Re: Imagine a plane is sitting on a treadmill...
 

The wheels are free to rotate, and will freely do so. They are not driven wheels, they are basically just a bearing system between the conveyor and the rocket/ plane.

The thrust of the rocket / jet engine is pushing against the air, not the ground, and the airframe will react in the opposite direction, pushing the aircraft/ rocket/ wheel assemble forward, regardless of how the wheels in contact with the conveyor decide to rotate...

Aw crap, Bill explained it so much better, although Hobbess' reply is brilliant thinking, but surely a club soda would have done just as well:)

Re: Imagine a plane is sitting on a treadmill...
 

Is the issue here still that you can't apply the engines and the plane to stay still? I think you can.

Imagine the plane with the engines off on the belt. The plane will move backwards because of the force T exerted by the treadmill. This will cause acceleration of the belt dependent on the mass of the treadmill and plane. The belt and plane will move at constant velocity until T is equal to the sum of the air resistance exerted on the plane (A - proportional likely the square of the velocity) and the mechanical resistance in the treadmill (M).

So the plane is moving backwards. If we are now to exert an infinitely small force on the plane F (i.e. the engines), will the plane start moving forwards? No because the force of the engines will only accelerate the plane enough until the frictional force (µR) between the wheels and the treadmill will equal the force provided by the engine. At this point the plane will be moving at a constant velocity forward due to the engine, but still at a constant velocity backwards - which is higher - due to the treadmill.

As you increase the force of the engines, F will increase, until exceeds µR. At this point, the plane will start to move forward from it's position on the treadmill, but will still be moving backwards overall. At this point F-µR (the net forward force) will be balanced by a reduction in A as it is moving backwards at a slower velocity.

Eventually, if enough force is provided the forces will balance out. That is F = T + µR. In this case, the plane is maintaining its position relative to the frame of the treadmill, but moving forward at the same velocity as the belt of the treadmill, but in the opposite direction. There's no airflow over the wings here.

An easier way to think about this is just to go back to the third paragraph. Engines off, treadmill on. The plane moves backwards. Switch the engines on to apply a minute force. What happens? The plane moves backwards more slowly. Eventually you can increase the engines until it moves at the same speed as the treadmill.

Re: Imagine a plane is sitting on a treadmill...
 

The wheels are not free to rotate when the plane is in contact with the ground as there is a frictional force, proportional to the mass of the plane.

Re: Imagine a plane is sitting on a treadmill...
 

Why does the plane move backwards? Have you locked the brakes? In the original problem as stated, the brakes are off and the wheels are free to rotate. Because of inertia, the plane will not want to move when the treadmill starts. The treadmill will simply spin the wheels on their axles. If the engines are applying thrust (force F on the airframe), how can a treadmill spinning the wheels counter that force? It can't.

Re: Imagine a plane is sitting on a treadmill...
 

Thanks! Karma on the way.

Unfortunately, not everyone is catching on...

Re: Imagine a plane is sitting on a treadmill...
 

Interesting, Bazzz, but really rather beside the point.

The treadmill is only going to move backwards, the problem states, to match the speed of the plane exactly. It isn't moving initially, and only starts to move once the plane starts to move. If the treadmill is only ever moving backwards with the same velocity as the plane is moving forwards, by definition the plane is moving forwards. Sure, more of the engines' power will be taken up overcoming the increased friction due to the wheels moving twice as fast, but that will be a tiny fraction of a percent of the total power available, so can be ignored for practical purposes.

The forces generated by jet engines sufficient to get a plane off the ground are orders of magnitude larger than the rolling resistance of the tyres and the bearing friction (your µR above). While one can't entirely discount friction forces , I humbly suggest that they disappear into insignificance against the thrust developed by aero engines at full throttle. The plane will take off.

Re: Imagine a plane is sitting on a treadmill...
 

Indeed. The original question was essentially "can the treadmill prevent the plane from taking off?" The answer is no. If you change the question to read "can we construct a scenario in which a small amount of thrust exactly balances a small amount of movement arising from the frictional loss in the bearings due to the movement of the treadmill?", that's an entirely different problem.

Re: Imagine a plane is sitting on a treadmill...
 

No, it wont, because the treadmill only moves backwards if the plane moves forward. No forward motion from the plane due to an absense of thrust means no backward motion of the treadmill.

So if the treadmill is to make any motion, there has to be forward motion of the plane, and if there is forward motion of the plane, eventually there will be sufficient airspeed to generate enough lift for take off.

This is not rocket science... Without forward motion due to the thrust there can be no backward motion of the treadmill. The treadmill is irrelevent.

Yes there will be some frictional forces, but not enough to counter the full thrust of a couple of jet engine...the brakes are stipulated as off remember.

Aw crap, Oakvillian is way ahead of me.

Re: Imagine a plane is sitting on a treadmill...
 

By that logic the tugs that move the planes around the terminal would never be able to move them. Wheels are generally attached to some sort of bearings designed to minimise the friction.

Re: Imagine a plane is sitting on a treadmill...
 

Re: Imagine a plane is sitting on a treadmill...
 

Hmm. I was assuming this was a powered treadmill.

Re: Imagine a plane is sitting on a treadmill...
 

It is?

Even if the treadmill was doing a constant 200mph backward rather than matching the aircraft speed, assuming the treadmill is long enough to stop the plane falling off the back of the conveyer first, once the engines were powered up and the nominal amount of friction in the wheel bearings was overcome, the plane would soon be thundering forward through the air fast enough to take off, although its wheels woud be spinning rather faster than normal.

Even my wife, who has a Masters of Arts degree for god sake, figured this out in only a couple of minutes:)

Re: Imagine a plane is sitting on a treadmill...
 

What makes you think it's nominal? If we assume a value for µ of say 0.7, for a 400 tonne plane, you'd require a force of 2.7 MN to overcome it.

Re: Imagine a plane is sitting on a treadmill...
 

I thought the point is that the plane wouldn't be going faster than the treadmill - i.e. it would be going forward at 200 mph relative to the treadmill - which is at rest compared to the surrounding air.

Re: Imagine a plane is sitting on a treadmill...
 

The spinning wheels are a red herring. The speed of the treadmill is a red herring. To take off, the plane needs to be moving at a specific speed relative to the air around it (air speed). The speed it is moving relative to the ground below it is irrelevant. If somebody wants a scenario in which the plane can't take off, here goes:

No treadmill. The plane is sitting in a special wind tunnel 100 miles long (or 100 parsecs long, that doesn't really matter). The air movement in the wind tunnel can be precisely controlled and is completely uniform from top to bottom, wall to wall. The plane throttles up its engines and starts to roll down the tunnel. (The tunnel is 5 miles tall and a mile wide, so the plane has room to take off). Now someone operates the controls and starts the air column moving in the same direction as the plane. The controls are precise enough that the moving air matches the speed of the plane. In this case, as long as the speed of the plane relative to the speed of the air around it is zero, there will be no lift and the plane cannot take off no matter how fast it travels. It could be travelling 1000 mph relative to the ground but if the air column is also travelling at 1000 mph relative to the ground, there can be no lift. The ground speed is irrelevant. Airspeed is what's important. In the original problem, the treadmill cannot prevent the plane from acquiring sufficient airspeed to take off.