:smack: I can’t believe I made that mistake! You’re absolutely right, of course.
Oh well, it has been 4 years since I last taught. I’m either getting rusty or stupid in my old age.
The OP is correct, the plane would not lift off, because there is insufficient wind force to generate lift if the plane is not moving forward. The constraints of the problem state the treadmill is moving in the opposite direction of the wheels at the same speed as the wheels. Therefore, the plane cannot be moving forward regardless of thrust, friction, or any of the other previously raised points. The originally posed problem does not mention anything about assuming the wheel bearings are frictionless. Instead of imagining a full sized 747, think of a balsa rubber-band propeller model airplane with really strong wheels.
Let’s look at it this way: The plane’s engine should accelerate it down the runway regardless of what the treadmill does. BUT we are given that the treadmill’s speed exactly matches that of the wheels, so the plane CAN’T be making any progress down the runway (because that would involve the wheels moving faster than the treadmill). THEREFORE the plane’s engine must not be working. Maybe it’s out of gas or something.
This is a clear and definite statement. It’s also an erroneous one.
The train can keep the rocketman motionless only if it can exert sufficient force on him. If it can exert not force, it can have no influence on his motion with respect to any other frame of reference (e.g. the earth). It can certainly influence his apparent motion with respect to itself, but that’s obvious and of no significance.
If the train can’t exert force on rocketman, then there’s also no significance to the fact that he happens to be quite close to it. The guy rollerblading in the park three miles from the train tracks is equally influenced by the train’s motion, which is to say: not in the slightest.
Your implication is quite correct - the two problems are the same. Depending on assumptions about friction, the treadmill either has a small effect on the plane, or none. In neither case can it prevent the plane from accelerating to flying speed.
This wrong, wrong, wrong. Stop posting erronous information.
Giraffe you can indeed do what you are asking and I will post how in a few minutes but I gotta go put laundry in the dryer.
I agree with this, depending on what is meant by “speed of the wheels”. Say the plane does take off; when it does, it has some positive speed X relative to the ground. The treadmill is going in the other direction at some speed Y, so the wheels are turning at X+Y. But the OP said that the treadmill always goes at the same speed as the wheels, so this is impossible. That’s unless you allow Y to be infinite, so that X+Y = Y.
Okay, let’s pick a sample case: No wind. Piper Cub (which flies happily at 45mph), is rolling north at 50mph, along with its wheels. Treadmill is thus moving south at 50mph.
Why not? The wheels are spinning at a rather brisk clip (equivalent to rolling at 100 mph) but that’s no big deal. The example above meets all your specifications and has the plane moving forward at comfortable flying speed.
Correct - and there’s no need to.
Might I ask that when you post something like this, you undertake to explain yourself?
What sort of sensible response is possible to someone who says nothing beyond “wrong, wrong, wrong”? Why did you post this?
Yes, basically what happens is that the wheels are continously accelerated. Its impossible to do continously becuase you will bump up against the limits of the speed of light and the physical charecteristics of the materials you are using.
What is necessary for the rocket to not move in relation to a person standing on the ground is that the force due to friction between the train and the wheel must counteract the force from the rocket. Lets call the force between the train and the wheel Ft and the rocket force Fr. Now our first relationship is Ft=Fr becuase we want no motion. To find out what Ft is and consequently Atrain we need to look at the wheel. The angular acceleration of the wheel is equal to FtRwheel/Iwheel. We need to look at the point of contact between the wheel and the train becuase those two accelerations must be equal becuase the wheel does not slip. The relationship between linear acceleration and angular acceleration is alpha=Atrain/Rwheel. Now we have three equations Ft=Fr, alpha=FtRwheel/Iwheel and alpha=Atrain/Rwheel. Combining them we end up with Atrain=Fr*Rwheel^2/Iwheel.
Its nonsensical to say that the treadmill is moving with the same speed as the wheels. What you are basically saying is that the units for rotation are a distance/time when they are in fact degree/time. The only thing that makes sense is to assume that the linear speed of the wheels are what they are talking about. That speed is the same as the planes speed forward.
To hopefully ward off anyone being confused by an incorrect answer.
Look guys, I’ve made a big post and a lot of pretty pictures about the question in the OP. My answer is right and the plane will take off.
Saying “wrong, wrong, wrong” clears up confusion and corrects errors?
I think you should post a specific and detailed explanation of what you believed to be wrong, and why.
Yes, I agree it’s nonsensical, that’s where the infinity comes from. I thought it might be meant this way because the OP specifically said the speed of the wheels as opposed to the speed of the plane. Also because there’s this big discussion, and if the OP means it your way, then yeah, the plane just takes off, no problem.
I posted the correct solution in my previous post.
Your Piper Cub is moving north at 50 mph relative to the ground.
Your treadmill is moving south at 50 mph relative to the ground.
The wheels of the Piper Cub therefore have a linear speed of 100 mph relative to their axles.
The problem with this is that the OP states that “the conveyer belt is designed to exactly match the speed of the wheels at any given time.”
Assuming that the speed of the wheels means “speed of the wheels with respect to their axles”, then in your situation above, the speed of the wheels is 100 mph, while the speed of the treadmill is only 50 mph.
No its nonsensical becuase its not the correct units. Its like if I asked you how much you weighed and you told me 3 feet. The units in your answer make no sense in relation to the question I asked. Its the same thing in this situation. I ask you how fast the treadmill is moving and you tell me 3 degrees/second. Its a nonsensical answer becuase a treadmill moves with units of distance/time not degree/time.
The wheels of the plane aren’t moving in relation to the axles.
Surely when it says the treadmill is moving the same speed as the wheel, that refers to the speed at which the wheel is rolling, i.e., a point on the wheels circumference is moving at 50 mph around the axle. With the treadmill also moving 50 mph, the plane has a net speed of zero. Otherwise the wheel would have to be skidding along the treadmill, and presumably that isn’t allowed (because it would make the problem trivial).
but if the wheel is rolling at 100mph, the treadmill has to also be going 100 mph.
No, it doesn’t. The OP says the treadmill matches the speed of the wheels (100), not the speed of the plane (50).
If the circumference of the wheel is 10 feet, and the wheel rotates once per second, it’s rolling at 10 feet per second. No problem at all.
The wheels have some definite angular velocity with respect to their axis of rotation, right?
A given point on the outer edge of the wheel has a tangential velocity equal to the angular velocity times the radius. The magnitude of this tangential velocity is the tangential speed of the wheel at that point, relative to the axis of rotation, which is the axle.
This what I meant by the speed of the wheels relative to the axle.
Right that makes sense becuase you are talking about the translational velocity of the wheel instead of its rotational. But when you say its spinning at X+Y that entails units of rotation. It does make sense to assume they are talking about translational velocity (its what I did). Then there is no problem becuase if your plane is moving 200 mph to the left relative to the ground the treadmill is moving 200 mph to the right relative to the ground and your wheels are spinning at the rate of w=400 mph/r.