Ok, looking at the interpretation of the question whereby the conveyor matches the wheelspeed (which leads to a stationary plane)
There are three ways in which the force of the treadmill acts on the plane in a fashion counter to its motion. As it’s a physics question putting in relevant formulae will give the answer. Why does nobody turn to the maths.
- Moment of Inertia. Using a Boeing 747 and information pulled from the internet
The 747’s 4 engines produce an approximate thrust of 772,000N
It has 24 wheels, each of which has a radius of 0.7112m and a mass of C.250kg (tyre is 110kg various numbers were found for the wheels but it was hard to be exact on which bits rotate)
This means each wheel has a moment of inertia;
Mass*Radius^2/2 = 63.22568
[This boils down to (thrust/wheels)/(WheelMass/2) so I’m no longer certain this maths is right]
The moment of inertia responds to the acceleration of the belt, not its speed.
Force per wheel is;
Thrust/wheels = 32,166.67N
The equivalent mass at the circumference for the same moment of inertia is;
Mass/2 = 125 kg
Acceleration of belt to counteract the engines
A = F/M = 257.33m/s/s
That acceleration of the belt alone would counter the engines and you get bonus resistance from
-
Friction in the bearings. this is correlated with speed and also vibration and heat.
I don’t understand the full physics of bearings but, the speed of the wheel affects the turbulence in the lubricant which affects viscosity and rolling friction (higher speed, higher friction). The heat in the bearing changes the viscosity of the grease and the kinetic friction between the rollers and the axle (higher speed higher friction) the vibration of the bearing affects viscosity, contact friction etc and this will go up as speeds increase due to 3. -
Deformation of the tyre. The Boeing 747 weighs between 178,750kg and 445,000kg, or 7,445kg to 18,420kg per wheel.
This causes serious deformation of the tyre.
To quote wikipedia
[QUOTE=Wikipedia]
Factors that contribute to rolling resistance are the (amount of) deformation of the wheels, the deformation of the roadbed surface, and movement below the surface. Additional contributing factors include wheel diameter, speed,[2] load on wheel, surface adhesion, sliding, and relative micro-sliding between the surfaces of contact. The losses due to hysteresis also depend strongly on the material properties of the wheel or tire and the surface.
[/quote]
So as the speed goes up, so does the resistance from factors 2 and 3 allowing acceleration to go down.
As 273m/s/s is an insane acceleration, we can look at how that can be affected. The easiest way is to change the mass of the wheels, I’m not confident in my wheel mass, if it was doubled the necessary acceleration of the belt would be halved.
Without the formulae for The bearings and the rolling resistance due to deformation anyone who rules out the ability of the treadmill to keep the plane stationary is simply being closed minded.
