We'll start by making two approximations, and then consider the effects of these effects more closely.
- Wheels. First we assume that there is no rolling resistance, that the wheels roll completely freely and that they have negligible mass: therefore it takes no force to turn the wheels, even at high speed. This is obviously approximate: because of their mass and thus their moment of inertia, the wheels do require a force (and thus a torque) to make them rotate; because of friction in bearings they require a force (and thus a torque) to keep them rotating; because of the deformation of the tires there is a rolling resistance and this requires a force as well. In principle, however, we could imagine making this effect very small. We might even use hovercraft or maglev instead of wheels.
- Air motion. Second, we assume that the conveyor belt does not create a wind with respect to the ground. In practice, the conveyor has to be very long and probably very wide and so will move air to some extent. In principle, however, we could imagine this to be very small: there could be one very narrow belt for each wheel: these would not affect the air much, though it might require precise steering. ([SIZE=-1]On the other hand, i[/SIZE][SIZE=-1]f we make the conveyor belt the size of a planet, we end up with Question 3.[/SIZE])
Let's start the plane's engines and the conveyor belt simultaneously. The moving conveyor belt starts to rotate the wheels but, according to our initial approximation of freely rotating, massless wheels and no rolling resistance, this requires negligible horizontal force from the ground.
So, subject to our approximations, the horizontal forces on the plane are only the thrust of the engines and the drag exerted by the air. Because we've assumed that the conveyor belt doesn't affect the air, these horizontal forces have their normal values. So the plane accelerates at its normal acceleration and, after a normal period, achieves its take-off velocity and then, with the usual values of flaps etc, takes off. In this case, with these approximations, the question and answer are both simple.
So now let's look at the approximations. The sketches at right show the horizontal forces at some point during the acceleration towards take-off: the propellor provides forward thrust to the plane. The drag is the plane passes through the air is in the opposite direction. There is also a small backwards force on the wheels.
Wheels. The runway and the conveyor belt both exert backwards forces on the wheels for three reasons. These effects are all reasonably small compared to the thrust of the engines, but not zero, and not equal for the runway and conveyor cases. What can we say about them?
- Bearings. The wheel bearings are not perfect so, just to keep the wheels turning at constant angular velocity on their axles, a torque is required. This torque is small: give a wheel a spin and observe that its angular velocity decreases due to this effect (and due to air resistance on the tires), but does so slowly. Nevertheless, though small, this is one of the key differences between the plane on the conveyor belt and the plane on the runway. This force increases with angular velocity and, at take-off on the conveyor belt, the wheels must turn twice as fast as on the runway. That's why, in the sketch, the force on the wheels is very small in the runway case and not quite so small in the conveyor belt case. So, because of this effect alone, one would expect that, in order to take off, the pilot would require very slightly more throttle on the conveyor belt than on the runway.
Angular accleration. The wheels also have mass, and therefore a non-zero moment of inertia. (See Rotation, torques, moment of inertia.) So a torque is required for the angular acceleration involved in making them rotate. Lift a bicycle and try spinning the wheel: you have to apply a force to get it spinning. For take-off from the conveyor belt, we have to achieve twice the angular velocity that we'd have on the runway. (Plane is going forward at normal take-off speed, conveyor belt is going backwards at the same speed.) Does this mean twice as much accelerating force? In fact no, because the conveyor belt is like an infinitely long runway: we can take as long as we like (fuel permitting) to spin the wheels up to the required angular velocity. So this is not an important effect in our problem.
Rolling resistance. Third, there is rolling resistance. The tire is deformed as it rolls, and the energy required to deform it where it touches the ground is a bit less than the energy you recover when it regains its shape after leaving the ground. This gives rise to rolling resistance. Now the amount of deformation of the tire depends on the downwards force. Initially, with the plane stationary with respect to the air, this downwards force on all wheels equals the plane's weight. As the plane accelerates with respect to the air, however, lift starts to support the weight of the plane, so the downwards force on the wheels becomes successively smaller, and so does the rolling resistance. As we approach take-off speed, rolling resistance becomes very small.
Air motion. We assumed above that the conveyor belt does not move the air. This can never be quite true, of course. The length of our conveyor belt has to be somewhat longer than the normal take-off distance of the plane. It probably has to be tens of metres wide, too. So it will have an effect on the air above it: it will tend to drag air
backwards: it will create a headwind. Pilots usually choose to take off into a headwind: it means that, when they have normal take-off speed with respect to the air, they have a lower speed with respect to the ground. So they achieve take-off speed (with respect to the air) more quickly and need less runway. So, if all other effects were zero, this effect would make it easier for the plane to take off from the conveyor belt than from the runway. The effect, however, is small. ([SIZE=-1]but again, see
Question 3 in which this effect dominates.[/SIZE])