Brachiostochrone Problem

Be careful with the term "shortest". The path of shortest DISTANCE between (A) and (B) is ALWAYS a straight line. The problem to which you are referring, the problem solved by Jacob Bernoulli, is the path of shortest TRAVEL TIME when the object travels from (A) to (B) under the influence of gravity and without friction -- the solution in that case is, as you said, a section of a cycloid. As for practical applications, I don't know of any other than the obvious -- situations where it is desired that an object roll or slide from a point (A) to a lower point (B) in the shortest possible time.

I agree with your observation "CMAC". It is the shortest time and not distance. While your taxi bill goes up the taxi driver spend less on fuel if he takes on the longer path.

Good idea. Use the acceleration to reach with less energy. Perhaps this is what NASA must be using to accelerate their satellite when passing near some planet. Free energy at work.

Did any one notice that this will also change the color (energy) of the light beam.

Just out of curiosity, does this have any similarity to what are called "Great Circle Routes?"

It is not a circle. It is something like bike jumpers use to go high in the air, a way to accelerate more than fuel can do.

"Great Circle Routes" have to do with the shortest distance between two points on the surface of a sphere. See here: Great circle route - Wikipedia

That's a valid comment cmac , though it's also the reason I included the picture . In general , cycloids crop up in other scenarios (Wankel combustion engine , design of coins ) . Also , how about a motive wheel that has no centre of rotation - is it possible ? I've yet to find an application for the shortest/quickest path solution , it may be just a mathematical curiosity. In the realms of Science Fiction perhaps a long train tunnel could be built (the Earths geothermal gradient would prevent this on a large scale ), but is there an advantage to be gained if a railway has to go down a gradient on more realistic scale ?

If you were into extreme skateboarding, it might maximize your thrill.

I was told once that the London Underground trains use this principle...

Between stations to maximise efficiency of the traction motors, there is a gradient down a slope and then up a slope to the next station. The gradients are hardly noticeable but they do exist to help accelerate the train from a standstill and then to help slow the train coming to a stop.

I guess that if the gradient was the other way round, you would need much larger motors to accelerate the train from a standstill and then to regeneratively brake the train to a stop in the same time.

John.

Hello John

That is pretty interesting one. Perhaps for train it may be OK but for cars one may need smooth road. Camel back ride is no good.

True John , though I would of phrased it as stations sitting on a hump (even though it amounts to the same ) . A major problem on the underground is the accumulation of debris from breaking - teams of 'fluffers' (honest , despite other meanings ) are employed to clear up every night . As such , the hump exists to facilitate slowing down without breaks . The actual (3D) shape of the underground in London is a lot more complicated than the fantastic map suggests ( and as you probably know was inspired by wiring schematics ) , being closer to a pile of dropped spaghetti. The evolution of the map is a great one in it's own right. A lot of the underground layout was dictated be the need to work around existing underground and surface features. It would be nice to work out an optimum system with a clean slate (so to speak). Chris

Hi Kris

Yes those hanging down sloping dropping ropes remind me of that sagging structure is formed easily than anything else. Spaghetti or noodles was another interesting way to think.

If you use the same principle to drive a rope trolley with free sliding ball bearing or self drive mechanism then greater chance of its coming out of the rope is at the sagging depth where velocity will be the highest. This may be so for the hill climbers also or those cross the bridge by putting that sliding hook on a rope like a glider in the air.

Hello Shyam , I like the idea of somehow applying this to a rope . The cycloid curve would have to be approximated since it's tendency is to hang in a Parabola , but the approximation would be quicker than the straight line . The approximation of (say) a 2 part ramp is one method to prove the Brachiostochrone problem. I suspect that friction loss will always take away any advantage gained. When I have time , it would be fun to set up a test rig to see if the speed difference can be observed in practice.

Taking another look at your diagram, I think I get it... The ball on the curve will beat the ball on the inclined plane. Correct?

Spot on Vermin ! Completely counter-intuitive.

Somewhere in web world there is a nice animation where you can manipulate a graph that shows various paths for the descending ball . The cycloid is always quickest . I can't get my head around what that may mean for a pipe of water (if friction loss was ignored ) - If the water jets out of a cycloid pipe at a faster speed than with a straight pipe , does that mean it effectively moves under a greater head of pressure ?!?

I think so. Being a cycloid, the part of the curve at A is a lot steeper than it is at B... At B, its curve is less than it is at A, not much rise at all. I assume that with a greater -y acceleration in the beginning, it'll shoot out the pipe a lot faster than a straight pipe connected between A and B. Huh?!

So , let's make the thing a hydro-electric scheme , and bolt a turbine to the end of each pipe . The cycloid pipe generates more power from a greater exit velocity, if we can forget the additional friction loss ??? That's got to be wrong ?

I don't think there's much loss due to friction (even though the path is longer) - the ball would feel the same friction, but it beats the other ball.

This is why Meso-Americans put honking big boulders in their downhill irrigation ditches every so many 1000-feet - to slow down the water and to keep it at a relatively similar speed.

The Brachiostochrone analysis ignores wind resistance and rolling resistance . For a small ball that is purely rolling I think there is no 'rolling resistance ' (ie that due to contact with the surface). Air resistance depends on velocity , and I suspect total loss to air may be the same in each case (higher speed = higher rate of loss , but is cancelled by the shorter travel time . This is a guess).

The turbine suggestion was a bit mischievous - for a closed pipe with constant id , cavitation must occur if exit speed from the pipe is greater then entry speed .Again it would be interesting to know what a fluid dynamics expert thought.

Ancient (and not so ancient ) peoples have done some terrific surveying/terracing work with terraced fields - all that and no laser in sight. Water engineering has to be one of the oldest and greatest human skills.

From personal experience...the subways in st. petersburg, russia are designed on this principle. They accelerate the train and cutoff power midway, the train rides on momentum for a while, then power is re-connected for the last stint. Interestingly enough, the subway system was also designed as a fallout shelter for the masses.