Will a plane on a treadmill take off?

Factual Questions
161

I’m talking about an accelerating treadmill. In that case, the treadmill exerts force on the tire, causing the airplane’s wheels to accelerate (spin faster). This force on the tire is a net force on the airplane+wheel system. This is a ridiculously small effect, but it’s there.

Imagine a drum lying on its side, on the bed of a pickup truck. When the truck starts moving, the drum starts to spin - rolling backwards relative to the truck, but “forward” relative to the ground. Eventually it will fall off the back of the truck, but that spot will not be where it was originally. It would have moved with the truck some distance. This is not analogous to a block of ice sitting on that truck, which would stay stationary relative to the ground and fall down exactly where it was.

162

If I stand behind the ice (I assume you mean frictionless), I can push it off the front of the truck. Even as the truck accelerates. Counter acting the movement of the truck. Just like the engines of the plane counter act the movement of the conveyer. The plane takes off.

163

Are you assuming the tire is frictionless, or the wheel bearings?

If the tire is frictionless (i.e. it slides across the runway/treadmill freely), then you’re correct. It’s analogous to a frictionless block of ice on the truck.

But I was assuming the tires do not slip. The wheel bearing may be frictionless, but the tire/runway contact is not. In that case, the situation is analogous to the drum on the truck. The truck can and does cause the drum to move forward. It takes a non-zero amount of force to counteract it.

164

Oh, dear Og, someone shoot this thread in the head. No court in the world would convict.

It would have been so much easier if the original hypothetical had been: "An airplane is chained to a mountain. Is the airflow created by the propeller/turbofan/whatever enough to generate the lift needed to make the plane lift off the ground and hover? Assume for the purposes of the question that all of the airflow generated by the engine flows directly over the wings: no pusher props, no underslung jet engines.

Instead we get all of this wheelspeed and conveyor confusion.

For the conveyor belt to keep the engine from moving the plane forward, the belt/wheel interaction must result in a net force through the axle that is equal and opposed to the thrust of the engine. For an ideal situation, where there is no friction at the bearings, no deformation of the tire, etc., this force can only come from constant accelleration of the wheel, since no energy is needed to keep the wheel spinning at a constant rate. Perhaps this reveals a deeper subtlety to the question posed in the OP: at what percentage of the speed of light would the rim of the wheel need to be going before its increase in mass cancelled out the lift from the wings? Back-of-the-envelope type calculations are acceptable. Assume a solid disclike wheel 1m in diameter, 10 cm in thickness, with axle of zero diameter, and total wheel mass of 10 kg.
-edited on preview-
scr4 is correct in that frictionless bearings do not mean a frictionless wheel, or a wheel that takes no energy to spin. If the wheel is frictionless, it slides rather than rolls. The frictionless bearing just means that no work goes into heating the grease, etc. Energy must still be put into the wheel to change its rotational velocity, unless the wheel is also massless.

165

I only read the first page, so forgive me if this was already stated.

I think I can see some of the confusion, and it’s purely due to the nature of this thought experiment, reality would not work this way.

The OP mentions that the belt matches the wheel speed, so the assumption is this is precise and instantaneous.

So, remembering there is a force in the forward direction, when I picture in my mind a wheel and a belt matching speeds INSTANTANEOUSLY, and I consider a stationary point 2 inches in front of the wheel, I can see that there is no possible way the wheel can move forward unless there is skidding, which is not allowed.

This is actually easy to draw on a wheel on graph paper, put a mark at 90 degrees counter-rotation, put another mark 1/4 * diameter * pi (hopefully I got that right) forward on the belt, and mark the current tire/belt contact point relative to a stationary object.

Then watch as you rotate, it doesn’t matter the speed, as long as they match, the marks will all continue to be synchronized, if and only if we have a magical belt and the rule that the tire can’t skid.

So the problem is really due to the fact it’s a thought experiment.

If it was worded “…such that the wheel does not move forward…” then it would be much more clear.

The reality is that the plane would move forward because the wheels would either skid, or the belt could not really match the wheel speed exactly, allowing for many, many, many fast small advances of the wheel relative to the belt and to the stationary observer.

166

It doesn’t matter if the treadmill is acclerating, if the bearing are frictionless there will be no horizontal force exerted on the plane and the vertical force will be equal in magnitude and in the opposite direction as the gravitional force acting on the plane.

For bearings with friction, the maximum force applied to the wheel by the treadmill will be equal to the static friction force which will be proportional to the mass of the plane. If the treadmill accerates faster, the force will be too great and the wheel will slip. If the wheel slips, the maximum force exerted by the treadmill will be equal to the kinectic friction force which is independent of velocity (or acceleration) and and is still proportional to the mass of the plane and less than the static friction force.

The force on the plane by the wheels will be equal to the kinetic friction force exerted by the bearings which is still proportional to the planes normal force and should be much smaller (smaller [symbol]m[/symbol][sub]k[/sub] for bearings than for tire/coveryor) than the forces mentioned in the previous paragraph. Bearings are designed to be almost frictionless, that is the point of bearings.

And finally, the wheels would NEVER turn. The magic conveyor belt would move forward (with the plane) in its attempt to move with the same speed as the wheels which would be stationary when the plane started out. As the plane engines thrust the plane forward, the wheels would start to turn due to the frictional forces applied by the stationary conveyor belt, and the conveyor belt would move forward to keep the same speed as the wheels (i.e. zero - the wheels are stationary remember). The engines would move the plane forward faster, the wheels would try to turn faster and the conveyor would move faster to compensate. Finally the conveyor, plane and all would be moving forward at 600 mph or whatever and the plane WOULD take off.

167

Both. No point in one or the other.

168

When I envisioned this problem I replaced the jet engines with a geartooth assembly attached to a bicycle chain. The bike chain was secured to solid ground separate from the treadmill.

As the geartooth (jet engine) operates, it exerts a force backward on the bike chain, and an equal but opposite force pulling the airplane forward. Nothing the conveyor belt can do will affect the operation of the geartooth assembly.

My analogy might not be good; I’m not a physics major. Is it an equivalent analogy to the jet engines?

169

If the tire and treadmill surfaces are not frictionless (i.e. tire is not slipping across the treadmill), then there will be horizontal force, even if the bearing is frictionless. Because a bearing does not need friction to transmit horizontal force.

Depends on how the treadmill is controlled. I was assuming the treadmill has a sensor to detect position of the object on top, and controls the speed so that the object is stationary. (After all, that’s what a treadmill is for - to provide a moving surface while keeping the object at rest.) That means as soon as the plane starts tomove forward, the treadmill starts to move backwards to try to stop the plane.

170

I pictured an analogy as follows:

You on rollerblades on a conveyor belt that goes 50 feet to a wall. Attached to that wall is a rope that you have a hold of.
No matter how fast that conveyor belt goes, you will still be able to pull yourself towards that wall.

171

If the tire is frictionless, the wheel would never turn… The only reason it turns is that the force of static friction between the ground (or conveyor) and tire keeps the tire from skidding.

172

enipla said:

Actually, there is a lot of point to one or the other. A frictionless tire is a skid and doesn’t need to spin at all, so you don’t need a bearing, or even an axle. While you want your bearing to be frictionless, you want your tires to have good frictiony contact with the road.

eyer8 said:

I think that you are forgetting the angular momentum of the tire/wheel combo. A force will be felt by the plane due to accelleration of the wheel even if the bearing is frictionless. No force is felt if the wheel rotates at a constant rate.

173

Right. The conveyor is almost a moot point.

174

Hoo boy. The answer you posted isn’t wrong, per se, because the “right” answer is a function of what assumptions one makes about the problem. However, I would argue that the assumptions contained in your post are somewhat incompatible.

  1. The poster assumes the existance of friction, but not wheel inertia. Nothing inherently wrong with that, if stated up front, but the since the poster does not explicitly assume massless wheels, I’m inclined to think he simply forgot to include inertia in his analysis. As treis has pointed out, a lot of energy is dumped into spinning the wheels.

  2. The poster assumes a constant coefficient of friction of 0.001. Since we’re talking about a real system, coefficients of friction vary with a lot of things: temperature and velocity being two of those things.

So how about we have massless wheels that are indestructible, but have a small, constant coefficient of friction, on a magic treadmill? Oh, magic air and no relativity. In that case, as soon as the thrust of the engine goes above the frictional force, the treadmill accelerates the wheels to infinite speed. A small amount of energy is lost in the wheel bearings, and the plane speeds up and flies away, leaving an infinite-speed treadmill behind.

175

Well, yeah, in the real world. But we are talking magical conveyor belts and frictionless bearings. :smiley: For this experiment, it’s pointless to have one without the other.

176

As I wrote earlier, a frictionless bearing can still transmit horizontal force. If the treadmill is in good contact (not frictionless) with the tire, and if the treadmill is accelerating, then it is exerting horizontal force on the tire. This horizontal force is not counter-acted by anything, so it is transmitted to the rest of the plane.

177

Again, the important value is acceleration not velocity.

178

Hmm, didn’t catch that. Ultimately accelerating will exert force against the plane because it’s accelerating the tire. Regardless of friction. I guess if the acceleration is ultimate, the force could be ultimate as well.

It’s all about the planes engines overcoming the acceleration of the tires.

Nothing was said about ultimately powerful engines. It would be pointless anyway. Irresistible force and all that.

I might just have to go and change my position.

179

Not true. You are saying that a rotating body applies a force on the axis of the body in a preferred direction? So a spinning gyroscope will push horizontally on the axel it rotates on?

That said, if you allow the wheel to deform, then you are correct.

Rolling friction = [symbol]m[/symbol][sub]R[/sub]mg/R

where [symbol]m[/symbol][sub]R[/sub] is the coefficient of rolling friction (constant), R is the radius ot the wheel, m is the mass of the object, and g is the acceleration of gravity. Nowhere in this equation is the angular velocity or accleration of the wheel. The treadmill can go at infinite speed and as long as the plane engines provide more thrust/force than the rolling friction force the plane will accelerate and eventually have the needed lift to take off.

180

Bah, how does the frictionless bearing apply a force (or torque) to the axel of the plane? Forget about the treadmill for now, hell lets even forget about the bearing being frictionless; how does a accellerating wheel apply a horizontal or vertical or for that matter any other unballanced force to the system. If it does apply a force, wouldn’t the object move and would’nt this violate conservation of linear momentum? I don’t get what the accelleration does. I know that the sum of the torques on an obect is equal to the moment of inertia times the angular accleartion, but how would this exert a horizontal force on the plane?