Son of Airplane on a Conveyor Belt
June 4, 2008 10:47 PM Subscribe
Apologies in advance for the groans this may elicit, but this is driving me crazy. It's the old airplane-on-a-conveyor belt question. (An airplane sits on a conveyor belt that is rigged with a computer system that moves it at the same speed as, but in the opposite direction of, the airplane. Will the plane take off?)
The answer is yes. This is not my question. It will definitely take off. This has been amply hashed out in the above-linked thread. Move along.
But what I want to know is...
...in the theoretical case in which the wheels are indestructible, and there is no limit to the conveyor belt's speed (the case in which the conveyor belt is allowed to move many many many times faster than the airplane, such that various frictional forces become large enough to stop the plane from taking off): a) how do I compute the rate of acceleration of the conveyor belt necessary to keep the plane from taking off? b) how would I compute the energy required to make the conveyor belt move that fast? With regards to a), it appears to me that what I need to figure out is: 1) the relationship of the conveyor belt's acceleration to the rolling resistance force of the wheels 2) the relationship of the rolling resistance force to the friction between the wheel and the axle (I'm pretty sure that this is the force that oppose the thrust from the jets--right?) With regards to b), I'm clueless. What I'm looking for here are equations. And kind, gentle guidance if I've already gone hopelessly off-course. Many thanks in advance.
...in the theoretical case in which the wheels are indestructible, and there is no limit to the conveyor belt's speed (the case in which the conveyor belt is allowed to move many many many times faster than the airplane, such that various frictional forces become large enough to stop the plane from taking off): a) how do I compute the rate of acceleration of the conveyor belt necessary to keep the plane from taking off? b) how would I compute the energy required to make the conveyor belt move that fast? With regards to a), it appears to me that what I need to figure out is: 1) the relationship of the conveyor belt's acceleration to the rolling resistance force of the wheels 2) the relationship of the rolling resistance force to the friction between the wheel and the axle (I'm pretty sure that this is the force that oppose the thrust from the jets--right?) With regards to b), I'm clueless. What I'm looking for here are equations. And kind, gentle guidance if I've already gone hopelessly off-course. Many thanks in advance.
This post was deleted for the following reason: This is basically Son of Hypothetical Chatfilter. -- cortex
Response by poster: I'm pretty sure that the friction between the wheel and the axle opposes the forward thrust. (Imagine the extreme example, that this friction is so great that the wheel doesn't spin at all. . .)
PS Apologies for large evil block of text. That's certainly not what my preview looked like. Maybe I should have used "More Inside"?
I always do Ask MeFi wrong.
posted by flotson at 11:01 PM on June 4, 2008
PS Apologies for large evil block of text. That's certainly not what my preview looked like. Maybe I should have used "More Inside"?
I always do Ask MeFi wrong.
posted by flotson at 11:01 PM on June 4, 2008
The limiting factor is of course the friction in the wheel assembly, and if you want to posit an infinitely fast conveyor belt why not completely frictionless wheels? Trying to overthink this scenario beyond what would *really* happen is kind of an arms-race of ridiculous propositions and caveats. As a thought experiment this one is filed under "Induces temporary insanity."
posted by kurtroehl at 11:17 PM on June 4, 2008
posted by kurtroehl at 11:17 PM on June 4, 2008
Friction in bearing assemblies increases roughly linearly with angular velocity, at least at sane velocities, but I don't know how you would determine the bearing torque in the first place. It'll be dependant on factors such as the size of the bearings, the load and the angle that the load is pitched at, heat dissipation mechanisms for when that comes into play, any sideways forces on the wheels, and the speed of the conveyor.
posted by Dipsomaniac at 11:25 PM on June 4, 2008
posted by Dipsomaniac at 11:25 PM on June 4, 2008
I'm in the camp that this question fundamentally misbegotten. If you're modeling a simple axis you're basically looking at a sliding or kinetic friction problem (wheel hub surface sliding over hub) and sliding friction is not dependent on speed, only on the normal force between the two surfaces. Once you overcome the static friction the resistance from sliding friction is basically constant. Think about it this way: you are dragging something on a string across the floor. As you speed up does it get harder to drag, eventually stopping you entirely? No. That is not how friction works. Once your idealized wheels get rolling they can proceed to infinite speeds, the resistance due to friction will remain constant and (as the experimental testing of the plane on conveyor belt demonstrated) negligible. If three quarters of physics 18 years ago means anything.
posted by nanojath at 11:43 PM on June 4, 2008
posted by nanojath at 11:43 PM on June 4, 2008
Flotson: It's not a question of acceleration, because no system can accelerate for ever, otherwise the treadmill would be traveling faster then the speed of light.
It's a question of how much friction is going to be produced in the wheels. Eventually it will reach an equilibrium state where the friction in the wheels is greater then the thrust created by the plains engines. When you get to that point.
If you want to figure out how quick the conveyor belt would have to accelerate in order to get to that equilibrium point should be easy. If you can figure out the force Fwheels, then just use the equation:
Fwheels = Fpropeller
So just figure out how fast the conveyor belt needs to spin to keep force greater then the mass of the plane M times the acceleration created by acceleration normally created by the engines (since F = MA)
posted by delmoi at 11:44 PM on June 4, 2008
It's a question of how much friction is going to be produced in the wheels. Eventually it will reach an equilibrium state where the friction in the wheels is greater then the thrust created by the plains engines. When you get to that point.
If you want to figure out how quick the conveyor belt would have to accelerate in order to get to that equilibrium point should be easy. If you can figure out the force Fwheels, then just use the equation:
Fwheels = Fpropeller
So just figure out how fast the conveyor belt needs to spin to keep force greater then the mass of the plane M times the acceleration created by acceleration normally created by the engines (since F = MA)
posted by delmoi at 11:44 PM on June 4, 2008
Unless I'm reading wrong, we're assuming the wheels have to spin to allow takeoff - at some point, the forward thrust will probably counter the friction of the tires against the tarmac, and the plane will just slide forward.
Plus, unless you have a conveyor that can spin up to the high speed necessary to build up that much friction in the hubs instantly, the plane will have already taken off. Possibly the wheels would have spun to pieces before then as well. If we're going to go to all 'theoretical possibilities', we've gone off the deep end.
I figured once we actually watched the plane take off the conveyor belt, this would have been over, but some beans must remain to be overthought, it seems...
posted by pupdog at 11:52 PM on June 4, 2008
Plus, unless you have a conveyor that can spin up to the high speed necessary to build up that much friction in the hubs instantly, the plane will have already taken off. Possibly the wheels would have spun to pieces before then as well. If we're going to go to all 'theoretical possibilities', we've gone off the deep end.
I figured once we actually watched the plane take off the conveyor belt, this would have been over, but some beans must remain to be overthought, it seems...
posted by pupdog at 11:52 PM on June 4, 2008
Friction in bearing assemblies increases roughly linearly with angular velocity
It's a question of how much friction is going to be produced in the wheels. Eventually it will reach an equilibrium state where the friction in the wheels is greater
Seriously? Did I study physics in Bizarro world? I mean, I know it's been a while, and a bearing problem is obviously more complex than simple kinetic friction of sliding surfaces, but it all boils down to kinetic friction, and kinetic friction is basically independent of speed, and indeed at extremely high speeds kinetic friction is in fact lower. Right? I hope you die, flotson. I hope you die in a conveyor belt plane accident and they put a bumper sticker on your coffin that says "More Inside."
posted by nanojath at 12:03 AM on June 5, 2008
It's a question of how much friction is going to be produced in the wheels. Eventually it will reach an equilibrium state where the friction in the wheels is greater
Seriously? Did I study physics in Bizarro world? I mean, I know it's been a while, and a bearing problem is obviously more complex than simple kinetic friction of sliding surfaces, but it all boils down to kinetic friction, and kinetic friction is basically independent of speed, and indeed at extremely high speeds kinetic friction is in fact lower. Right? I hope you die, flotson. I hope you die in a conveyor belt plane accident and they put a bumper sticker on your coffin that says "More Inside."
posted by nanojath at 12:03 AM on June 5, 2008
The fact that you're asking this question means you still don't understand the answer to the last one. It has nothing to do with how fast the treadmill spins, or how indestructable the wheels are. Imagine a rocket, just laying on the ground. It has no wheels. Now start that rocket up. will it go forward, even without wheels? Yes, because the reason it is moving is because of the ball of flame coming out its back, not because it has wheels that are turning.
Now move that rocket onto a conveyor belt. Again, no wheels. Now start it up. Will it go forward? Yes. Again, it has fire coming out of its back. If it were nailed to the conveyor belt, something would have to give. But it's not nailed down. It can fly. That's the point.
Now, lets put that rocket on wheels, on a conveyor belt. Will it move forward? Yes. Again, it is a flying thing. It just has some wheels coming out the bottom.
The problem that makes this hard for people to understand is that people think planes work like cars, where there is energy being put into the wheels that makes them turn and move on the ground. But planes are not cars. The power comes from the flying engines, the things that will keep pushing it forward, in the air, no matter what the air or ground underneath is doing.
posted by kingjoeshmoe at 12:32 AM on June 5, 2008
Now move that rocket onto a conveyor belt. Again, no wheels. Now start it up. Will it go forward? Yes. Again, it has fire coming out of its back. If it were nailed to the conveyor belt, something would have to give. But it's not nailed down. It can fly. That's the point.
Now, lets put that rocket on wheels, on a conveyor belt. Will it move forward? Yes. Again, it is a flying thing. It just has some wheels coming out the bottom.
The problem that makes this hard for people to understand is that people think planes work like cars, where there is energy being put into the wheels that makes them turn and move on the ground. But planes are not cars. The power comes from the flying engines, the things that will keep pushing it forward, in the air, no matter what the air or ground underneath is doing.
posted by kingjoeshmoe at 12:32 AM on June 5, 2008
Flotson, seeing your example about the time when friction is so strong that the wheel does not spin at all, my response is, the wheel doesn't have to spin. It will jerk up and down as the force of the plane moves it on and off the ground.
posted by kingjoeshmoe at 12:34 AM on June 5, 2008
posted by kingjoeshmoe at 12:34 AM on June 5, 2008
The basis of the question is flawed - Dynamic friction does not increase in line with acceleration. Think of friction this way. You have two saw blades facing each other. When the two saws are stationary, the teeth are locked together. This is static friction, and is quite high. When enough force is applied to move the saws in opposite directions to other, overcoming the static friction, the saw tips brush past each other, bouncing along the top of each other. Less saw surface in contact means dynamic friction, with the objects moving past each other, is LOWER than the static friction in the first place - and since it's already moving, the surfaces bounce more as they move faster, meaning dynamic friction generally *falls* as velocity increases. It certainly doesn't linearly increase indefinitely as the materials slide past each other faster and faster.
Compare push starting sled; the inital push to get it started is hardest, then as the sled starts to slide, it gets easier and easier.
Depending upon the materials, the worst that would happen is so much heat builds up from the constant dynamic friction, or the surface degrades so much they melt or otherwise weld themselves together.
Let's examine that case with the plane. The bearing spin so fast they melt and jam solid. The plane is now effectively on rubber tipped skids. There would be a jerk as the wheel surfaces switch from rolling friction to static friction - briefly. Static friction is a lot higher than rolling friction. If it were sufficiently high (pretty unlikely!) it would stop the plane dead. But otherwise, the locked tyres would start to skid, and again the dynamic friction of the skidding tyres would be lower than the static friction, and continue to be so as either surface accelerates, and the plane would continue to move.
If the static friction in either the bearings to the wheels, or the tyres to the conveyer belt were so high as to stop the plane at some point, the plane would never have enough power to start moving in the first place.
Effectively, you either nail the rocket to the ground with big enough nails to stop it ever moving, or you use nails that allow it to start moving, at which point it pulls the nails out and drags them along the ground and can never be stopped by the drag, since the dynamic friction is always lower than the static friction.
One friction that does increase with speed is air resistance. The plane is flying in an invisible thin fluid (sort of). As the plane goes faster, it hits more air, has to move that out of its way, and encounters ever higher air friction acting as a retarding force. However, that is a function of the plane's speed, not the conveyer belt's. If the plane did not have enough power to overcome air resistance it wouldn't take off from tarmac either; the conveyer belt makes no difference.
Let's finally examine rolling friction. Rolling friction is complex, and does increase with velocity. A bit. But rolling friction between the tyre and the conveyer surface can only increase so far as the conveyer gets faster and the wheel spins faster; eventually, it will be easier for the wheel to skid (dynamic friction) than roll (rolling friction). This is what happens when you drive on ice; because of the lower surface friction, it's often easier for the tyre to skip across the surface than roll with it. And as already examined, once the tyre starts to skid with dynamic friction, that is effectively a fixed drag that doesn't change regardless of the speed of the conveyer.
In anything like real world physics, the plane is simply not affected sufficiently by the conveyer to stop it taking off. If it were, it would never move in the first place.
As long as we're purely examing the relationship between plane, wheels, bearings and conveyer it all works out. In the real world, explosive material failures due to heat buildup are rather to be avoided due to the secondary effects. All that work done to overcome the constant friction has to go somewhere. Having it go on for long enough gives you an awful lot of heat to deal with. Think formula 1 car ceramic brake discs glowing - it might be dynamic friction invariant due to velocity, but they still make it nice and high through material choice, and use it to slow a fast car down in a very short space of time. Oddly enough, ceramic brake discs are relatively low in friction when cold, they're designed to work at high temperatures, and are rubbish when not used heavily, which is why you don't often get them in road cars.
posted by ArkhanJG at 1:38 AM on June 5, 2008
Compare push starting sled; the inital push to get it started is hardest, then as the sled starts to slide, it gets easier and easier.
Depending upon the materials, the worst that would happen is so much heat builds up from the constant dynamic friction, or the surface degrades so much they melt or otherwise weld themselves together.
Let's examine that case with the plane. The bearing spin so fast they melt and jam solid. The plane is now effectively on rubber tipped skids. There would be a jerk as the wheel surfaces switch from rolling friction to static friction - briefly. Static friction is a lot higher than rolling friction. If it were sufficiently high (pretty unlikely!) it would stop the plane dead. But otherwise, the locked tyres would start to skid, and again the dynamic friction of the skidding tyres would be lower than the static friction, and continue to be so as either surface accelerates, and the plane would continue to move.
If the static friction in either the bearings to the wheels, or the tyres to the conveyer belt were so high as to stop the plane at some point, the plane would never have enough power to start moving in the first place.
Effectively, you either nail the rocket to the ground with big enough nails to stop it ever moving, or you use nails that allow it to start moving, at which point it pulls the nails out and drags them along the ground and can never be stopped by the drag, since the dynamic friction is always lower than the static friction.
One friction that does increase with speed is air resistance. The plane is flying in an invisible thin fluid (sort of). As the plane goes faster, it hits more air, has to move that out of its way, and encounters ever higher air friction acting as a retarding force. However, that is a function of the plane's speed, not the conveyer belt's. If the plane did not have enough power to overcome air resistance it wouldn't take off from tarmac either; the conveyer belt makes no difference.
Let's finally examine rolling friction. Rolling friction is complex, and does increase with velocity. A bit. But rolling friction between the tyre and the conveyer surface can only increase so far as the conveyer gets faster and the wheel spins faster; eventually, it will be easier for the wheel to skid (dynamic friction) than roll (rolling friction). This is what happens when you drive on ice; because of the lower surface friction, it's often easier for the tyre to skip across the surface than roll with it. And as already examined, once the tyre starts to skid with dynamic friction, that is effectively a fixed drag that doesn't change regardless of the speed of the conveyer.
In anything like real world physics, the plane is simply not affected sufficiently by the conveyer to stop it taking off. If it were, it would never move in the first place.
As long as we're purely examing the relationship between plane, wheels, bearings and conveyer it all works out. In the real world, explosive material failures due to heat buildup are rather to be avoided due to the secondary effects. All that work done to overcome the constant friction has to go somewhere. Having it go on for long enough gives you an awful lot of heat to deal with. Think formula 1 car ceramic brake discs glowing - it might be dynamic friction invariant due to velocity, but they still make it nice and high through material choice, and use it to slow a fast car down in a very short space of time. Oddly enough, ceramic brake discs are relatively low in friction when cold, they're designed to work at high temperatures, and are rubbish when not used heavily, which is why you don't often get them in road cars.
posted by ArkhanJG at 1:38 AM on June 5, 2008
Apologies, I did make a mistake above - the cost of starting rolling friction is lower than the cost to start skidding (the difference between a plane with its brakes off vs on when first trying to move) so the plane may well be prevented from taking off once the wheels lock due to bearing failure and it starts to skid along the conveyer, even though that skid friction doesn't change based on the speed of the conveyer - if the force from the skid friction of the rubber exceeds that provided by the propellers/jets, with fixed wheels, the plane will eventually slow and stop, then go backwards with the conveyer. Since you specifically removed this as a possible outcome in your hypothetical though, the general points still stand:
When dealing with the contact surfaces of two solids, static friction is greater than dynamic friction, and dynamic friction cannot exceed static friction as pretty much by definition the surfaces will have less surface in contact when moving than when stationary, regardless of the relative speeds.
Failure modes of real materials can cause two moving surfaces to revert to static friction, or at least dramatically increase the dynamic friction.
However - assuming our plane bearings cannot fail, then there is a fixed limit the plane can be slowed down by, and that's the dynamic friction of the wheels on the axle which is speed independant, again ignoring material failures.
posted by ArkhanJG at 3:34 AM on June 5, 2008
When dealing with the contact surfaces of two solids, static friction is greater than dynamic friction, and dynamic friction cannot exceed static friction as pretty much by definition the surfaces will have less surface in contact when moving than when stationary, regardless of the relative speeds.
Failure modes of real materials can cause two moving surfaces to revert to static friction, or at least dramatically increase the dynamic friction.
However - assuming our plane bearings cannot fail, then there is a fixed limit the plane can be slowed down by, and that's the dynamic friction of the wheels on the axle which is speed independant, again ignoring material failures.
posted by ArkhanJG at 3:34 AM on June 5, 2008
F = μN
Is the basic equation for this, by the way. F is the force exterted by friction, N is the normal force exerted by the object perpendicular to the surface, and μ is the coefficient of friction.
the normal force (N) is the mass times the acceleration due to gravity, in newtons, on a flat surface relative to earth. On a slope, the normal force will be smaller.
the coefficient of friction is a property of the two surfaces it's basically how grippy the two surfaces are; and in solids the static coefficient is usually (much) higher than the dynamic coefficient, in some rare cases they can be the same.
The coefficient is down to the material properties, with changes in the material making the coefficient rise or fall, usually based on heat build up - which can be speed dependant, the most obvious being complete melt down.
posted by ArkhanJG at 3:52 AM on June 5, 2008
Is the basic equation for this, by the way. F is the force exterted by friction, N is the normal force exerted by the object perpendicular to the surface, and μ is the coefficient of friction.
the normal force (N) is the mass times the acceleration due to gravity, in newtons, on a flat surface relative to earth. On a slope, the normal force will be smaller.
the coefficient of friction is a property of the two surfaces it's basically how grippy the two surfaces are; and in solids the static coefficient is usually (much) higher than the dynamic coefficient, in some rare cases they can be the same.
The coefficient is down to the material properties, with changes in the material making the coefficient rise or fall, usually based on heat build up - which can be speed dependant, the most obvious being complete melt down.
posted by ArkhanJG at 3:52 AM on June 5, 2008
Screw the treadmill, just bolt the plane to the tarmac or put chocks under its wheels. That's how to keep it from flying. Alternately, apply the brakes to the wheels so that they can't spin, and make sure the weight of the plane is sufficiently high to generate friction against a treadmill surface.
The speed of the treadmill won't matter, now can we please close this before PaulSC comes to muck it all up again?
posted by explosion at 5:58 AM on June 5, 2008
The speed of the treadmill won't matter, now can we please close this before PaulSC comes to muck it all up again?
posted by explosion at 5:58 AM on June 5, 2008
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the conveyor belt is allowed to move many many many times faster than the airplane, such that various frictional forces become large enough to stop the plane from taking off
See, there is no speed of conveyer belt that will prevent the plane from taking off. The plane ignores the relationship between the wheels and the conveyer belt, and the forward motion comes solely from the propeller pulling the plane. If the belt was moving at a million miles per hour, the wheels would just spin backwards that fast while the plane went forward and took off.
posted by Fuzzy Skinner at 10:56 PM on June 4, 2008