that the engine is decoupled from the ground by the wheels; the plane will move roughly independent of how fast the wheels are spinning, if at all.

Adding that the treadmill can match the speed of the wheels is really just a red herring, and it’s illogical to boot. Why?

  1. The engines will push air out the back, moving the plane x distance forward.
  2. The treadmill will attempt to move the plane an x distance backward by rotating the wheels.
  3. However, the plane didn’t use the wheels to move forward; it used air. All the treadmill does it make the wheels move faster.

Now if you blew a really, really strong wind at the plane - to counteract the force - or put it in a vacuum - so that it had nothing to act against - then it wouldn’t move.

This isn’t the case with a car, because a car engine pushes against the wheels and the wheels turn against the ground. It’s only if the ground pushes back that the car moves forward.

133

I can’t believe this thread is still going. for anyone still in doubt, it just isn’t a simple question of reference frames, because the treadmill only affects a small (and relatively insignificant, in terms of flight) part of the plane.

134

Herein lies the problem. What is meant by matching the speed of the wheels? If you take it to mean the speed at which the wheel’s center is moving with respect to an outside observer, the plane will take off with little more difficulty than it would from a normal runway. If you take it to mean the rate at which the wheels are rotating, you have a problem. Any movement of the wheels relative to the ground will put the treadmill into a runaway acceleration. The wheels turn, which causes the treadmill to accelerate to match them, which causes the wheels to turn faster, resulting in an unbounded positive feedback loop that only ends when the rolling friction and bearing drag from the wheels equals the thrust from the plane’s engine. In practice, the wheels will disintergrate long before this can happen.

I’ve seen this riddle elsewhere on the net in a slightly different form:

This form of the riddle seems to make it clear that it is the speed of the plane that matters, not the rotation of the wheels. Here the plane will clearly take off.

135

You guys seriously here now. I drew up some pretty pictures that illustrate what happens exactly. It is possible to hold the plane on a treadmill like this for a short period of time. What will happen is that your wheels will in essence be taking the full force of the Jet and will be torn apart very quickly.

Those of you who say the VELOCITY of the treadmill are correct. However, you ignore the important part which is the ACCELERATION of the treadmill. ACCELERATION of the treadmill causes a frictional FORCE on the plane which can counteract the jet FORCE.

Look guys I drew up FBDs for both possible situations of what is meant by wheel speed (although its train and a rocketman for the second case). Only 5 of you have downloaded it thus far. The answers are if you take wheel speed to mean forward linear velocity of the plane then your plane takes off but needs a bit more energy. If you take the wheel speed to mean rolling speed then your wheels spin faster and faster until they fail but the plane will go nowhere. The proof is in the two .pdf files I linked to.

136

Can I jsut say that I regret posting this thread? :smack:

I actually tried to go back to the thread on the message board I found it at, because the OP came back and gave the answer (the answer being that yes, it did take off) and even poste dsome equations stating why. But, alas, since that forum is much less moderated, and inhabited by a larger percentage of idiots and flamers, it turned into a flame fest and was not just locked, but deleted.

But, to try to clear some things up, here is what I do remember about his solution. There IS rolling friction of the wheels that has to be overcome, so it is not some kind of “perfect scenario” where there is no friction. Sorry if that was implied (or stated outright) in the OP.

137

Bouv, you ought to not have regret from starting this. However, maybe you can start a less controversial topic next time.

May I suggest:

  • Telemarketing is a noble pursuit

  • Shoes are to be taken off at the door

  • I’m not doing anything illegal by driving at the speed limit in the left lane

While contemplating this thread, I’ve had a few zen moments – I’ve been in the zone – I can see the light clearly – I’m absolutely confident in my theory. The rest of the time, I’ve had a strange sense of anxiety that I can’t grasp this seemingly simple question.

I resolve to leave this problem behind me now and go to work with clear head.

Go to work at the airport where I will watch airplanes take off all day long.

138

Different people are getting different answers to this question because the basic assumptions can be interpreted in different ways.

Suppose we actually built a treadmill like that described in the OP and put a 747 on it. Would the 747 take off? If “exactly matching the speed of the wheels” means that the treadmill matches the hub speed of the wheels, then yes. The plane takes off, and the treadmill’s going twice the speed in the opposite direction.

OK, as Manduck say, that problem is trivial. Let’s assume that “exactly matching the speed of the wheels” means “matching the outer diameter surface velocity” Would the 747 take off? Almost certainly it would, but only because the treadmill wouldn’t be able to keep up with the thrust transmitted to the plane by the engines–in other words, we violate the spirit of the question, because the treadmill isn’t matching the wheel velocity.

OK, that’s stupid. It’s a thought experiment. Posit a magic treadmill that can accelerate as fast as desired. All right. Multiple things could happen. To begin with, the plane would stay stationary as the thrust power was dissipated in the wheel bearings (as friction), tires (hysteresis), and in accelerating the wheel to ever-increasing speeds. Since all the power is dissipated in the wheels, eventually either the bearings would overheat, the tires would blow, or the wheel would rip itself apart due to inertial forces, and the plane crashes and burns. Then you’ve destroyed a rather expensive magic treadmill.

Thought experiment, I said! Let’s posit ultra-strong and heat resistant tires. All right. Again, multiple things. If the treadmill is a long, runway-sized treadmill, it will eventually, running thousands of miles an hour, pull in air at high enough velocity that the plane will lift off at zero ground speed (but substantial air speed). However, now you’re running into trans-sonic compressibility effects…

No speed of sound effects! And assume magic air that doesn’t become entrained with the treadmill motion. In that case, the treadmill speeds up until it and the wheels are running near light speed, and relativistic effects takes over. The wheels get smaller, I suppose…

None of that! No relativity-- Hey, wait a minute. Back up. Suppose we have zero friction bearings and tires. And while we’re at it, make them massless, so there’s no inertial effects. Well, zero friction tires would mean they just skid on the runway, since nothing turns them. So the plane will take off, tires motionless, and the treadmill won’t move.

OK, then. There’s friction between the tires and treadmill, but not in the bearings or sidewall. No energy is lost in the wheels and tires, in other words. Now you’ve got an unstable runaway system. There’s no resistance to treadmill motion, and a positive feedback circuit. Imagine, now, the poor mechanic who bumps a wheel, setting it in motion. A very slight roll by the tire is sensed, and the treadmill luches forward. The tire goes faster, the treadmill goes faster, the tire goes faster… Since we’ve posited an instantly-accelerating treadmill and no relativity and no air resistance and no wheel inertia, the treadmill goes from zero to infinity in no time flat. Try to keep your balance on that. The same thing happens, of course, when the engines light off. There’s nothing coupling the plane to the treadmill–no bearing friction, no inertial effects, no air resistance, and no way for the treadmill . So the plane takes off, leaving the infinite-speed treadmill behind.

Did I miss anything?

139

Just get on a treadmill and start walking. The treadmill matches your forward velocity, so you appear to be standing still. Now toss a paper airplane.

You are the undercarriage, your arm is the jet engine, the plane is the plane. It flys.

Or look at it like this. A treadmill is like a runway moving backwards relative to the plane. The same thing happens on landing. The runway is moving backwards relative to the plane, matching it’s speed. But the plane can still take back off. To me, this seems like a bad analogy since the airspeed is not considered, but it’s not considered in the original treadmill question, either.

140

The plane would move forward under thrust. An airplane on a stationary surface uses wheels to minimise friction that would hinder it’s forward motion. The wheels turn because of friction against the surface and a small amount of forward thrust is disapated with friction, and no other reason. The engine pushes air for thrust and remains just as effective reguardless of the wheels. The belt only matches the speed of the wheels and not the power of the plane’s thrust. Only gravity connects the planes force to the surface. The plane will gain speed and take off reguardless of the belt matching the wheel speed in the opposite direction, because you don’t applied forward force through the tires turning, like a car. All the forward thurst is applied by moving the air with the prop. The moving belt would apply no more force in the opposite direction when the plane taxied than if it was on the ground. Please let me know if this isn’t clear enough people.

Thrust using tires to go forward like a car, would leave you setting on the belt, like in the car labs.