Banked Highway: CR4 Challenge (02/09/10)

OK. Now explain to me how the car gets up to 70 mph on a frictionless roadway. Wouldn't the drive tires just spin in place?? Is this a trick question, or is there some info missing??

Who said tires? and who said wheel driven? Who needs to read before posting?

Here is you answer buddy....

The car achieves 70 mph on a straight roadway, which HAS friction, connected to the above mentioned curve... I hope that explains things, or furthir clarifications needed???

"furthir clarifications needed???"

DougRH will be after you for the ∞ avatar choice - my advice is resist as Doug talks ∞ rubbish - whereas you may be a smart-ass "buddy", but at least get the concepts - and only need to learn to spell check.

Yours Truly

RĂZVAN HAVOC

Just to be picky, you can't make the transition.

Neither did quite a few in the luge - if you been watching the Olympics - quite a few things 'not included' are quite fascinating.

I'm probably being even denser than usual, but I don't see why there is necessarily (or even usually) a problem with making the transition from linear (or indeed almost any) motion.
If 'you' are already going round the curve on a rough surface that is banked so that the force on all contacting surfaces is orthogonal to the surface, there will be no transition to make. Similarly, if 'you' have your centre of gravity at the centre of a sperical surface.
In many other cases it may be impossible to avoid gyrating all over the place, and the surface may have a complex shape (i.e. no single well-defined "angle" for the surface if you want an instantaneous transition. However, in most cases I don't see a problem in routing your centre of gravity around a horizontal curve (which is presumably what is intended).

Obviously I'm missing something that you consider obvious! More details as to the issue, please.

Thanks

Fyz

Notice that I was answering Krahul. I can make the transition, at least in principle. I suspect you could too. But, Krahul's transition won't work.

Understood. Thanks.

(I'm afraid I read it as "no transition is possible".
I doubt in practice that I could design the transition for any but the simplest of bodies. More complex shapes and rotating wheels would require specialised tools - and the skills to use them. )

Fyz

By the way, just in case you have nothing to do now that Arsenal has gone 3D, consider the problem of defining the bank given that the right hand wheels are at a different radius of curvature than the left hand ones. With a small amount of friction, that's not a problem; with NO friction, it becomes entertaining.

No problem if 'you' (the vehicle?) have had the path of motion set up before moving onto the friction-free surface, as no torque is needed, only a force passing through the centre of gravity. Curiously, if we allow spring loss, the mere fact that the surface is conical is enough to ensure that the motion is stable - provided that the surface-orthogonal through the CofG lies inside a "conical polygon" that is defined by the contact points.

Well, that is a problem. You simply cannot "move onto the friction-free surface" except in a level straight path. Then you have the problem of going from that path to a curved, banked path WITHOUT any da/dt.

But, coming back to my original comment, look at this as a two-wheeler (side by side) so that you don't have an indeterminate problem and look at the forces for the two wheels. BTW, I didn't say it was difficult, only entertaining.

And, yes, there is a unique "track" that you have to folllow as you point out. Almost like the luge runs in Vancouver, eh?

The official answer will probably be tan^-1(31.298²/200/9.81), or 26^O31' between surface normal and the vertical.

But reading the question literally it could equally be the solution to
tan(θ)=31.298²/200/sin(θ)/9.81, or 38^O38'

Either would be stable if you were going under your own momentum, but only the second if your speed were maintained constant by longitudinal force only.

Wall of death, anyone?

P.S. 70-miles per hour is about 31.298 m/s, and I'm taking g to be 9.81-m/s.

Oh, and the second solution takes 200-m to be the (spherical) radius of the road.

Sorry - the second comment on stability is nonsense - both are stable under both sets of conditions.
_______________________
I'm not all that worried about thinking slower than I would like. But publishing faster than I think remains a constant embarrassment. (Adapted from E. Fermi).

Well yes that's a point - you'd want the road spherical as otherwise you could exit the corner ass first.

Well observed

To be more practical, the problem can be stated as: "shoot a bullet at 70mph onto a 200m radius frictionless track, determine the tilting angle of the track so that the bullet won't fall out of the track."

Maybe the Frictionless highway is a down hill stretch of road. Then the banked curve should be 90º to catch the vehicle and sling it around.

If I were you, I'd let the idea of a 90^O banked curve drop before the vehicle does.

Explanation (maintaining the mixed units): with an angular velocity of 0.156 rad/sec, it would take about 1/358 hours (10-seconds) for the vehicle to go through a right angle; but if the the road was banked at 90^O, the vehicle would also have fallen about 497 metres, and the vertical component of its velocity would have reached about 221 miles per hour (ignoring air resistance).

Does frictionless road mean just the road, because there could be air resistance added into the equation as well? Mind you, I think the whole point was to answer the question as you "know" the more simplistic sense in which it was asked. 65.5 degrees eh :-)

Highway should be banked at 90 deg, and car would have to have jet engine or ICE driving a propeler, as the traction will not be possible on frictionless road...

If the road is frictionless, how did the car get up to 70 mph in the first place?

hello tank you my friend but i don't speak about cars my question is about marine engines

That is a good question, and I answered to it my posting. It would take a jet or propeller driven car to get to speed... Theoretical bank angle would have to be 90°... to avoid any lateral forces acting on the center of gravity of the car.

A 90-degree bank would give zero vertical force from the bank. 'You' (there's nothing in the challenge about a car) would of course automatically follow the curve as projected vertically from the horizontal plane - but 'you' would be unrestrained vertically, so would simultaneously go into free-fall.

As pointed out by others, 'you' could reach the specified velocity by sliding downhill - a bit like before a ski-jump.

Would this level of understanding of dynamics be why North American cars generally have such abominable reputation for their 'handling'?

You are right! I forgot about the mass of the car... It would have to be zero... Otherwise the car would indeed fall from 90° bank.

This is as farfetched as frictionless road...

Up the skid resistant on ramp

HHO!

If the road is truly frictionless, there will be no centripital force. The road will have to be vertical. The question is whether the radius is tight enough to keep the car suspended in the turn at 70 mph.

F_c=Wsin(a) => m*υ²/R=Wsin(a) => sin(a)=m*υ²/(R*W)=υ²/(R*g)=

=(1,88m/sec)²/(200m*9,8m/sec²)=0,0018 => a=0,162^o

Pretty picture, interesting implicit interpretation (did you know that you had taken the radius of the circle of motion parallel to the road surface rather than horizontal?).

Shame about the units conversion...

Hi guest. You are right. (I was, also, wrong about the units conversion). So, it is as following:

F_c*cos(a)=W*sin(a) => sin(a)/cos(a)=F_c/W => tan(a)=(m*υ²/R)/W=

=υ²/(R*g)=(31,286m/sec)²/(200m*9,8m/sec²)=0,499 => a=26.52^o

If the road is frictionless were you shot of a cannon or do you have a rocket powered car? How would you achieve any velocity without friction between the tires and the road surface? Question is moot! There, now I have solved the puzzle with minimal effort.

Maybe he mislaid the wings to the Cessna 140?

Maybe it's a hover craft?

All fair enough Guest - I like relating it to real world too - but this is one of these "moments in math" independent of reality.

For instance were it a pure radius, so uniform bank, there would be a sudden stop, or a heart stopping launch, at the transition - due to hitting the bank.

Equally, if there is a transition blend, there is a need for drift provision into banking greater than the 'moment in math' - or - to get onto the curve, it can't be a radius but must be hyperbolic or parabolic or sim.

So the bank indicated is only at a specific instant and the bank angle is only correct at that steady state instant.

Luge would be closest to seeing the bank angles and paths required, straight to curve, are non 'steady state'. Fastest time is the path of least exerted lateral g on blades = least frictional loss.

Fortunately the OP hasn't asked for that plot.

Or maybe unfortunately - cause it might occupy a few annoying quick folk a bit longer.

"if the road is frictionless?"

Place an object , like the moon, at the apex of curb and use the gravity to sling shot you around.

if space is limited use a black hole. They might be able generate one for you Switzerland.

angle of banking=arctan(v^2/(rg)); where g=accleration gravity r=radius,v=velocity angle should highway be banked around curve=26.55degree

Ninety degrees.

It is my understanding that highway and railroad curves are not spherical but they in fact consist of two spirals, one leading into the point of maximum curvature and on leading away from the point of maximum curvature, the second spiral would be a mirror image of the first if you were to draw a line from the center of the point of maximum curvature through the point of maximum curvature. If it were not so the starting point of the banking for a simple spherical curve would be instantaneous and would inflict one awful jolt on the vehicle and its occupants.

The next time you are driving on a freeway with a reasonable speed limit (say 60 mph or greater) notice how you begin with a gentle turn of the steering wheel which then becomes tighter until you reach the center of the curve and then the turn on the steering wheel will become less until you are steering on a straight line again.

My little bit below.

I'm still unable to figure why the challenger wanted the road to be frictionless. Maybe (s)he just wanted to emphasise that the bank angle should be such that the ground reaction is normal to the road, since lateral friction forces can otherwise ensure stability over a range of inclinations.

I would take a road here to mean a smooth paved surface generally used by people to get around on foot or using two- or three- or more-wheeled contraptions. I doubt whether any self-respecting animal (in its natural undomesticated state, if such a condition still exists!) will care to use a road, especially a banked one. Maybe you could train a cheetah to run on a banked curve.

Assuming we are talking about a wheeled vehicle, the inner and outer wheels of a for-wheeler would be riding at different radii with respect to the centre of the road curve. wouldn't that introduce complications? The 200-metre radius should presumably apply to the centre-of-mass of the vehicle-cum-occupant(s).

Even with a two-wheeler, I think gyroscopic and coriolis effects of the rotating wheels would complicate the simple analysis, because there is a secondary angular motion as well. Fortunately that kind of dynamics and mathematical analysis is far beyond my capabilities. The field is now open for the real gurus to enter the fray seriously! =TeeSquare=

Tee, why not the hovercraft analogy? Why not the luge analogy? Why 'assume in' rotating wheels? If so: Why assume 4?

"Why not ...."

Just because.

I prefer to take words like road with their conventonal meanings wherever they 'seem reasonable to me', or else there is no end to possible objections and weird interpretations and why-nots. Actually I have never seen a hovercraft, and don't know whether they can be stable on banked highways, frictionless or otherwise. Nor do I have any familiarity with road vehicles having magnetic levitation or rocket propulsion or whatever.

The problem is interesting enough as stated, and I did mention that a two-wheeler may be easier to deal with ( not by me though ! ) if one has to consider position of CG and rotating wheels. Fyz has already commented about the contact between 'object' and road surface in#50

The steady condition of moving around the curve is the basic problem. Solutions to the point-mass situation have already been posted, and the official answer will probably be on those lines.

The transition to that steady state is another issue, which is being discussed in a qualitative way already.

I hope this discusson continues, and I'll look in again probably after a fortnight of little or no internet access. =TeeSquare=

Here is a link;

HowStuffWorks "How Luge Works"

Here is the relevant passage, (click on "The Physics")

"Centrifugal force pulls the slider outward in the turn. To maintain speed, the slider must perfectly balance the centrifugal force with the force of gravity pulling him downward through the course. This means finding the "sweet spot" and staying there. If the forces are balanced, the sled will smoothly move through each turn and back into the straightaways. If they are unbalanced, the slider will have to steer too much, slowing down the run."

So "sweet spot" is the analogy I was making - fits?

But hey! - Welcome to CR4

_{<sorry about the baptism by fire, it's the way of things around here -hope you come to enjoy it>}

Sensible answer, but steering will not work as we are dealing with frictionless road. The balance between force of gravity and centrifugal force so that the resultant is always perpendicular to the road can only be done by adjusting speed of the vehicle (as gravity force is constant). Obviously the vehicle must be jet or propeller driven, again because frictionless road. That would imply that on the straight line, the vehicle speed would have to be brought to zero, and then, once bank angle=0, vehicle could be accelerated to any required speed – again, with jet or propeller... Similarly, to enter the curve, the car would have to be stopped first, before the bank angle is adjusted...

I think steering was mentioned to illustrate what was needed if the banking did not match properly.

On the other matters, the challenge says frictionless. As most contributors tacitly assumed, this is normally taken to imply other losses (such as air) are also zero.
If you were to allow jets or propellers, we could also use these for steering, and the problem becomes void.

Then, you write "The balance between force of gravity and centrifugal force so that the resultant is always perpendicular to the road can only be done by adjusting speed of the vehicle". This is missing even the trivial point of the challenge: which is that the road itself is curved and the banking is exactly correct to keep you on the road at the stated speed.
No need to stop first either - just adjust curvature and banking simultaneously - though if you were riding a small-wheeled bicycle you would need a significant transition region to make the change (unless you made the "banking" so curved that it acts as a positive restraint).

Personally, I feel it's always a good idea to sort what is happening "within the box" before venturing too far outside.

Fyz

Amen

An interesting comment - my take:

If the vehicle is already going around the curve before it reaches the smooth surface, its angular velocity will already be as required; so there will be no change to the bank angle. The same applies while the base of the object that contacts the road is a sphere centred at the CofG

Any other object will require the supporting surface to have a complex transition region if the path of the CofG is to instantaneously change its curvature while staying at the same height.

The transition to the smooth road surface can perhaps be done in the following manner:

Have a circular track with the 'proper' bank angle as required by the challenge, but with 'thin' friction carpets in two or more segments firmly fastened to the edges of the (frictonless) road at inner and outer radii. Let the driver accelerate to the stated 70 mph and reach steady state. Then as the vehicle passes each segment, undo the clamps by some suitable mechanism and yank the carpet off the track. Once all the segments are off, the driver has a choice of going on for ever or smashing up. =TeeSquare=

There's a small problem. You can reasonably assume a friction-free surface (We can get down to perhaps 0.00001 coefficient ), but you cannot assume a friction pad of 0.15 since you cannot guarantee that by any theoretical means.

I don't follow your objection to "... a friction pad of 0.15 ..." Maybe you intended to reply to some other post, rather than my #52.

I did not suggest any particular friction factor.All I meant was that you need a friction surface to get up to speed in the manner specified, and then the surface can be made frictionless. I only suggested a plausible method of achieving this!

I've just re-established internet access, and will have something more to say on this thread in a day or so. But maybe everyone would have moved on by then! =TeeSquare=

No, I replied correctly. We cannot hypothesize a particular value of fiction because there is no theoretical basis. All friction is something like "0.13 as reported by Horowitz and Blankenstern, 1958, using dry aluminum on wet steel". The same situation would be reported differently by another person doing the measurement.

Although TeeSquare's method is not necessarily the one I would choose, I'm with him on this. He doesn't mention any specific coefficient of friction, and neither is the value important. If the road is a cone then any finite friction that is sufficient to overcome air resistance (which will of course magically vanish as soon as you remove friction) will allow you to reach the desired speed and trajectory. All that is need is to apply the appropriate acceleration and steering, and the longitudinal and lateral components of force will self-adjust to match.

It is not the specific value that I see as the issue; it is the fact that any value of friction is pure pragmatic instrumentalism. Friction, because it is a wastebasket term, is always concretum, never abstractum. Yeah, if you want to, you can always argue many worlds, but that's asking a bit much.

Sorry, not buying that.

I cannot see that what you write here bears any relation to what either TeeSquare or I have written.

"Any value of friction is pragmatic instrumentalism"?? I've no idea what you mean. If you had written "any exact value of coefficient of friction is pragmatic instrumentalism" I would both understand and be inclined to agree* - but clearly such a value is not required.
*Although even this, like zero friction, would an acceptable artefact as a basis for a conceptual problem

Maybe your intention is relevant, but it is simply not coming across. Try to be specific and concrete about where you see a problem and I will try purposes of the challenge) is not.

(Is it perhaps the idea of introducing controlled steering and acceleration as precursors to the passive world of the challenge that you can't accept?)

Perhaps I don't understand your solution. Are you saying that any value of friction at all is acceptable? There is a solution where that holds, but that was not my understanding of what was proposed.

I'm trying to be specific about the difference between coefficient of friction and frictional force - though I fear I will still be ambiguous from time to time.
I'm saying that any value of coefficient of friction above zero is acceptable if the surface is a cone that extends inwards from the final region. You can then accelerate and steer so that the actual frictional force matches what is needed to go in some sort of a spiral. Once you reach the desired speed you can steer and provide corrective acceleration until you are on the desired circular trajectory. If you remain on that trajectory thereafter, the banking will provide any force that is needed to keep you on path, so the actual frictional force will automatically reduce to zero. At which point you can move onto a section of the road where the friction is zero without noticing any effect. If TeeSquare then chooses to remove the frictional pads from the road behind you (later to be in front as you come full circle) so that you would go around for ever without friction, that is his hypothetical privilege.

That's interesting. It might work. I'm still unclear what happens when the front wheels quit rolling while the back ones are generating torque, but that's a darn good solution! If you ever get that trouble with the fan dancers and the "borrowed" double-decker cleared up so you can get a visa, you'll have to stop by the house. We can sit on the front porch, drinking bitter (You will remember to bring that, won't you?) and argue this.

Of course it would never work - friction-free surfaces don't exist (at least not anywhere 'you' could access)

Returning to hypothee-land: in the absence of any friction, why do the front wheels stop rotating? But of course you are right that all losses need to vanish before control is lost, otherwise we will need a hard-to-calculate offset in the trajectory immediately before we run onto the friction free surface.

I'll try to remember the bitter. Given that it's merits are somewhat climate-dependent, what time of year is best where you live?

Any time when it's not snowing! &^**$#!!

Good point about the wheels, though. Are we assuming everything goes friction free? It helps if they don't stop.

I think that non-physical "simplifying" assumptions such as zero-friction usually carry the associated (non-physical) assumptions of zero other losss and zero aerodynamic forces (otherwise we could perhaps have an unbanked curve and use wings...)

Terrible diagram.

And what is meant by "this, as you know, is too a steep an angle!" (It's exactly what is needed for the purpose. Clearly you couldn't park your car on this slope, but then with zero friction you couldn't park on any slope. Maybe we could regard sliding sideways onto the hard shoulder as a safety feature?)

Maybe the author's point is that although a 200-m radius bend sounds quite large, at 70-mph (and even under dry conditions) it's within a factor of two of the limit for most roadgoing vehicles.

Firstly, thanks to Fyz for the sustained supporting comments in the last few posts.

Some much-belated thoughts on the subject, especially since the official answer has already appeared, and if anyone has the patience to wade through my verbosity:

The challenge only tells us that the road is frictionless, so I take it that everything else is "normal", and the inevitable air friction can be a significant drag at 70 mph. Other than some jet or rocket thrust acting at the centre of mass, I don't see what else can sustain the equilibrium beyond the "moment in math" suggested in post #26. I still can't see how such a thrust can be applied in a direction which is constantly changing -- tangential to the circular path.

The alternative may be to reduce the problem to that of an object skidding around a banked circular track in vacuum. Even in this case a non-spherical, non-homogeneous, non-rigid body is liable to have its motion disturbed in some way due to forced changes in tilt angle as it moves around the curve.

Coming back to a real vehicle on a frictionless track, if the bank angle is constant (simple inverted right cone with vertical axis), the motion will be a slow inward spiral which reaches the vertex after infinite time since air resistance would be roughly proportional to the travel speed. Any friction in the bearings and transmission will have no influence on the frictionless road. I expect a two-wheeler would topple sideways pretty soon, whereas a car ought to remain upright throughout, though it may spin around and wobble a bit.

If the track were not conical but (say) doughnut-shaped, tangent to the 'theoretical' cone, my guess is that the deceleration will be hastened (?) though coasting to a complete stop at the bottom of the 'groove' will still take infinite time in theory.

Another line of thought is whether a slight decrease in bank angle can result in some sort of quasi-stability (at least for a while) if the increased 'fling' can somehow compensate for the speed decrease due to air resistance. Maybe some kind of hunting effect about a mean circular path. Just wild thinking, purely qualitative, and totally beyond my modest mathematical capabilities.

I must admit that I don't drive any kind of vehicle, and have never even owned a bicycle (!) though I can ride one. So I have little 'feel' for the parameters in the challenge. I'd be obliged if someone can give me an idea of how far and for how long a 'typical' two-wheeler and four-wheeler will roll on a level straight smooth road without braking, if the engine is cut out at 70 mph or 100 kmph. =TeeSquare=

I don't know the answer - but the relevant parameter is probably the distance over which the speed will halve. And you will need to at least double any measured distance, as friction and flexural tyre losses will dominate at lower speeds.

BTW, even if we restrict our attention to air resistance, I think we are looking at infinite time before you stop, but finite distance of movement.

I agree whole-heartedly with you about the "hunting" solution. I think intuitively that's what it would take.

Anyway, I'm not attacking your solution; I'm just arguing cause I'm an engineer and I have no other social skills.

Assuming losslessness: if you don't make the transition absolutely correctly, 'your ' CofG will of course oscillate about the centre value. But the frequency of this oscillation is of some interest...

If we were talking inverse square law (e.g. planets), we would have an elliptical orbit. The potential energy of a gravitational field is proportional to -1/R. So the equivalent surface would approximate a hyperbolic cone. You may consider the following is too obvious to say, but I'll highlight it anyway: for sufficiently small deviations from the nominal circle, all smooth conical shapes may be regarded as equivalent. Which means that if you get things nearly right the very small oscillation will be coherent with the cycle, and the orbit will be elliptical. For finite (but still small) errors, the ellipse will precess.

Anybody have objections to my statement in post #79: "I expect a two-wheeler would topple sideways pretty soon, whereas a car ought to remain upright throughout, though it may spin around and wobble a bit."?

No strong objections, but the following is worth considering:
If the two-wheeler becomes lossless in all respects, it could precess aroud a vertical axis (as a gyroscope) rather than falling*. Indeed, you could in principle have some control via the handlebars (using the gyroscopic effect of the front wheel).

*This depends on how near equilibrium it was originally

By the by, I am almost ready to concede that your method works. But, then you brought this up and I have to ask: What about the gyroscopic effect of the spinning wheels as you go around the curve? What does that do? And, is it possible to steer the front wheels so as to counter the back wheels (after all, steering is not really a factor here.

Fyz, jump in, please. Settle this for me and I'll do my now famous mea culpa to TeeSquare.

If the motion is correct before friction is removed, the wheels are already rotating as required. By whcih I mean that not only are they rotating about their axles, but the axles and wheels are jointly rotating about the surface normal. For a vehicle with three wheels or more, the rotation of the surface normal about the vertical axis will automatically be corrected by the wheels providing torque around the direction of motion; for a bicycle you would need to lean the cycle in at exactly the correct angle. I'm not completely certain, but I beleive that in all cases a small departure from the ideal would simply result in the position oscillating about the intended nominal.

Notes:
a) If it was a straight line, the bicycle would definitely be unstable - it's the variation of speed with height on the road that provides the restoring forces.
b) The reason that I'm uncertain about the stability is that my experience causes my line of thought to be guided by the effects of castor on a frictional surface. If 'you' (the rider) were to move the handlebars appropriately you could undoubtedly remain upright and pointing in (approximately) the correct direction.
I think that the curvature of the cone will have similar effects, but this is purely on the basis that stable dynamic equilibrium generally coincides with the unstable equilibrium of the static case.

"... for a bicycle you would need to lean the cycle in ...'

Wouldn't that presuppose the presence of road friction? (I guess we're all on rather slippery ground here!) =TeeSquare=

We have a typical gyroscope situation here - the gound force (and so the road angle) has to be at the correct angle to counter the centripetal force, but the requirement to provide a torque to drive the precession (=turning of the bycicle) means that they do not pass through the centre of gravity.

The only mismatch we have between the initial (frictional) condition and the final one is that under frictional conditions the rear wheel always points slightly outwards relative to the direction of travel. With friction, this would exert a torque around the surface normal, and its sudden removal would result in (small) oscillations of the attitude of the cycle. If they were large enough for you to notice, hopefully you would soon learn to use the "steering" to compensate; actually, your breathing etc. would probably cause larger variations, so your attitude will inevitably be variable.

So - clearly TVP's reservations are justified: perfect transition (without attitude wobble) is unlikely. On the other hand, the transition will only institute a small oscillation around the ideal orientation - not a fall-over. [But, I'm not certain whether normal 'cycle steering strategy is appropriate for friction-free conditions; if not you would have to learn fast, as initial attempts at self-correction could have "interesting" results]

Hi,

No need to concede anything! This is just an anonymous voluntary discussion forum. Sometimes we get convinced by someone else's arguments, and gain new insights thereby. Often we encounter those who struggle to express themselves due to conceptual or linguistic limitations. Some regulars like Fyz are fantastic at explaining matters clearly and patiently to those who are confused (and sometimes rude as well). I've certainly gained much from reading his comments in various challenge topics and perhaps more from the inevitable digressions. I've been in the mea culpa category often enough when my misconceptions have been corrected,

I'm here in CR4 despite my severe limitations in coping with computers and internet. Like you (post#83) I'm just an engineer, and don't claim any social skills. Maybe I should say 'was', because the engineers I meet today seem to speak mainly computerese, in which I am functionally illiterate.

I found this particular challenge rather intriguing because some of the secondary issues like angular momentum of rotating parts have a relevance to real-life situations. I don't have the analytical capabilities for exploring these but it helps to have some intuitive notions about various physical influences. Of course it is difficult to judge how far one can rely on intuition.

I think there is something of deeper or universal relevance in what Fyz has stated in #85. Maybe there is something about real systems which ensure that disturbances upto a point will die down eventually. An extension of the simple mechanical concept of static equilibrium? Many aspects of dynamics are still like black magic to me, just because I'm too lazy to wade through all the daunting theory! =TeeSquare=

Oh, I don't mind conceding arguments; I'd think myself a damn poor engineer if I were always right. And part of the job of arguing for a point is admitting if that point is wrong. I'm even thinking of changing my signature line to : Often wrong. Never in doubt.

But, back to the problem. (I could beat this really, really to death so long as you and Fyz don't stop talking to me.), suppose we had a rectangular solid zooming along through deepest space, far from any g fields of significance, could we, with 5 impulse rockets, get the same motion in a stable manner? One rocket for each wheel and one for the weight. Then, if we added the four angular momenta (the spinning tires), what would happen? We'd use magic rockets that don't change the mass of the block.

On the smooth surface I think we only need three wheels - which would be one rocket for g and three rockets for the wheels. The reason that I believe this can work is that both the magnitudes and the relative directions of the surface forces change as the vehicle rotates.

You can see that the space problem becomes somewhat different. Firstly (as I take it you already assumed) we would need a control system. Second, because the directions of all forces track the orientation of the module rather than the location of the wheels or (for the G-force) being constant.

Although in principle full-control would require six rockets in total, because we only want a constant total linear acceleration we might get away with just four non-parallel fixed-direction rockets* - but I'm not confident that we could do this with rockets that are in directions that are equivalent to the nominal surface forces on the wheels. Work [ugh] would be needed.
*Giving three angular degees of control and controllable acceleration in a fixed direction relative to module attitude.

Fyz

Three wheels! Do you mean you'd drive this at 70 mph? OMG!

Certainly not, even if it could manage the speed (and not even if the surface is smooth and optimally banked; I'm not suicidal - yet).

But no problem with one of these (provided the surface is reasonably smooth, as I'm getting a bit fragile in my dotage)

(borrowed from here)

I agree with you about right and wrong in engineering. If I don't make mistakes as a designer every now and then, I'm probably not learning anything and maybe playing too safe. It's the lessons from one's own blunders that really sink in!

I'm afraid I'll have to exit from ths thread as I'll be out of town again for a week or so. Anyway the foray into rocketry is getting beyond familiar terrain, and I doubt I can make any meaningful remarks. And let's give Fyz a break as well -- he's been been accomodating us admirably for quite some time, and I'm sure his talents can be put to better use elsewhere.

It's curious that you mentioned a rectangular block. I did consider that the problem could be simplified if the (supposed) four-wheeler were replaced by a rectangular homogenous object skidding around the banked curve. Its moments of inertia about the principal axes are known. It would make point contact with the conical track only at the four bottom corners (provided it doesn't skew), though its inclination would not exactly match the bank angle. Maybe some day I'll find the time to try writing out the equations of motion for such an object and see if it can maintain a steady predictable path and orientation.

Anyway it's been fun, and hopefully we'll meet in some other challenge thread, with or wthout friction. =TeeSquare=

Rectangular block - and mistakes - both nicely put. These forums get me away from my current specialism for a bit - which is all to the good, as I (for one) tend to go stale.

(They also remind me how irritable I can get - takes a load off my colleagues).

Fyz