A harmless little riddle

KnowItAll wrote:

But the force required to hold an accelerating wheel in place is proportional to the rate of acceleration.

I don't know man I've done a lot of looking around (on google and wikipedia) and I haven't found any indication that this is true. Do you have an equation or something that it comes from?

KnowItAll wrote:

But the force required to hold an accelerating wheel in place is proportional to the rate of acceleration.

admire the persistance anyway wrote:

I don't know man I've done a lot of looking around (on google and wikipedia) and I haven't found any indication that this is true. Do you have an equation or something that it comes from?

I don't have the equation in front of me, but you can go throw a cueball on a treadmill and turn on the treadmill and watch it go off the back. A cueball doesn't have the internal resistance of bearings, but there is still a force required to keep it in place if the treadmill is accelerating. That force is proportional to the rate of acceleration. I think some dude actually posted an equation on here several pages back that showed the force required by the engines is proporational to the acceleration of the treadmill, as well as some physical characteristics of the wheel.

KnowItAll wrote:

I don't have the equation in front of me, but you can go throw a cueball on a treadmill and turn on the treadmill and watch it go off the back. A cueball doesn't have the internal resistance of bearings, but there is still a force required to keep it in place if the treadmill is accelerating. That force is proportional to the rate of acceleration. I think some dude actually posted an equation on here several pages back that showed the force required by the engines is proporational to the acceleration of the treadmill, as well as some physical characteristics of the wheel.

I think "admire the persistance" is right in this case. I don't think the forces in the airplane wheels are dependant on the speed that the wheels are turning.

There are only two forces at work - rolling resistance and friction.

Rolling resistance is defined by the deflection of the tire, the surface that the tire is riding upon, and the weight of the plane.

The equation is here:

The other force is friction, which IS dependant on speed only because of the difference in the static versus dynamic coefficients of friction - HOWEVER, the dynamic coefficient of friction is LOWER than the static coefficient of friction, and so, there will be LESS frictional force when the plane's wheels are turning.

However, once the plane's wheels turn faster and faster, there will be more heat engergy disappated through the bearings, and the lubricant will eventually breakdown causing metal to metal contact. This will significantly increase the coefficient of friction, resulting in a much higher frictional force.

That's the best I can do to tie the force that the treadmill imparts on the plane to the speed that the plane is moving.

In the cueball example, you have three forces: 1) the inertial force of the cueball as a result of the ACCELERATION of the treadmill, not velocity, and the mass of the cueball (or more correctly, moment of inertia) as well as 2) the frictional force holding the ball in place, which is a function of the coefficient of friction between the ball and the surface, and the mass of the ball. The third force is the rolling resistance between the ball and the treadmill surface, which in this case, is really just a function of the cueball's mass and the deflection of the treadmill surface.

The ball will move relative to the ground ONLY if the acceleration of the treadmill is slow enough that the resulting inertial force is LESS than the rolling resistance and frictional force combined. Interestingly, whether the ball will roll or slide when the inertial force is greater than these two is dependant on whether the rolling resistance force is greater than the frictional force.

If you accelerate the treadmill fast enough, the ball won't move backwards with the treadmill surface, but will most likely spin relative to the treadmill surface.

I didn't really explain why the ball would spin in the cueball example. The ball will spin because the inertial force acts through the center of mass of the cueball, and the inertial force is always in the direction opposite the acceleration. Therefore, the inertial force in combination with the frictional force will produce a moment that causes the ball to spin.

Hmm, I'm not sure about your conclusion. I admitted it wasn't the speed, but the acceleration that increased the inertial "force". That is the main force I was considering. Perhaps if there were no rolling resistance, than the treadmill would supply all of the inertial force? Would the engines only be countering frictional and rolling resistance forces then?

KnowItAll wrote:

Hmm, I'm not sure about your conclusion. I admitted it wasn't the speed, but the acceleration that increased the inertial "force". That is the main force I was considering. Perhaps if there were no rolling resistance, than the treadmill would supply all of the inertial force? Would the engines only be countering frictional and rolling resistance forces then?

There are two inertial forces in the plane - 1) the inertial force in the wheels and 2) the inertial force acting on the entire airplane itself.

The inertia in the wheels is a result of angular acceleration, which in a rotating object, is directed radially inward. This causes an inertial force in the direction opposite, so the inerial forces is directed outward from the center of mass in ALL directions. Did you ever see the Mythbusters episode where they spun a CD so fast that it exploded? This inertial force literally "pulling" the rotating object apart is the reason it exploded.

The other inertial force is the force acting on the entire plane in the direction OPPOSITE the acceleration. So in our case, the acceleration of the treadmill would cause an inertial force that would tend to act on the plane in the direction opposite the acceleration. So, this inertial force would not impede the FORWARD motion of the plane relative to the ground, but rather keep the plane from moving in reverse. Think of accelerating a car in reverse - any person in the car would tend to be pushed toward the front of the car due to inertia.

So basically, inertial force caused by the treadmill's acceleration does nothing to impede the forward motion relative to the groundspeed of the airplane.

Ok, I was misunderstanding your use of the word inertial force. Thanks for the explanation.

Let's see if I can phrase this question correctly.

To spin a wheel up to speed, you must give it kinetic energy. You have to impart a force on the wheel to supply this kinetic energy. Is this force completely supplied by the treadmill or is it split between the treadmill and the force of the engines equally? I know the actual force is supplied by the surface of the treadmill. But is that a direct force, or is it split by a reactive force from the engines trying to push the wheel forward with respect to the treadmill surface.

For example, on solid ground, if you started the engines, they would push the plane forward, and because of friction between the ground and the wheel, there would be a counter-force to the forward movement that imparted the wheel to spin. This force would be equal and opposite to the force of the engines (minus the force of the engines put into the inertial energy of the body of the plane).

But on a treadmill, the equation gets trickier because the treadmill is imparting a force on the wheels and the engines are imparting a similar force as if the plane were on solid ground. Yeah?

Oh, and this was the force that I was saying would increase proportionally to acceleration. But if the force is supplied entirely by the treadmill, then it's irrelevant. That's why I'm trying to determine that.

no it will crash right into the huge television screen that they attach to treadmills nowadays and blowup before takeoff.

KnowItAll wrote:

For example, on solid ground, if you started the engines, they would push the plane forward, and because of friction between the ground and the wheel, there would be a counter-force to the forward movement that imparted the wheel to spin. This force would be equal and opposite to the force of the engines (minus the force of the engines put into the inertial energy of the body of the plane).

If this was the case, then no aircraft could take off from any surface. But the treadmill and the wheels are red herrings. Some aircraft don't need or have wheels, seaplanes for instance.

That would have to be one big ass treadmill.

From thestraightdope.com

Dear Cecil:

Cecil, always enjoy your column, however you've got this [airplane and conveyor belt business] absolutely wrong. . . --strafe, via the Straight Dope Message Board

It's all about the interpretation of the question. Unfortunately, Cecil commingled two different interpretations in his column. --zut, via the SDMB

My confidence in Cecil has taken a gigantic hit. . . Cecil has fallen into the common trap of believing that the velocity of the treadmill in this case is what is important. It's not. What is important is the acceleration of the treadmill. I swear, on pain of retaking physics before I graduate as an ME, that if I accelerate the treadmill at a rate of 2 x (force from engines)/(mass of tires) that the plane goes absolutely nowhere. treis, via the SDMB

Cecil replies:

I knew this was going to happen. Everyone else, forgive me. This week's column is for the geeks.

Here's the original question: "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?" (The Straight Dope: 060203.)

The implicit assumption is that if the conveyor belt's speed backward exactly counteracts the airplane's "speed" (whatever that means) forward, the plane remains stationary relative to the earth and, more importantly, to the air. (We assume the winds are calm.) With no wind moving past its wings, the plane generates no lift and can't take off.

But the assumption is false. While the conveyor does exert some modest backward force on the plane, that force is easily overcome by the thrust of the engines pulling the plane ahead. The plane moves forward at roughly its usual speed relative to the ground and air, generates lift, and takes off. Many people have a hard time grasping this (although it can be easily demonstrated in the lab), but eventually they do, smack their foreheads, and move on. We'll call this Basic Realization #1.

Message-board discussions of this question tend to feature a lot of posters who haven't yet arrived at BR #1 talking right past those who have, insisting more and more loudly that the plane won't take off. Then there's a whole other breed of disputants who, whether or not they've cracked the riddle as originally posed, prefer to reframe it by proposing progressively more esoteric assumptions, refinements, analogies, etc. Often they arrive at a separate question entirely: Is there a way to set up the conveyor so that it overcomes the thrust of the engines and the plane remains stationary and doesn't take off?

The answer is yes. Understanding why is Basic Realization #2.

The conveyor doesn't exert much backward force on the plane, but it does exert some. Everyone intuitively understands this. To return to the analogy in my original column, if you're standing on a treadmill wearing rollerblades while holding a rope attached to the wall in front of you, and the treadmill is switched on, your feet will initially be tugged backwards. Partly this is due to friction in the rollerblade wheel bearings, but partly--this is key--it's because the treadmill is accelerating the rollerblade wheels and in the process imparting some angular (rotary) but some linear (backward) momentum to them. You experience the latter as backward force. Eventually the treadmill reaches a constant speed and the rollerblade wheels cease to accelerate. At this point you can easily haul in the rope and pull yourself forward.

But what if the treadmill continues to accelerate? Different story. In principle it's possible to accelerate the treadmill at a rate that will exactly counteract any forward force you care to apply. (This is a departure from the original question, which said the conveyor belt compensated for the plane's speed,, not its force.) The only mathematics needed to demonstrate this is the well-known physics axiom F = ma--that is, force equals mass times acceleration. Given that the conveyor exerts some backward force F on the plane, we simply crank up the acceleration as much as necessary to equal any forward force F generated by its engines. Result: The plane stands still and doesn't take off. Welcome to BR #2.

You may say it's impossible to build a constantly accelerating treadmill, that eventually we run into the limitation imposed by the speed of light, etc. True but irrelevant--BR #2 has an intrinsic elegance that transcends such practical concerns. Why didn't I bring it up in the first place then? You've got to be kidding. It took an entire column to get BR #1 across, and a second one to convey (I hope) BR #2. One fricking thing at a time.

--CECIL ADAMS

The riddle states that the plane is moving. The riddle might have said that the wheels were moving in relation to the treadmill, but it didn't. As the plane itself is already known to be moving it doesn't matter how fast the wheels and/or treadmill are spinning.

A plane can only move from the thrust of it's engines, roll down an incline(wheel movement), fly, or freefall. As we know the plane is already moving due to the trust of the engines, the thrust of the engines will move the plane beyond the treadmill. Treadmills don't move per se, treadmills just sit there and spin. We have two types of movement that can funcion exclusive of each other.

Easy 10 wrote:

All of you are wrong. The plane does not move forward by powering its wheels. It moves forward by using its ingines to pull it through the air (which is basically stationary). The wheels are free spinning and exert a negligible amount of friction on the plane itself.

So, although the wheels will be turning twice as fast, the plane will still move normally. Which of course means it will be able to take off.

correct answer.

But with powerful enough engines and an indestructible fuselage, it could even take off without wheels. The treadmill having an equal magnitude velocity in the opposite direction doesn't mean it will produce a force equal and opposite to the plane's engines; those two forces are independent of each other. The force the treadmill can exert on the plane depends on the plane's mass, the surface area in contact, the materials involved, their relative speed, and other factors. Given those factors, it has an upper bound somewhere over the interval between 0 and the plane's takeoff speed. All the engine must do is exceed that upper bound and there is nothing the treadmill can do to stop it.

A plane does not need a treadmill nor an engine to take off.

https://www.youtube.com/watch?v=cHhZwvdRR5c

I never took physics and I flunked calculus.

As requested by Raysism, here it is!

the riddler wrote:

An airplane is sitting on an enormous treadmill. As the plane starts its engines, the treadmill runs in the opposite direction at the same speed the plane is moving. Can the plane take off?

But doesn't a plane need lift, so the propeller propels the plane to keep up with the treadmill but the plane isn't moving relative to the Earth. Thus the air pressure below and above the wings is the same, thus no lift occurs, thus no vertical force, thus no flying.

Saying the plane can take off is like saying a plane can generate lift in a vacuum, it cannot