Plane On A Treadmill

[DC_DEEP]

Now here is a well framed problem with a small but sufficient amount of information for it to be solved:

An air bubble rises from the bottom of a fresh water lake, doubling its volume when it reaches the surface. How deep is the lake?

PV=NRT. Are we discounting temperature?

 

[JustAsking]

PV=NRT. Are we discounting temperature?

DC,
You are on the right track, but that doesn't surprise me. You may assume the air and water temperatures are the same.

Dudley

 

[JustAsking]

... The only interpretation which seems to make sense is that the aircraft is free to move forward at some velocity V with respect to .... what? .

Yes, there are four inertial frames:

- The earth on which the conveyor belt is sitting.
- The surface of the conveyor belt.
- The airplane.
- The air in the vicinity of the plane.

And the planes "lift" is a function of the difference of velocity between the airplane frame and the air-in-the-vicinity frame.

By the way, the problem forgot to specify whether relativistic affects can be taken into consideration. If so, we have a big problem because I think the relative velocity between the plane and the belt will be so high that both the belt and the plane will be going towards infinite mass (at least for an observer standing on the belt). Also, the flight attendants will be a little bit younger than they should be after the experiment is done. The observer on the belt might like that outcome.

... Then all we have to do is realize that the conveyor speed has no effect on the aircraft's motion, except to spin the wheels faster. Since the aricraft isn't driven by the wheels, that's inconsequential.

That's right, if the rolling friction is negligible or zero.

 

[DC_DEEP]

DC,
You are on the right track, but that doesn't surprise me. You may assume the air and water temperatures are the same.

Dudley

Alright, in that case, without looking it up or doing any "math-on-paper", I'm gonna guess... 33 ft deep?

 

[JustAsking]

Alright, in that case, without looking it up or doing any "math-on-paper", I'm gonna guess... 33 ft deep?

Ding! No more callers, please, we have a winner. If the volume of the bubble doubles at the top, the pressure must be half of what it was at the bottom (since PV = nRT). The pressure at the top is 1 atmosphere, so the pressure at the bottom must be 2 atmospheres. An "atmosphere" of pressure will raise a column of water to 33.95 feet (at STP) in a water barometer. DC, you are quick. This is an excercise in cutting to the chase, and you are always good at that.

In college there was a guy who would write one of these kinds of problems on the board in the "physics lounge" once a week. At first it always seemed like not enough info was provided. So people would start doing some elaborate stuff to try to come up with the answer (like big white-board drawings of parabolic lake beds, and associated math, etc. for this particular problem) But the trick was to be able to see what was not important and zero in on the one necessary relationship.

This bubble problem is a simple one, but there were really tough ones that took everyone a long time to figure out. You knew you had it right when the answer turned out to be a one liner. It was getting there that was the problem.

 

[DC_DEEP]

Heheh, I can't believe I remembered that. I did well in class, but it was SO long ago. Way back in 1991, in fact, and taking Organic I and Biology in the same semester as Physics I, then Organic II, Physics II, and Botany the next semester.

"Dr. Anderson, may I be excused? My brain is full."

"How things work" has intrigued me since before I could read, so chemistry and physics were always my favorite topics.

 

[B_big dirigible]

Yes, there are four inertial frames:

- The earth on which the conveyor belt is sitting.
- The surface of the conveyor belt.
- The airplane.
- The air in the vicinity of the plane.

Wh-oa, inertial frames need not enter into it at all. They just complicate things without clarifying them.

Some frame is necessary for the concept of velocity to have meaning, but inertial space isn't one of them, as an inertial frame can have any constant velocity. It can't be accelerating (or spinning, which necessarily involves accelerations). That's why I postulated Officer Friendly lurking behind a tree as a useful frame, and noted that the wind is assumed to be zero (ie, not moving in that frame).

 

[rawbone8]

Wh-oa, inertial frames need not enter into it at all. They just complicate things without clarifying them.

Some frame is necessary for the concept of velocity to have meaning, but inertial space isn't one of them, as an inertial frame can have any constant velocity. It can't be accelerating (or spinning, which necessarily involves accelerations). That's why I postulated Officer Friendly lurking behind a tree as a useful frame, and noted that the wind is assumed to be zero (ie, not moving in that frame).

Big
There have been many hints of it in the past, but I want a confirmation.
You literally are a rocket scientist, aren't you? :biggrin1:

 

[JustAsking]

Wh-oa, inertial frames need not enter into it at all. They just complicate things without clarifying them.

Some frame is necessary for the concept of velocity to have meaning, but inertial space isn't one of them, as an inertial frame can have any constant velocity. It can't be accelerating (or spinning, which necessarily involves accelerations). That's why I postulated Officer Friendly lurking behind a tree as a useful frame, and noted that the wind is assumed to be zero (ie, not moving in that frame).

Yes, big_d, you are totally right. Inertial frames imply that there is no acceleration so they are not useful here. Officer Friendly is a much better analogy. I needed a way to get to the special relativistic question, though. I am trying to work up a case for the flight attendents getting younger as the treadmill speeds up. My Physics degree is 30 years old, so it takes me a few tries.

 

[B_big dirigible]

Big
There have been many hints of it in the past, but I want a confirmation.
You literally are a rocket scientist, aren't you? :biggrin1:

Umm, ahh .... at one time, yes.

I was able to extricate my butt from the DOD/aerospace business a few years back.

I still have the user's manuals for a couple of the boosters somewhere around here. The Titan II and the Delta. Good rocket-type stuff.

 

[SpeedoGuy]

Is she required to solve physics problems? How about someone with a mere physics degree? Even better, someone with a physics degree who's flown an airplane or two himself?

Claire's insight would help. I minored in math and physics and I've been piloting powered aircraft and sailplanes since I was 13.

Regardless, I'm disappointed so many clever people here are conflating irrelevant facts in this case. I fear the detailed discussions about physics principles are making the problem seem far more complex than it is. It need not be so.

I think the best analogy was made by LeeEJ: Run at a good clip on a treadmill and try to fly a kite by tugging on it. The kite will settle right back to the floor every time.

 

[kamikazee_club]

Ding! No more callers, please, we have a winner. If the volume of the bubble doubles at the top, the pressure must be half of what it was at the bottom (since PV = nRT). The pressure at the top is 1 atmosphere, so the pressure at the bottom must be 2 atmospheres. An "atmosphere" of pressure will raise a column of water to 33.95 feet (at STP) in a water barometer. DC, you are quick. This is an excercise in cutting to the chase, and you are always good at that.

Basic dive school one that (abient pressure increases 1 atmosphere per 10m) therefore if a bubble doubles in volume from bottom to surface it's 10m down, if it quadruples it's 30m down etc.

That's the reason one should NEVER hold one's breath underwater and definately not when ascending, especially as one nears the surface but should relax and allow the expanding air to escape. Exploded lungs are very messy.

 

[B_big dirigible]

DC,
You are on the right track, but that doesn't surprise me. You may assume the air and water temperatures are the same.

Could be, but that'd have to be at a rate much slower than a bubble wants to travel in water. The other necessary relationship here is the ratio of specific heats. For an isothermal expansion (constant temperature, not a bad model for gasses in water if time is allowed), PV=a constant (NKT or mRT, either will work). For adiabatic expansion (an insulated system - water's not a good thermal insulator but air is), PV**gamma=a constant. Gamma is about 1.4 for air.

___________
And for the peanut gallery - ** is exponentiation, same thing as P V(exp)gamma, so the relationship is "P V to the gamma".

 

[B_big dirigible]

Claire's insight would help.

I know she's full of surprises, but I doubt that she's taken off from any more moving conveyors than I have.

Consider a slightly different experiment. Postulate that the plane's wheels are genuinely frictionless. And the plane isn't trying to take off - it's just sitting on the runway on its frictionless wheels, and no brakes. Now run the runway conveyor at some speed, in either direction. The plane won't move. Inertia keeps it in place until some force is exerted. But the conveyor can't exert any forward-or-back force on the plane, because it's only touching at the wheels, which are frictionless. The conveyor can move at any speed, in either direction, and it will have no effect on the plane, aside from spinning the wheels. (There is a slight inertial force effect from the wheels, but I can make that go to zero by postulating that the wheels have zero moments of inertia - in real life, that's actually fairly close, certainly when compared to the inertia of the entire plane.)

Now start the plane's propulsion system - propellers, jets, rockets, it doesn't matter. This exerts force on the plane, and it accelerates. A=F/M - the acceleration depends only on the mass of the plane and the applied forces. The only horizontal forces acting on the plane are the thrust from the propulsion system - which has nothing to do with the runway - and aerodynamic forces (which at zero speed are zero). These are the same forces which act on it during a normal takeoff from a normal (fixed) runway. The conveyor runway still exerts no horizontal force whatever on the plane through its frictionless wheels. (Of course the runway, fixed or moving, exerts a vertical force on the plane equal to its weight until it leaves the runway, but that's not what makes it take off.) The conveyor runway can move forward or backwards at any speed and have no effect on the plane (except wheel speed) - the exact same situation as when the plane was just sitting there with its propulsion off.

The problem as specified at the beginning is just a special case of the situation I outlined above. The conveyor speed is constrained to equal the plane's speed, but that isn't necessary - the conveyor speed could be anything at all and the plane would take off as normal.

Now if the plane wasn't a plane at all, but was a glider being towed behind a vehicle, that's a different story, because the glider's speed would equal the tow vehicle's speed relative to the conveyor plus the speed of the conveyor, which as originally posed would add up to zero.

 

[B_gagger]

I believe in the case of the bubble, the change in gas solubility due to pressure change would have to be factored in (since the bubble was stated to be an air bubble and not water vapor) as the number of moles of gas "in" the bubble would not be constant at the changing pressures.

 

[SpeedoGuy]

I know she's full of surprises

lol, agreed. She's full o'surprises, that one. And charms too.

I respect all the technical points points you're making, big d. I guess I still feel as if I've failed to make mine.

Here's another way I'll try to couch it:

but I doubt that she's taken off from any more moving conveyors than I have.

lol, and me as well. And that brings me to my point. If the conveyer belt idea really worked, wouldn't the navies of the world have long since installed them on aircraft carriers to save space and reduce hazard aboard ship? It really matters there. To wit: Why fling an F-18 off a carrier deck at perilously high speed with a risky steam catapult when the same F-18 could much more safely levitate off a rapidly moving conveyor belt (in a manner much as variable thrust A/V-8 Harriers do)?

Further, if the conveyor belt idea is valid, that same F-18 could just as equally land on a rapdily moving conveyor belt and really reduce risk in the most hazardous part of carrier operations: the approach and trap.

But none of these have ever been attempted because they're not possible.

 

[Gillette]

Certain most of you have seen this in one variation or another but since this is the fun with physics thread I thought I'd drop it in here.

Thermodynamics_Exam

 

[DC_DEEP]

I believe in the case of the bubble, the change in gas solubility due to pressure change would have to be factored in (since the bubble was stated to be an air bubble and not water vapor) as the number of moles of gas "in" the bubble would not be constant at the changing pressures.

Ah, here we go again, adding in variables that don't belong. I'm sure Mr. Avogadro would be happy that you want to add in his contributions, but the theoretical question deserves a theoretical answer. The question, as stated by JA, implied the caveat "disregarding solubility curves and friction, and assuming standard temperature and pressure, ..." In other words, "considering only the variables given, and disregarding any other principles of physics, ..."

 

[ClaireTalon]

An Jet aircraft is on a runway that is also a conveyor belt. The conveyor exactly matches the aircrafts movement BUT is going in the opposite direction of the aircraft. Will the aircraft be able to get airborne?

No, it won't. The fact that a plane flies is based on the physical law of lift, Bernoulli's equation, and therefor you need air flow around the wings to cause lift forces. A quickly moving conveyor belt would probably turn the plane's wheels, but not cause air flow.

However, if you put an airplane in a headwind of 130 - 150 knots, it might lift. That's why take-off's and landings are conducted under headwind conditions.

 

[crescendo69]

Like, you guys are smart, or something; but, I agree that wind is needed.