This is where I’m getting the problem. I do not understand how the plane gets airflow over the wings.
The plane is being thrusted forward by the engines. The engines themselves don’t push air over the wings of the plane - the engines push the plane forward, which then causes air to move over the wings, which then causes the plane the lift.
Now, add in the treadmill. The plane is thrusting forward and the treadmill is matching the plane’s wheel rotation speed, only in reverse, so the plane looks stationary.
The planes thrust is forward, but it isn’t actually moving forward relative to the air around it, so the air is NOT flowing over the wings, which means the plane doesn’t lift off of the ground.
Someone please correct me, because I’d like to know the answer.
Again, how does the conveyor belt place a force on the plane? It doesn’t matter how fast it spins, there’s no way it can counteract the external force from the engines. Once you overcome the inertia, the belt cannot keep the plane from moving.
That’s what a lot of people seem not to understand: that never happens. The faster the conveyor belt goes in reverse, the faster the wheels spin. What you’re doing is creating a feedback loop such that both the belt and the wheels will keep going faster and faster to compensate for each other. But as long as the wheels can turn, there’s nothing preventing the plane from moving forward WRT the ground, and taking off.
The conveyor belt only reacts with regards to the wheels and the ground.
The propeller (or jet engine) acts with regards to the air mass around the plane for which the conveyor has no control.
Lets take that same plane and place it in a wind tunnel with the wind blowing from the back of the plane. If the plane were to roll forward relative to the ground at 60mph and the wind blowing from the back of the plane were to be blowing at 60mph, the resulting airspeed would be 0 mph. Yes, the plane would be traveling down the runway relative to the ground but would never have enough airspeed and air flow over the wings to actually create lift.
The speed at which the wheels rotate does not affect the forward movement of the plane. Only something that pushes back on the plane as hard as the engines are pushing forward can affect the forward movement of the plane.
Suppose you were watching from the side and held up a sheet of paper so it covered the runway, and you couldn’t see it moving. The plane starts to roll forward. The wheels seem to be spinning a little faster than you expect, but the plane will still roll forward and eventually take off, because there’s nothing acting on it other than the mighty engines in one direction and the tiny little wheels in the other.
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Now, add in the treadmill. The plane is thrusting forward and the treadmill is matching the plane’s wheel rotation speed, only in reverse, so the plane looks stationary.
The planes thrust is forward, but it isn’t actually moving forward relative to the air around it, so the air is NOT flowing over the wings, which means the plane doesn’t lift off of the ground.
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The plane doesn’t look stationary. It looks like it’s moving down the runway, which it is. The treadmill may be matching the plane’s speed. But that’s a red herring - all the treadmill does is make the wheels spin faster as the plane moves forward relative to the air around it. The reason is that a plane is propelled by its engines, not its wheels. So additional wheel-spin doesn’t significantly slow the plane - the plane moves down the runway (wheels spinning madly), air flows over the wings, and it lifts off.
The key to this is while the conveyer belt could possibly slow down the plane’s forward acceleration relative to a stationary observer, the very fact that the wheels turn would still let the plane’s engines move the plane forward, regardless of how fast the conveyer or wheels turn. The engines still are thrusting air out the back of the plane at a very high rate of speed. For every action there is an equal and opposite reaction. The wheels are just there to provide a more-or-less frictionless surface for the plane to rest on until it takes off. And while the conveyer belt and wheels turn faster and faster, it doesn’t matter how fast as long as they can stand the RPMs, the plane will still move forward and nothing short of tieing it down with very strong cables will stop that.
So the initial question’s assertion that a conveyer belt by going backwards faster and faster could keep a plane from moving foward, is flawed.
'Course the plane moves forwards. The only way to get the plane to be stationary is to have the treadmill move at such a blisteringly fast speed that the wheels spin so fast, they fuse the bearings, and turn the wheels into non-rotating stubs.
The plane does not move forwards because the wheels are spinning; the wheels are spinning because the plane is moving forwards. If the ground is moving backwards under the plane, the wheels have to spin faster. But the plane still takes off.
First, the wheels have a moment of inertia which will cause a backwards force on the plane when the belt is accelerating. Besides that, there is friction in the wheel axle, and rolling resistance due to deformation of the loaded tire. Sure, those forces are normally small when comparing them to the thrust of a jet engine, but you could imagine that the belt is turning so incredibly fast that those forces exactly offset the thrust. Or that the engine has very little thrust, or whatever. However it happens, there is one interpretation of the problem’s wording that says it does happen. If you take that interpretation, the plane stays stationary wrt to the Earth, so doesn’t fly.
I can’t help throw out another thought experiment for people to think about. We have a conveyor belt. On the conveyor belt is a toy car. The car is connected to a rope that is connected to a wench. The wench does not rest on the conveyor. We start the wench, the rope shortens and the car is drawn forward. No matter what speed we crank up the conveyor, the wench will pull the car off of it. Forward motion, air moves over wings, takeoff.
VINSIN, although you are correct in much of what you say, your starting equality is incorrect. Velocity is not force. Ask yourself what force the treadmill will exhert on the plane (not the wheels) that will counterbalance the forward thrust of the engines (regardless of type).
The wheel axle friction and rolling resistance are NOT proportional to wheel speed; they are constant (except for the static case before things start moving). Therefore these forces are no greater than during a normal takeoff.
(Please note that the “boat in a moving stream” is not a valid analogy. Yes, a 60 mph current requires more force from your motor to overcome than a 30 mph current. That’s because you’re dealing with fluid flow, and the force due to friction in that case DOES vary with speed; this is why “terminal velocity” exists.)
There is a little something in the “rotational inertia” concept. But, as you point out, this requires very little force to overcome. And the belt will be doing part of that work anyway!
No, you can’t, because the problem says that the belt matches the airplane’s speed.
In that case the airplane wouldn’t be able to take off on a normal runway either.
See, that’s another part of the problem–it’s stated that the belt is only going as fast as the plane is, just in the opposite direction, and a reasonably-built plane is still going to be able to take off against a mere doubling of whatever resistance the turning of the wheels would normally cause.
OK, how about a different thought experiment? We have a plane on a perfectly normal runway, but next to the runway are a bunch of people whistling. The plane starts its engines, but the more the engines rev up, the louder the people whistle, so the plane can’t move at all. If it did move, the people would just whistle even louder, to prevent it. Since the plane can’t move, being prevented by all the whistling, there’s no airflow over the wings, so the plane doesn’t take off.
What’s stated is “The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction).” I agree that the first interpretation that came to my mind is the one where the wheels turn twice as fast as the plane’s ground speed, but I can see that the control system phrase could be interpreted as being wrt to the conveyor belt. And for all those people who are saying that it couldn’t take off, that’s the interpretation they’re using. Just face that little fact and I think this discussion could be retired.
I think part of what people get hung up on here is the image of an ordinary conveyor belt moving something from one place to another. Yes, this even works for an airplane with its gear down and its brakes unlocked, due to static friction. Static friction is normally a greater force than dynamic friction, as anyone who has ever tried to push a car knows – it’s a lot tougher to get it moving than it is to keep it rolling.
Now keep that image of the conveyor simply moving an airplane (engines off, wheel brakes unlocked) backwards. But before we reach the end of the belt, we tie a rope to a hardpoint on the airplane and secure the rope to some point on the ground, beyond the “forward” end of the belt. The airplane no longer is carried anywhere by the belt; instead the wheels turn under the airplane. And of course no matter how fast the belt moves, the airplane isn’t going to lift off. No air movement over wings, no lift, exactly. No one questions this.
However… if we put a force gauge (like a fish scale) in the rope, thus measuring the force applied by the belt to the airplane, we’ll find it’s TINY (unless the wheel bearings fail).
We will also find the surprising fact that this force doesn’t change for different belt speeds. Rolling drag is independent of wheel speed.
( The formula for the force of friction is simple: coefficient of friction multiplied by the normal force. The “normal force” is the airplane’s weight. You’ll note that “velocity” does not enter into it. )
Because of this, it isn’t at all necessary to assume frictionless wheels.
And as rhettro said, if you have a winch aboard the a/c and start hauling in that rope, you WILL progress down the runway, despite the belt’s matching your speed – or even if it accelerates to “ludicrous speed”.
Now… replace the rope with the airplane’s engines, with throttles barely cracked open.
YES, there is (theoretically, anyway) a point at which the force from the airplane’s forward thrust would exactly balance the force applied to the airplane through the tires and wheel bearings. At that point the airplane would be standing still, relative to any fixed position on the ground, despite the belt’s movement… and of course it would not take off.
The mistake people make is assuming that this scenario applies at any throttle position.
(In a strict interpretation of the problem the belt would STOP in this scenario, as the airplane’s ground speed, relative to the true ground, is zero…)
Reality is that since the frictional force of the tires and wheel bearings is tiny, and does NOT increase with increased wheel rotational speed, this balance point would be reached with only a tiny fraction of the engine’s output. Throttle up beyond that “balance point” and the airplane WILL progress down the belt, despite the belt’s matching its speed… exactly as if you started pulling on the rope in the previous case. And if you maintain takeoff power it can reach takeoff speed and rotate into flight.
We all agree and seem to understand that no airflow over wings = no flying. If a bystander can’t see the plane moving forward (ignoring wind), it won’t fly, no matter what happens to the wheels. And, making airflow requires the wings to move relative to the air. Tether a jet’s tail to prevent forward motion, it won’t take off no matter how you rev the engines. But, tether the nose and give it enough headwind, and it could potentially lift off without engines. (Like a kite or a glider)
The problem says that the conveyor belt moves at a certain speed determined relative to the speed of the plane’s WHEELS, not to the speed of the plane itself. So, if you are assuming that part of the question as posed requires the plane to be motionless relative to a bystander, wipe that thought from your mind - it’s not anywhere in the question and indeed, it is ONLY that incorrect but natural assumption that makes the question seem difficult.
It does help to think of the ‘conveyor belt’ as being the length of a runway, NOT the length of the plane. Now, you have it all right in your first 5.5 sentences, right up through ‘only in reverse.’ It’s the next step that falters - ‘so the plane looks stationary’ … why? What is holding it back? That conveyor belt isn’t PUSHING the plane back while the engines push it forward, it’s just making the wheels spin.
Maybe this will help - imagine the plane is backed up directly against a brick wall (or a thrust deflector, if it makes pilots happier). Now spin that conveyor belt ‘backward’ (moving nose-to-tail) at the plane’s takeoff speed. What happens? Does the plane bust through the wall, or take off? Now speed the conveyor belt up by a factor of 2 (or 5, or 50). Any change? No - the wheels spin faster and faster, but as long as they spin with anything approaching low friction, but that plane goes nowhere. Do we agree so far? Wall is behind plane, belt is making the wheels spin madly.
Now, with our runway-length conveyor belt still running madly, convert our stationary brick wall into an aircraft carrier’s catapult, or whatever - i.e., have the wall physically PUSH the plane forward at takeoff speed. The conveyor belt can’t overcome the force of a brick wall pushing the plane, no matter how fast they may make the wheels spin. Plane is moving forward at takeoff speed and takes off, because it has airflow over the wings, right? Are we still in agreement? Now, simply mentally replace the pushing wall/catapult with the pushing jet engines, because, relative to the tiny backward force exerted by the spinning wheels, the jet’s engines are functionally the same as the wall - an irresistible force.
So, as the question is stated, Cece is clearly right. If we change the question and require that the plane remains motionless with respect to a bystander, first we have to add why/how this is so (e.g., tethered at tail), and it doesn’t fly. That result might be slightly counterintuitive, but it isn’t the question presented.
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