It’s not taking any effort at all. Basically you are bad at Math. You are setting values that literally do not add up. The “A=A+5” comment in the column is just a variation of your misunderstanding.
“No slippage allowed” means you welded the plane to the ground. Did you weld the plane to the ground, strongly enough that the engines can’t produce enough force to break free? If so, it’s obvious the plane isn’t going anywhere. But you literally have to bypass the wheels entirely for that to happen.
Yes, I’m positing that world, or something really close to it. So if I had two 100-tooth gears that are exact copies of each other, one is mounted and can rotate and the other can rotate and revolve around the first (sort of like a sun & planet gear). So if I attach an axle to the planet gear and spin up the sun gear, I can hold the planet gear stationary with very little force. The two are now rotating at the same speed. I can easily push the axle forward with my hand and move the planet gear forward regardless of the rotational speed of the two gears (this is analogous to the plane moving forward on the treadmill from the plane’s thrust). But me pushing the planet gear forward is increasing the rotational speed of the planet gear by what could be a minuscule amount, therefore violating the original premise.
There is a way around this by mounting the gears on opposite ends of a pivot bar so that my pushing on the planet gear’s axle is splitting my force between revolving the planet gear around the sun gear and revolving the sun gear around the planet gear. In this case, the planet gear never can move around the sun gear, any small increase in speed of one gear is copied on the other gear because the reference plane is no longer the center of the sun gear but the pivot point in between the two gears. This scenario is analogous to the plane on the treadmill problem for me, in that regardless of the thrust produced by the plane’s engines, the force is split between the treadmill accelerating and the plane moving forward.
I wish I could find a way to illustrate this online. It would be a lot easier than trying to explain it.
But if you’re mounting the wheels on a pivot bar, then you’re shackling the plane to the ground. Obviously a plane shackled to the ground can’t move, but that has nothing to do with a plane that’s only contacting the ground via wheels.
Yes, but A + 5 = A + 5, which is what I’m getting at.
In that case, the plane stands still, and the ground takes off.
The pivot bar (ideally) splits the force applied to the planet gear’s axle so that the increase in speed of the planet gear from this force is instantly copied to the sun gear and end result is the sun gear’s rotational speed always matches the planet gear’s speed. It’s a way around the infinite acceleration problem of the treadmill speed always matching the wheel speed. I don’t think this is shackling the plane to the ground, but maybe I’m missing something.
The wheel is attached to the pivot bar, and the pivot bar is attached to the ground. The pivot bar is the shackle.
In this case, the sun gear is set spinning, the planet gear spins at the same speed and any force I place on the axle is split between speeding up the planet and sun gears an equal amount. This split comes from the use of the pivot bar. Removing the pivot bar allows me to move the planet gear (plane wheel) relative to ground and, you’re right, the pivot bar is preventing this but still allows for force to be applied or removed from the axle (plane) and for the wheels and treadmill (planet and sun gear) to speed up and down. I can apply all the force I want to, the wheels/treadmill/gears will speed up to whatever limit, but will never move relative to ground.
If I physically shackled the plane to the ground, I don’t think the wheels would ever spin from thrust. I believe this is the difference between your and my interpretation of the gear analogy.
What is causing the wheels to turn? Sounds like you’ve basically just set up a system where everything is locked up, and nothing can move.
Nope. Recheck your statement. You’re specifying something that generates an impossible equation. You can’t have a treadmill that matches the speed of something that has a 2nd component of change.
You can have the plane not moving forward (no thrust) and the treadmill and the wheels are matched.
You can have the plane moving forward (thrust) and the treadmill and the wheels do not match.
Pick one!
Right, so as a fix I just stated that the treadmill speed should always be greater than the wheel’s speed. Or use a device like I described above that splits the thrust force between the plane and the treadmill.
Finally, someone has asked the right question! This expletive deleted discussion about airplanes and treadmills always gets bogged down in the details of whether you can remain stationary or whether you can be forced to remain stationary, all of which misses the point. The POINT is, could you replace a 2,000-ft runway with a 200-ft conveyor belt? The answer is a resounding NO. That would never work. The conveyor belt would need to be the same length as the runway you’re attempting to replace.
Each aircraft has a minimum speed required for takeoff. This minimum applies to air speed, i.e. the relative motion of the wings through the air. Ground speed is irrelevant. Wheels are irrelevant. Conveyor belt speed is irrelevant. The only thing that matters is the speed of the air moving past the wings (or the wings moving through the air, depending on your point of view).
A headwind helps. Suppose your airplane has a minimum takeoff speed of 50 knots and you are facing a 40 knot headwind. Then you only have to get the plane up to 10 knots ground speed and voila, you have 10+40=50 knots air speed, and the plane can take off. That doesn’t take much runway. Tailwinds hurt. If you have a 30 knot tailwind, you need to get the plane up to 80 knots ground speed to have 80-30=50 knots air speed before the plane can take off. That takes a lot more runway.
Other than a headwind, the only way to shorten the amount of runway needed is to make the airplane accelerate faster, so that it reaches minimum takeoff speed in a shorter time. You can do this by giving it a more powerful engine with more thrust, or by attaching the plane to a catapult. A treadmill is useless because it does absolutely nothing for air speed or acceleration. Yes, an airplane can take off from a treadmill but this fact doesn’t do you any good in trying to shorten the takeoff distance.
To simulate the treadmill, where the treadmill speed > wheel speed, the sun gear (the treadmill in this analogy) will spin using whatever, it doesn’t really matter, but if it helps imagine an electric motor spinning it. This will cause the planet gear to somewhat rotate but mostly revolve around the sun gear. Putting a little force on the axle attached to the planet gear will allow you to move the planet gear around the sun gear regardless of the rotation speed of the sun/planet gears. Placing these gears on a bar that can freely rotate around its center will split any force placed on the planet gear’s axle causing the speed increase to appear on the sun and planet gears nearly simultaneously.
I’m trying to find a free physics simulator where I can illustrate this, mainly to make sure I’m not missing something due to a huge blind spot.
If the plane is moving forward at 1 ft/sec, and the treadmill is moving backward at 1 ft/sec, then the treadmill is making the wheels turn at 2 revolutions per second–the speed they would be turning if the plane was moving 2 ft/sec on a solid runway.
The problem as written assumes that the treadmill is driven by some external motor to run in the opposite direction that the plane is moving at the same speed as the plane. The friction between the wheels and the treadmill, in conjunction with the plane’s forward movement, make the wheels spin at a speed (at the edge) of twice the airplane’s forward speed, relative to their axle fixed to the airplane.
If your imagined problem has the treadmill being driven such that its speed is equal to the speed of the wheels, then you’re imaging something self-contradictory. The plane is not coupled to the treadmill, so it’s going to start moving forward; then, by this definition of the treadmill’s rules, it has to instantly accelerate to a speed of infinity in the opposite direction.
V1: airplane’s speed in the forward direction
V2: speed of the wheels
V3: speed of the treadmill in the backward direction
V2 = V1 + V3 (since the wheels are attached to the plane, and coupled by friction to the treadmill)
If V3=V1, then V2=2xV1 (OK)
If V3=V2, then since V1>0 (nothing in the design can stop the plane from accelerating) V2=V3=infinity (there is no finite value for V2 and V3 that makes V2 = V1 + V2 when V1>0)
the plane will never split its thrust between moving the aircraft and the treadmill, the planes thrust affects the plane and only the plane
Planes aren’t cars.
Even if the treadmill applies your mystical “more speed” than the plane is moving at the plane will simply take off, because the plane isnt pushing or being dragged or in anyway tied to the treadmill.
There is nothing anchoring the plane to the ground other than gravity, a force planes are specifically designed to overcome.
your hang up is that you are under some weird illusion that the plane cares at all what the treadmill does.
On a related note I did once read a book with the odd idea that airports could be made a lot smaller and more fuel efficient at the same time.
run ways now have a hill in the center, with the airport at the top, incoming planes land into the wind and lose speed as they climb the hill, out going planes save fuel by accelerating down the otherside into the wind.
Makes a lot of sense if you live in a word where nothing ever goes wrong with your aircraft or pilots during take off and landing.
I’m assuming the treadmill is always spinning faster than the wheels. This solves the instant acceleration problem.
My hang up is if the plane is moving forward, the wheels are turning faster than the treadmill is moving. I don’t see how you can get around that.
I don’t know how many times I have to explain that I completely understand the jet exhaust is pushing the plane forward. That is trivial. The torque applied to the plane’s wheels is 0. I am not confused on that fact.
Right. My planet & sun gear analogy is a mechanism not unlike the original question where the speed of the treadmill would keep up with the speed of the wheels. It is unnecessary when you stipulate the treadmill will always move faster than the wheels.