What if the pilot was the director of an all swine choral group?
Just wondering.
Neither sides of this abortion of a question will ever convince the other.
Sorry; I was phrasing my statement for the case where everything begins at rest and the treadmill accelerates up to the given speed.
You’re talking about the case where a stationary and unpowered “plane” (or whatever rollable object) is dropped onto a treadmill that is already at a constant velocity. In that case, the treadmill imparts a frictional force on the tire. Since that force isn’t infinite, the tire will slip. However, the force will cause the wheel to begin rotationally accelerating until it stops slipping. The force is also passed on, through the axle, to the fuselage of the “plane”, and the whole thing will translationally accelerate.
There are two factors that couple the treadmill to the wheel to the fuselage, creating the force on the fuselage. The first is the force due to the rotational inertia of the wheel. This force exists only when the wheel is rotationally accelerating. If this were the only coupling, the system would reach equilibrium when the tire stopped skidding, and the plane would from then on travel at a velocity that is less than that of the treadmill (but greater than zero) and the wheels would continue to turn on the axle.
The second force is friction. The frictional moment exists as long as the wheel is turning on the axle, so the force on the fuselage would continue to accelerate the fuselage until it matched the speed of the treadmill (and the wheels ceased turning on their axles). Note that IF the frictional moment were independent of rotational speed (as Chronos suggested), the acceleration of the fuselage would be the same whether the treadmill was going 5mph or 1,000,000,000 mph (because the force would be the same). It’s just that the acceleration would stop earlier in the case of the slower treadmill.
Note also that I’m ignoring air resistance and lift and all the other forces here to concentrate solely on the forces between treadmill and wheel and fuselage. When other forces are present, such as engine thrust or the tension in a rope, then the resulting accelerations and velocities would change.
Eh. Depends on how you define a “side.” There’s a whole bunch of people that simply don’t realize that other people interpret the question differently than they do, and some people that don’t understand the phyics. Neither group seems to be traditionally organized on a “side.”
Zhing said:
You seem to have a conceptual problem. If the plane engine is on, then the treadmill must be turning backwards for the plane to be stationary. Unless you tie the plane to the ground with cables or something, but that’s not the setup. The setup is a free-standing plane that is sitting on a treadmill, and the plane engine is powered to attempt takeoff. Ergo, the treadmill is on, running backwards.
Think of it this way, if the plane and the treadmill are not moving, then the wheels cannot be moving. The only way the wheels can move is if the plane, the treadmill, or both are moving.
Exactly correct. If the plane doesn’t move, the treadmill doesn’t move.
Once the plane starts moving, the treadmill starts moving.
Neither device has any ability to affect the speed of the other, so the plane and treadmill will both merrily speed up until the plane reaches takeoff velocity, at which point it takes off. What the treamill does at that point is anybody’s guess.
My (and other but not all posters) goal is better understanding of an interesting physics problem. The complexity makes it fun. No sides, doesn’t matter what the answer is, as long as it’s correct.
I just realized, when I read your answer I misinterpreted “based on acceleration not velocity” as meaning only the acceleration of the belt is interesting the velocity is not. But I believe what you actually meant was that the speed differential between the wheel and the belt causes the wheel to accelerate which allows a force to be imparted, once the speed is matched (whether 5 mph or 1,000,00,000 mph), no more force is imparted due to that component.
Because I misinterpreted I re-asked the simpler case of treadmill moving and setting an object on it to clarify.
Many people have made this similar observation but the original question put no limits on the speed of the treadmill. Take for example a Mythbusters type experiment. If one were to use a toy plane without the huge power output of a real plane it seems a rather easy task to run a treadmill up to a speed whereby the friction would pull the plane backwards.
The thing that surprises me about this question, in regards to Dopers answers, is that no one has done the math. You guys are famous for that. I don’t have the skills but it seems like it would be easy enough to calculate the thrust of say a small prop plane and then the (approximate) drag in bearing friction and ‘tire to surface’ friction so you could then calculate how fast the treadmill would have to be spinning to overcome the thrust. I’m guessing anything short of the SOL is allowed.
The Mythbusters did that experiment. They put a toy plane on a treadmill. It took off.
Here’s some math. Treadmill + airplane = airplane takes off.
Zhing is correct. If the plane is not moving forward then the belt, which is moving in the opposite direction and speed, wouldn’t be moving at all.
But, I don’t think those that don’t understand the concept, and thus, don’t understand that the plane will take off, will find that point very convincing.
Thanks for the snark. If you re-read my posts you will see that I reference the Mythbuster test and why I feel it was flawed. It did not meet the requirements of the original question - The treadmill was not variable and did not match the opposing wheel speed and it wasn’t going fast enough.
Here’s some higher math:
Fast enough treadmill + airplane = no take off
A solid point; there’s no reason to assume the “plane” is a 747. In fact, constructing an RC plane with small engines or huge wheels or sticky bearings would be interesting.
David Simmons, RIP Seems like I recall someone else doing additional calculations, but I looked through a dozen pages and didn’t see them, and I didn’t feel like going through the other three dozen searching for something that might or might not be there.
Doesn’t it, though? I thought the treadmill could only go as fast as the plane was moving in the other direction. That sounds like a limit to me.
He shoots. . . he scores!
I guess the treadmill will only go as fast as the plane’s required “lift-off” speed.
That is not correct. The question poses that the treadmill’s speed matched the wheel’s speed not the planes speed. What I was addressing was Chronos’ (and several other poster’s) point:
“Now, assuming that everything’s textbook-standard, it is possible for the treadmill to keep the plane stopped, but it has to be continually accelerating, and at a truly ludicrous rate”
The dismissive point seems to be ‘this could only work if the the treadmill was going insanely fast’. I was pointing out that there is nothing in the original post that prevents the treadmill from going insanely fast. Indeed the bit about matching the speed of the wheel seems to predict a really fast speed - it should be obvious as we are talking about matching the take off speed of the plane at the very least. No standard treadmill does that.
It depends on your interpretation of the original question (as well as the phrasing of the question you’re interpreting).
Well I suppose that’s true… I first heard the question that the treadmill matches the plane’s speed. I didn’t realize there were still people toying with it matching the wheels’ speed. Bear with me because I’m a bit slow, but if the plane were able to taxi at 1 MPH, then the wheels would be moving 1, so the treadmill would go 1 backwards. But now the wheels are going 2, which means the treadmill is going 2, which means the wheels are really going 3, which means… that such a wording of the question is basically retarded, because it requires a treadmill to instantly ramp up to infinity as soon as the plane has any non-zero forward velocity.
Correct thinking process, but incorrect conclusion (not unusual–Cecil himself came to the same incorrect conclusion at the end of his first article). If you interpret the question as relating to wheel speed, it just requires the treadmill to hold the plane at zero speed with respect to the ground. The plane can still move with repect to the conveyor belt, though.
I don’t believe this is true. I don’t think I agree with your reasoning either (I am really tempted to build a treadmill that would work at speeds sufficient to hold back an RC plane to prove (or disprove) this once and for all). Suppose for example we were using a car instead of a plane. Using your above numbers, if the car was driving at 1mph, the belt would counter by going 1mph backward. The car is now stationary with the wheels still moving @ 1mph - not 2.
Regardless, the increased thrust of the plane will increase the speed of the belt but not to infinity. As soon as the friction of the bearings and the tire-to-surface friction is equal to the thrust output of the plane the belt will stabilize.
Johnny did a great job of listing a lot of the plane/treadmill threads but not the one that started it all (if memory serves me well). I believe the original thread predated Unca Ceeses column.
It’s really weird that so many smart people have twisted around this question and omitted the belt’s “control system”, which is central to the question.
The top belt speed is exactly equal to and in the opposite direction of the top plane speed before take-off. The belt could never get so fast as to stop the forward motion of the plane, because the belt speed is directly tied to the forward speed of the plane.
There’s a staggering amount of selective reasoning going on in the thread.
Already covered:
As well as: