Conceptual Puzzle: Runway Belt

[st.cronin]

Originally Posted by Huckleberry

Which is the angular velocity option you already nixed. Now you are trying somehow to tie in the belt's linear velocity with the wheel's angular velocity. I'd like to see how that can be accomplished. In your example you can't say that the wheel is rotating one foot per second, the wheel is rotating one revolution per second. 360 degrees per second.

No, it isn't.

Picture Earth and Mercury. Imagine that their angular speeds are exactly the same. This means that from 12 oclock to 6 oclock they take the same amount of time. BUT, in terms of what I am talking about, Mercury is moving slower.

[BrianD]

Originally Posted by Huckleberry

Absolutely.

Does it make sense to determine if we've got enough lift to make the plane rise in a world where we have zero friction?

[Huckleberry]

Originally Posted by BrianD

There has to be friction, or the wheels wouldn't move. If the plane can still move forward on the moving runway, then the runway is not moving as fast as the wheels. It means that the moving runway will be spinning the wheels in addition to the plane's thrust is moving the wheels. This means we will have to have a runway moving at infinite speeds which seems to make the puzzle's premise impossible.

It seems to me the component you're missing is that there is a difference between friction between the belt and the wheels and friction between the wheels and the plane.

Yes there is friction between the belt and the wheels. That is why they will spin. A frictionless wheel refers to the internal mechanism of the wheel. So there is no friction between the wheel and the plane.

[JHandley]

Originally Posted by albionmoonlight

Let's try one more thing for the "plane ain't gonna move" crowd.

I'll grant you a suped up runway. Your conveyor belt is moving backwards at 1,000,000,000 mph.

Now, I have a plane whose engine thrust is such that, if it were on solid ground, it would go forward at 400 mph.

When I drop my plane (with engines running) on the conveyor, does it go backwards at 1,000,000,000 mph (less 400 mph). If not, then can you see why the speed of the wheels just does not matter?

If so, then you and I just differ on our understanding of free rolling wheels.

I've been saying all along that the speed of the wheels doesn't matter. My contention is the distance that the wheels are travelling. It doesn't matter whether it's 1,000,000,000 MPH or 20 MPH. A 1 foot circumference cannot travel more than 1 foot in one revolution. It doesn't matter if it takes a ms or a year for that revolution to take place.

Therefore, in order for the 1 foot circumference wheel to travel more than 1 foot in 1 revolution, it has to move without revolving. I still can't see how that's possible.

[Toddzilla]

Originally Posted by BrianD

There has to be friction, or the wheels wouldn't move. If the plane can still move forward on the moving runway, then the runway is not moving as fast as the wheels. It means that the moving runway will be spinning the wheels in addition to the plane's thrust is moving the wheels. This means we will have to have a runway moving at infinite speeds which seems to make the puzzle's premise impossible.

I'm not sure what you are getting at, here.

There has to be friction where? Betweeen the plane's axle and the wheels? No there doesn't. There is (very little) but there should ideally be none. It is the friction of the wheels against the runway that makes then roll.

[Huckleberry]

Originally Posted by st.cronin

No, it isn't.

Picture Earth and Mercury. Imagine that their angular speeds are exactly the same. This means that from 12 oclock to 6 oclock they take the same amount of time. BUT, in terms of what I am talking about, Mercury is moving slower.

The term that you are talking about is linear velocity.

[st.cronin]

Originally Posted by Huckleberry

It seems to me the component you're missing is that there is a difference between friction between the belt and the wheels and friction between the wheels and the plane.

Yes there is friction between the belt and the wheels. That is why they will spin. A frictionless wheel refers to the internal mechanism of the wheel. So there is no friction between the wheel and the plane.

Ok, but if the friction between the runway and the wheels could not be overcome, what would happen? The plane would not be able to take off.

[Toddzilla]

Originally Posted by JHandley

I've been saying all along that the speed of the wheels doesn't matter. My contention is the distance that the wheels are travelling. It doesn't matter whether it's 1,000,000,000 MPH or 20 MPH. A 1 foot circumference cannot travel more than 1 foot in one revolution. It doesn't matter if it takes a ms or a year for that revolution to take place.

Therefore, in order for the 1 foot circumference wheel to travel more than 1 foot in 1 revolution, it has to move without revolving. I still can't see how that's possible.

It is indeed not possible. And no one has suggested that it can otherwise. I do not understand where you are interpreting the facts presented that the wheels are moving more than 1 foot in 1 revolution.

[st.cronin]

Originally Posted by Huckleberry

The term that you are talking about is linear velocity.

Ok, but the point is that it is the same for all points on the wheel.

[Huckleberry]

Originally Posted by JHandley

Therefore, in order for the 1 foot circumference wheel to travel more than 1 foot in 1 revolution, it has to move without revolving. I still can't see how that's possible.

In one revolution the wheel travels one foot relative to the belt. It travels half a foot relative to the ground. Where is it necessary to move more than one foot in one revolution?

[Toddzilla]

Originally Posted by st.cronin

Ok, but if the friction and the wheels could not be overcome, what would happen? The plane would not be able to take off.

What friction is this? Between the wheels and the plane or the wheels and the runway?

[st.cronin]

Originally Posted by Toddzilla

What friction is this? Between the wheels and the plane or the wheels and the runway?

Sorry, dropped a few words - edited for clarity. I mean the friction between the wheels and the runway.

[JHandley]

Originally Posted by Toddzilla

It is indeed not possible. And no one has suggested that it can otherwise. I do not understand where you are interpreting the facts presented that the wheels are moving more than 1 foot in 1 revolution.

Because in order for the plane to move forward, it has to move faster than the conveyor.

If the conveyor moves one foot backwards, the wheel moves one foot forward. In order for the plane to move 2 feet forward, and the conveyor to only move 1 foot backward, the wheel has to either cover more than 1 foot in 1 rotation or it has to have skidded the extra foot.

[Huckleberry]

Originally Posted by st.cronin

Ok, but if the friction between the runway and the wheels could not be overcome, what would happen? The plane would not be able to take off.

Okay, now you have moved on to a completely different problem. The only way that the plane will have to overcome friction between the runway and the wheels is if there is friction between the wheels and the plane. And that means they are no longer free-rolling wheels.

[Huckleberry]

Originally Posted by JHandley

Because in order for the plane to move forward, it has to move faster than the conveyor.

If the conveyor moves one foot backwards, the wheel moves one foot forward. In order for the plane to move 2 feet forward, and the conveyor to only move 1 foot backward, the wheel has to either cover more than 1 foot in 1 rotation or it has to have skidded the extra foot.

Not true. The plane moves one foot forward when the belt moves one foot backward. Take me through the process whereby the plane has to move two feet forward when the belt moves one foot back.

[Toddzilla]

Originally Posted by st.cronin

Sorry, dropped a few words - edited for clarity. I mean the friction between the wheels and the runway.

Well, that is what we have in reality. The rubber wheels and the tarmac on the runway have an enourmous amount of friction, and so the wheels don't skid on the runway*, they roll. Thats what we want to happen. Eventually, during takeoff, there is sufficient lift on the wings to lift the plane into the air and the wheels lose contact with the runway.

*they skid a little when the plane lands, but they don't keep skidding, the friction makes them start rolling.

[KWhit]

Originally Posted by JHandley

Because in order for the plane to move forward, it has to move faster than the conveyor.

If the conveyor moves one foot backwards, the wheel moves one foot forward. In order for the plane to move 2 feet forward, and the conveyor to only move 1 foot backward, the wheel has to either cover more than 1 foot in 1 rotation or it has to have skidded the extra foot.

Right. That's the part that doesn't make sense.

In fact, I think once you start to apply force to the plane from the jet, it becomes impossible for the wheel speed to match the conveyor speed - making the puzzle impossible.

[Toddzilla]

Originally Posted by JHandley

Because in order for the plane to move forward, it has to move faster than the conveyor.

If the conveyor moves one foot backwards, the wheel moves one foot forward. In order for the plane to move 2 feet forward, and the conveyor to only move 1 foot backward, the wheel has to either cover more than 1 foot in 1 rotation or it has to have skidded the extra foot.

Incorrect.

In your example, the conveyor moves 1 foot backwards, the plane 2 feet forward. The plane is now 3 feet ahead of the starting point, and the wheel covered 3 feet and rotated 3 times.

[KWhit]

Originally Posted by Huckleberry

Not true. The plane moves one foot forward when the belt moves one foot backward. Take me through the process whereby the plane has to move two feet forward when the belt moves one foot back.

You stated it wrong, I think. We're saying this:

The WHEEL has to move two feet forward when the belt moves one foot back (in order for the plane to move 1 foot forward).

[Huckleberry]

Originally Posted by KWhit

In fact, I think once you start to apply force to the plane from the jet, it becomes impossible for the wheel speed to match the conveyor speed - making the puzzle impossible.

I already showed how the wheel speed matches conveyor speed.

[st.cronin]

Regardless of what causes the wheels to move ... in order for the plane to achieve lift, it has to move forward. By moving forward, the friction between the wheels and the runway will cause the wheels to move. The runway will move in response to this movement of the wheels.

Now, I think there are two choices about what happens next. If the treadmill has no ability to carry the plane, as some people think, all it will do is speed up the wheels, creating a feedback loop that results in infinite rotational speed for the wheels and the treadmill.

If it is capable of carrying the plane, it will exactly cancel out the plane's forward motion - the force of the jet will result in motion of the treadmill, not motion of the plane.

[Huckleberry]

Originally Posted by KWhit

You stated it wrong, I think. We're saying this:

The WHEEL has to move two feet forward when the belt moves one foot back (in order for the plane to move 1 foot forward).

No it doesn't. Explain why.

[KWhit]

Originally Posted by Toddzilla

Incorrect.

In your example, the conveyor moves 1 foot backwards, the plane 2 feet forward. The plane is now 3 feet ahead of the starting point, and the wheel covered 3 feet and rotated 3 times.

Right. And I'm saying that:

Conveyor moved 1 foot.
Wheel moved 3 feet.

Those aren't the same speed.

[KWhit]

Originally Posted by Huckleberry

I already showed how the wheel speed matches conveyor speed.

[Huckleberry]

http://www.operationsports.com/fofc/...&postcount=151

Starting with that scenario, let's analyze what happens in one revolution of the wheel.

One wheel revolution:

The wheel revolves one time. It travels one foot relative to the belt based on the zero velocity at the belt point of contact. When the wheel travels one foot relative to the belt it has traveled half a foot relative to the ground. Therefore the belt has traveled negative half a foot relative to the ground. 1/2 - (-1/2) = 1 foot which equals the relative movement between wheel and belt. Therefore it takes two revolutions of the wheel for the wheel to travel one foot relative to the ground. In two revolutions the belt travels one foot backward relative to the ground, the wheel and plane travel one foot forward relative to the ground, and the wheel and plane travel two feet forward relative to the belt.

[st.cronin]

The puzzle does not say "ground speed." The wording of the puzzle implies a rotational speed.

[JHandley]

This might be theraputic as I try to resolve this in my own head.

Ok, if the wheel makes one revolution, then the wheel has travelled one foot. If the conveyor matches the speed of the wheel, then the conveyor will move backwards one foot. 1 foot forward = 1 foot backwards, however there is no airflow under the wings as relative to the ground, the plane has not moved.

In order for the wheel to move one foot, relative to the ground, it has to move an extra foot. That is, 1 foot forwards = 1 foot backwards, now you have to add a foot relative to the ground. Since the speed of the conveyor exactly matches the speed of the wheel, the only way to accomplish that is for the force of the jets to push the wheels forward an extra foot without the wheel rotating.

[KWhit]

Originally Posted by st.cronin

The puzzle does not say "ground speed." The wording of the puzzle implies a rotational speed.

That's my take too.

[Toddzilla]

Originally Posted by KWhit

Right. And I'm saying that:

Conveyor moved 1 foot.
Wheel moved 3 feet.

Those aren't the same speed.

No, you're not saying that. CAPS added for emphasis, not shouting

Conveyor moved 1 foot RELATIVE TO THE GROUND
Wheel moved 3 feet. RELATIVE TO THE CONVEYOR BELT

You cannot compare those.

Wheel moved 3 feet. RELATIVE TO THE CONVEYOR BELT
and
Conveyor Belt moved 3 feet RELATIVE TO THE WHEEL.

[st.cronin]

Originally Posted by JHandley

This might be theraputic as I try to resolve this in my own head.

Ok, if the wheel makes one revolution, then the wheel has travelled one foot. If the conveyor matches the speed of the wheel, then the conveyor will move backwards one foot. 1 foot forward = 1 foot backwards, however there is no airflow under the wings as relative to the ground, the plane has not moved.

In order for the wheel to move one foot, relative to the ground, it has to move an extra foot. That is, 1 foot forwards = 1 foot backwards, now you have to add a foot relative to the ground. Since the speed of the conveyor exactly matches the speed of the wheel, the only way to accomplish that is for the force of the jets to push the wheels forward an extra foot without the wheel rotating.

I think the way some people are working around this is to claim that the treadmill is actually moving in relation to the wheel. I think this is absurd, since even a runway will match the wheel's speed in this way, therefore the puzzle makes no sense.

[Huckleberry]

Originally Posted by st.cronin

The puzzle does not say "ground speed." The wording of the puzzle implies a rotational speed.

No it doesn't. You still don't understand that you can't describe rotational speed in feet per second. You are talking about translational speed even though you don't realize it.

[Huckleberry]

Originally Posted by JHandley

Ok, if the wheel makes one revolution, then the wheel has travelled one foot. If the conveyor matches the speed of the wheel, then the conveyor will move backwards one foot. 1 foot forward = 1 foot backwards, however there is no airflow under the wings as relative to the ground, the plane has not moved.

This is your error. I edited my above post, please review.

If the wheel makes one revolution then the wheel has traveled half a foot relative to the ground.

[st.cronin]

Originally Posted by Toddzilla

No, you're not saying that. CAPS added for emphasis, not shouting

Conveyor moved 1 foot RELATIVE TO THE GROUND
Wheel moved 3 feet. RELATIVE TO THE CONVEYOR BELT

You cannot compare those.

Wheel moved 3 feet. RELATIVE TO THE CONVEYOR BELT
and
Conveyor Belt moved 3 feet RELATIVE TO THE WHEEL.

Why does it have to be relative to each other? Why can't both move in relation to some other object (say, the ground?)

If you allow the movement to be measured in relation to each other, it doesn't matter at all what the treadmill does. It can move forwards, backwards, be made of conrete and not move at all. I don't think that's in the spirit of the puzzle.

[st.cronin]

Originally Posted by Huckleberry

No it doesn't. You still don't understand that you can't describe rotational speed in feet per second. You are talking about translational speed even though you don't realize it.

Of course you can. How fast does the earth move around the sun? That's ROTATIONAL speed.

[Huckleberry]

Originally Posted by st.cronin

Why does it have to be relative to each other? Why can't both move in relation to some other object (say, the ground?)

I just translated (get it?) the movement to the ground. Hopefully that's easier to understand.

[Huckleberry]

Originally Posted by st.cronin

Of course you can. How fast does the earth move around the sun? That's ROTATIONAL speed.

No, that's the earth's linear velocity in its revolution around the sun. The earth's average rotational speed around the sun is 360 degrees per year.

[st.cronin]

Originally Posted by Huckleberry

http://www.operationsports.com/fofc/...&postcount=151

Starting with that scenario, let's analyze what happens in one revolution of the wheel.

One wheel revolution:

The wheel revolves one time. It travels one foot relative to the belt based on the zero velocity at the belt point of contact. When the wheel travels one foot relative to the belt it has traveled half a foot relative to the ground. Therefore the belt has traveled negative half a foot relative to the ground. 1/2 - (-1/2) = 1 foot which equals the relative movement between wheel and belt. Therefore it takes two revolutions of the wheel for the wheel to travel one foot relative to the ground. In two revolutions the belt travels one foot backward relative to the ground, the wheel and plane travel one foot forward relative to the ground, and the wheel and plane travel two feet forward relative to the belt.

Your error, to me, is starting with "relative to the belt". I think that's not in the spirit of the puzzle.

[st.cronin]

Originally Posted by Huckleberry

No, that's the earth's linear velocity in its revolution around the sun. The earth's average rotational speed around the sun is 360 degrees per year.

Whatever you want to call it, it can be measured. It's how a (non-digital) clock works.

edit - it is similar, but not identical with, angular speed. Angular speed is constant regardless of how large/small the wheel is. Since the wheel is a fixed size it doesn't really matter whether you call it angular speed or something else. But, the wheel is not the same size as the treadmill - so you have to use this different measure when comparing them. You can't compare the speeds of Saturn and Mercury using angular velocity - well, you could, but that would be for different purposes.

[albionmoonlight]

Originally Posted by KWhit

That's my take too.

But then don't we just end up with an absurd result? The wheel spinning makes the ground move faster (because of the way that you read the problem), which makes the wheel move faster (because of physics), which makes the ground move faster (because of the way that you read the problem), which makes the wheel move faster (because of physics), which makes the ground move faster (because of the way that you read the problem), which makes the wheel move faster (because of physics), which makes the ground move faster (because of the way that you read the problem), which makes the wheel move faster (because of physics), which makes the ground move faster (because of the way that you read the problem), which makes the wheel move faster (because of physics), which makes the ground move faster (because of the way that you read the problem), etc.

Don't we end up with an infinite belt pretty quickly there? (Which belt, by the way, still would not keep the plane from taking off, but it would have those wheels spinning pretty fast by the time that the plane did take off).

[Huckleberry]

Originally Posted by st.cronin

Your error, to me, is starting with "relative to the belt". I think that's not in the spirit of the puzzle.

What are you talking about? The belt is designed to correlate its movement relative to the wheel. But somehow describing the wheel's movement relative to the belt is not in the spirit of the puzzle? That's absurd.

If a 1-foot circumference wheel rotates one time on a belt the wheel will have moved one foot relative to the belt (no skidding). That is a fact. Not sure what the spirit of anything has to do with it.

[KWhit]

Originally Posted by albionmoonlight

But then don't we just end up with an absurd result?

Yes. But I think this is a pretty absurd puzzle, anyway.

[Huckleberry]

Originally Posted by st.cronin

Whatever you want to call it, it can be measured. It's how a (non-digital) clock works.

That's fine, I'm willing to work with this incorrect assumption. So explain to me what the belt is matching. Exactly.

You seem to be saying, correct me if I'm wrong, that the belt is designed to move backward one foot when the wheel rotates one foot. Is that correct? If that's correct then the belt will move backward one foot when the wheel rotates one time. Which is exactly what my scenario describes.

[st.cronin]

Originally Posted by Huckleberry

What are you talking about? The belt is designed to correlate its movement relative to the wheel.

NO! The belt is supposed to make it's movement EQUAL to the wheel's movement, relative to, I assume, something else. Two objects in relation to each other will ALWAYS have identical, but opposite, movement. So saying the movement of the belt is measured in relation to that which it is supposed to be equal to, it doesn't matter what it's motion is.

[KWhit]

Originally Posted by Huckleberry

One wheel revolution:

The wheel revolves one time. It travels one foot relative to the belt based on the zero velocity at the belt point of contact. When the wheel travels one foot relative to the belt it has traveled half a foot relative to the ground. Therefore the belt has traveled negative half a foot relative to the ground. 1/2 - (-1/2) = 1 foot which equals the relative movement between wheel and belt. Therefore it takes two revolutions of the wheel for the wheel to travel one foot relative to the ground. In two revolutions the belt travels one foot backward relative to the ground, the wheel and plane travel one foot forward relative to the ground, and the wheel and plane travel two feet forward relative to the belt.

I was almost with you, but then I relaized that I don't think you're actually calculating speeds equally. You're NOT comparing them both relative to the ground (the wheel, yes, but not the conveyor).

Forgive my lack of using the correct terms, but you're calculating 2 different things here, aren't you? For the wheel, you're calculating how far the entire wheel traveled through space (relative to the ground). But for the conveyor, you are measuing how far a point on the conveyor traveled - how far did the point rotate? It seems to me if you use that calculation for the conveyor, you must use the same for the wheel which gives you a totally different result.

[st.cronin]

Originally Posted by Huckleberry

That's fine, I'm willing to work with this incorrect assumption. So explain to me what the belt is matching. Exactly.

You seem to be saying, correct me if I'm wrong, that the belt is designed to move backward one foot when the wheel rotates one foot. Is that correct? If that's correct then the belt will move backward one foot when the wheel rotates one time. Which is exactly what my scenario describes.

I realize the belt is not a circle, but imagine that it is, and imagine that its circumference is 360 feet.

The wheel (1 foot circumference) rotates one time. The belt rotates 1 degree the opposite direction.

A point on both surfaces has moved one foot, in opposite rotational directions. This, to me, is what is meant by equal and opposite movement.

[MJ4H]

This may or may not help some of you:

http://www.straightdope.com/columns/060203.html

[Huckleberry]

Originally Posted by st.cronin

NO! The belt is supposed to make it's movement EQUAL to the wheel's movement, relative to, I assume, something else. Two objects in relation to each other will ALWAYS have identical, but opposite, movement. So saying the movement of the belt is measured in relation to that which it is supposed to be equal to, it doesn't matter what it's motion is.

I'm not gonna lie. This is getting painful. Setting yourself equal to something is a pretty strong way of correlating yourself to it, wouldn't you say?

At this point I would give up if I were wiser. It's just too frustrating seeing people struggle with something like frames of reference.

[albionmoonlight]

Originally Posted by KWhit

Yes. But I think this is a pretty absurd puzzle, anyway.

Actually, according to relativity, as the belt and wheels approached infinite speed, their mass would increase. Eventually, their mass would become so great as to create a black hole, which would suck everything into itself, including the plane.

So, in that instance, the plane would not take off.

Having given a bit more thought to the absurd result, I like it.

[KWhit]

AH HA!

[st.cronin]

Originally Posted by MattJones4Heisman

This may or may not help some of you:

http://www.straightdope.com/columns/060203.html

In other words, you phrased the question wrong.