please read my post in #163. I am mistaking nothing.
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Interesting physics/logic riddle
- Thread starter sahtt
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please read my post in #163. I am mistaking nothing.
Let's take another look at the actual question I presented so long ago. This is exactly how the question on the 1st page reads:
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"A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?"
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There is no vagueness. You added vagueness. You put wheels where it says 'plane'. If you want to make it more complicated or get theoretical that's great and brings interesting discussion, but don't make it as if that somehow changes the intent of the question. Of course I understand the paradox of if it matches the wheels' speed; if that was the case the plane 'couldn't' move in the first place and the FIRST SENTENCE of the question I proposed would destroy the scenario. I chose this specifically to avoid that.
Is a car on a dyno that's speedo says 60mph going 60mph or 0mph? How much relevency does a plane's wheel speed on an oversized treadmill have to do with the speed of the plane?
zero. cero. ling.
we are in agreement. i just chose to cover all bases and not take anything for granted.
however, there are others like vegasnsx who claim the plane would take off even if the belt matches wheels' speeds.
vega$ nsx said:
Not sure what you mean by 'not take anything for granted' but I get the jist of what you are saying.
Mr. Vega$ has been quite clear in his evaluation of the problem. A couple of his posts earlier in the thread are probably the most detailed/accurate/thorough analyzations I've read on the topic; certainly worth a few minutes of your time to read if you haven't done so yet.
i've read all his posts. he sounds quite educated but it doesn't change the fact that you and he are not in agreement. isn't your position that if the belt matches the wheels' speeds then the plane can't possibly move forward? and that the wheel speed has to be greater than that of the belt in order for it to take off? or did i misunderstand you?
If the belt strictly matches the wheels' speeds than an instant paradox arises as if the plane begins to move foward, in relation to the ground, the wheels have spun faster than the belt has moved, as odd as that sounds.
I didn't notice Vega$ claiming otherwise. However, this thread is quite old, and I haven't read anything besides page 4 in at least a couple weeks so you could be right.
Nick,
The belt can never match the speed of the wheels, unless the plane isn't producing thrust!!! The speed of the wheels will be dictated by the thrust of the engine + the speed of the belt. If you increase the speed of the belt you will increase the wheel speed, but the total will only be the belt speed + thrust of the engines.
Example:
Engines generating enough thrust to go 200mph + belt moving at 400mph = 600mph wheel speed.
Engines generating enough thrust to go 200mph + belt moving at 40,000mph = 40,200mph wheel speed.
The plane will take off!
Now if you want to say that the friction of the wheel bearings could be great enough at some theoretical speed to stop the plane from taking off then I would agree. In reality many things would fail long before this could ever happen so we have to make some assumptions.
Assuming:
Frictionless bearings
Tires that won't explode
Belt that won't explode
etc...
Please realize that I'm not trying to cocky or indignant. I do understand what you are saying and all I’m trying to do is help you understand the solution better.
First off let me clarify a few definitions to get you on the same page. Let’s look at the problem in a 2-D view. Picture a graph with an X and Y axis. The X axis is the direction of the plane moving forward or backwards, while the Y axis is the up and down motion of the plane. Let’s put the plane at 0 point of the X axis facing right. That means the positive position along the X axis means the plane moves forward, while the negative side of the X axis means the plane has moved backwards.
Ok, now let’s take care of some terminology:
MPH is miles per hour. It is a measurement of DISTANCE over TIME . It is a velocity vector which denotes a direction and speed.
RPM is revolutions per minute. It is a rotational speed not a velocity vector. It is a measurement of how many times an object ROTATES over TIME. Note that it has absolutely nothing to do with distance or position in the X or Y axis. Do not make the mistake that RPMs have anything to do with motion or direction in the X or Y axis. A wheel at the 0 point on an X axis can spin at a very fast RPM and yet it can move anywhere on that X and Y axis or not move anywhere at all. Its rotational speed has absolutely nothing to do with its motion or position.
Now let’s go back to our plane example:
Your first mistake is to give a wheel a velocity vector by saying it is spinning at 200 MPH. Wheels don’t spin at an MPH. Wheels spin at RPM. Why is that important? Because there is no correlation to a wheel’s RPM to its motion (as stated above). Sometimes there is a correlation, like a wheel spinning at a certain RPM will result in a car going a certain MPH, but not always. A car on the street with a wheel spinning at a certain RPM will go some MPH in the X direction. However, another car on a free rolling drum, like a dyno, can have its wheels spinning at the same RPMs yet be going 0 MPH. What that means is that you can’t use the wheels RPMs to make any judgment on the vehicle’s speed. You can’t say a wheel spinning is at 200 MPH because that makes no sense. You also can’t say a wheel that spins at 200 MPH will result in an object that is attached to it will move at 200 MPH either because that too makes no sense. You can’t even say a wheel is spinning at an RPM that will result in 200 MPH because as I showed earlier, two identical wheels can be spinning at the same RPMs with two different MPHs. If fact get your mind off of any relationship between a wheel’s RPM and its motion or the motion of the object it is attached to. In a car, the only correlation is that a wheel spinning may impart a force only if there torque applied to the wheel. That force may move the car a certain MPH, but it is the force of torque that moves the car not the wheel’s RPM and there are other factors that affect the force generated on a car other than the wheel’s RPM such as friction. Forces move objects not RPMs.
Ok now with that out of the way, on to the second part. You are probably thinking that if you measure the circumference of the wheel, it will correspond to a certain length on the belt. So if the wheel tries to make one revolution the belt would move that same distance back, essentially making the plane stationary. That is correct except for one problem, a force is pushing the plane causing the wheel to want to try to spin faster. As a result the belt will move faster to counter act the rotational speed of the wheel. Essentially you have a cat chasing its tail, so that both the belt and wheel will reach infinity RPMs. The wheel will try to move faster, the belt will move faster to keep up with the wheel. Assuming the belt and wheel can handle an infinite number of RPMs the plane will still take off because in my earlier post, I’ve shown that there is a fixed amount friction force a wheel will impart on a plane not matter how fast the wheel or belt spins. The friction force is not a function of velocity or rotational speed. If the thrust of the engine is greater than this friction force it will move despite the belt and wheels are moving at infinity RPMs.
So it’s basically a theoretical question because there is no such thing as a belt or wheel that can move at infinite speed. However, in theory, if there was, the plane would still take off.
i've been stressing acceleration from the beginning. you are only thinking in terms of constant speeds. your examples are only valid at constant speeds. as the belt's acceleration increases the wheels' speeds decrease. try not to confuse acceleration with velocity.
i do enjoy your posts because they are lucid and well thought out. i envy your ability to put your thoughts together.
HOWEVER:biggrin: i still respectfully disagree. as you stated, the wheels and belt will increase in rpm's infinitely. this implies that they will accelerate infinitely. as discussed earlier, friction and forces are tied directly to acceleration (not velocity). therefore the thrust applied is negligiable compared to the infinite friction.
How is it that the wheels speed decrease as the belt accelerates? That makes zero sense, you are trying to over complicate a simple problem.
I am not thinking about constant speeds, I was giving examples at set speeds hoping that you could extrapolate the rest of the data.
In my previous example, as the belt accelerates from 200mph to 40,000mph the wheels will always be going faster than the belt at the exact speed the engines would be pushing the plane on a normal runway.
the math in your examples ARE based on constant speeds... and are over simplified.
if 200mph is capable at max thrust, it won't be once the belt accelerates (increased friction). are you saying that acceleration of the belt has no effects on the plane whatsoever?
if you are inside of a semi truck, you can run up and down the length of the truck at the speed your legs choose regardless of how fast the truck is moving... as long as the truck is moving at constant speed. for the same reasons, a plane CAN take off on a belt moving at constant speed.
once the truck accelerates, your speed is no longer dictated purely by your legs. the greater the acceleration, the more your body has to fight to move (I can demonstrate in your CGT :biggrin: ). it is theoretically possible for the truck to accelerate at a rate in which your legs can't move you forward at all. for the same reasons, it is possible for the belt to accelerate at a rate that will negate the thrust of the engines.
edit: i just re-read your post
Carguy! said:
i was not making this assumption. i can only guess you made the assumption thinking the "theoretical speed" is something astronomical and pointless to this excercise. i assumed the bearings to have friction because it does not take great speeds to hold the plane back.
but yes, if the bearings are frictionless... of course the plane will fly.
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Lol, well as long as we are having fun! :smile:
Sorry about this, but you've made an incorrect assumption. Friction forces are tied directly to acceleration but only in the perpendicular direction. The Newtonian formula for friction or rolling friction is:
F = Crr * N
F = Friction or Rolling Resistance Force
Crr = Coefficient of Friction or Rolling Resistance
N = Normal Force (Force = mass * acceleration (aka gravity))
Note that N is the Normal force. It means that it is perpendicular to the direction of travel. As such, the only acceleration is the force of gravity which is constant. Which means that the friction or rolling resistance is a constant number. It doesn't depend on velocity or acceleration in the direction of travel. I know that sounds a bit counter-intuitive but its true. Which means as the wheel and belt accelerates forward, the acceleration has nothing to do with the friction force holding the plane back. It is a set fixed number and will remain a set fixed number even if the wheel and belt go to infinite velocity and acceleration. I know you may have doubts about this formula and equation and intuition may tell you otherwise but modern day dynamics is built around this formula. The only way I could show you otherwise is to actually build a wheel and belt and a torque measurer and show you that as the wheel and belt accelerate, the friction force (torque) remains constant.
i will concede knowing that I can only go off of my intuition, logic and common sense... all of which is flawed i'm sure. i can't argue with science.
having read through the posts, i do believe that you are alone when you say the plane will take off AND the wheels will match the belt's speed.
Well I got to applaude you for taking such a keen interest in the discussion and having a thirst for knowledge. Your questions were certainly thought provoking and had me really reaching deep into my brain to respond.
Since it is clear that you have a pretty good technical aptitude, you should look into getting a college level course or book in Dynamics since few people can grasp it tangibly. If you have a knack for it, I will say it can be quite a lucrative field. :smile:
Ping
Excellent! I totally understand everything you are saying.
Please indulge me and answer a few more questions to see if what you are stating makes sense.
Remember those old fashioned exercise bikes? The ones with a brake caliper on the flywheel on the front. And you could adjust the brake pressure by cranking down on a spring.
Lets say I had one and it was geared so that for every revolution of the pedals the flywheel went around one tenth of a revolution. And you had an identical bike set at the same brake pressure except yours was geared so that for every revolution of the pedals your flywheel went around ten times.
Our personal trainer requires us to both pedal at a constant sixty pedal revolutions a minute. After we both get out flywheels up to speed, who is doing the most work and why?
Regards,
Patrick
Hmm, I might have to crack open my old dynamics textbook for this one. It's been a while since I've done a dynamics problem. This is a pretty generic dynamics question, I just need to remember or look up the equations for torque and work.
Well it's late and I need to get to bed, but let me see if I can take a stab at it really quick. I'm thinking on the fly so if I think out of turn, please give me till the morning to revise myself.
Ok for constant torque, rotational work is defined as:
W=t*theta
W = Work
t = torque
theta = total angle covered in motion
torque = (moment arm) x Force
Ok, so let’s assume that the flywheel has a 10:1 ratio with the wheel, so that every revolution results in 1/10th revolution of the actual wheel. I’m also going to assume that these are stationary bikes so that the wheels are suspended and so I don’t need to look at any additional forces related to the ground.
Bike #1: Each pedal revolution = 1/10th flywheel revolution
Bike #2: Each pedal revolution = 10 flywheel revolutions
Now let’s translate 60 RPMs of the pedal to revolutions of the actual wheel. For calculation's sake let's assume that we are observing this for 1 minute. It is inconsequential as it just takes the time factor out (i.e. per minute) but the theory will hold true for any duration of time or even rate of time.
Bike #1: 60 pedal revolutions * 1/10th flywheel revolution *1/10th wheel revolution = 0.6 revolutions of the wheel (or 216 total degrees)
Bike #2: 60 pedal revolutions * 10 flywheel revolutions * 1/10th wheel revolution = 60 revolutions of the wheel (or 21,600 total degrees)
Ok now let’s look at the forces on the wheel. Since both wheels are identical and the brakes are set at the same pressure (assuming a dynamic coefficient of friction)
Bike #1 torque = Bike #2 torque = ConstantXvalue
This is because the friction force is the same (constant dynamic pressure) and the moment arm (length of spoke) is the same.
Now looking at the Work of each wheel:
W = torque * theta
Bike #1 = ConstantXvalue * 216 degrees
Bike #2 = ConstantXvalue * 21,600 degrees
So by my calcs, clearly Bike #2 will have produced more work and will have “travelled” (i.e. rotated) many more times than Bike #1.
That's just off the top of my head. I think it's right but it's late and I'm not 100% sure until I re-read it in the morning.
How does the explaination sound to you? :smile:
Correct!
I was confident that you would be able to see that.
I am sure you can see where I am going with this.
So if we release the friction brake on both bikes and forget about any friction in the system. Once we get our flywheels up to speed we are doing the same amount of work. But when we apply the friction one of us is going to have use a lot more energy to keep the bike at speed. And the only difference is that you are having to drag your brake pad over a further distance of steel than me. We could also do this experiment in a linear fashion where I have to drag a slab of steel with an elephant standing on it 10 yards in 10 seconds over concrete and you would have to drag the same device 10 miles in ten seconds. You would have to gear your winch differently from mine and you would be doing a lot more work and your poor elephant would have hotter feet than mine. But if I gave you lots more time to complete your task and you could drag at 1 yard per second then we would both be doing the same work per second and we would have the same friction. So it seems like distance and time are a factor when calculating the amount of heat and friction we are dealing with and as you know distance divided by time is velocity.
As you ponder this I want to encourage you to leave the textbooks on the shelf. Let go of the formulae and rules and laws of “classic dynamics”. Just like the true scientist I know you are. Figure this out ab initio using first principals. Through thought experiment and by using the power of your mind. So you can re derive the laws and then we know they are good and we own them.
You might want to do that over the weekend. :smile:
I want to add that I have a lot of respect for you and at the risk of insulting everyone else, I don’t think there are many other people on this thread that I would even consider trying to have this conversation with.
Regards,
Patrick
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