[ "Whimsy" What is? ]
"Bizarre" is to know the king to bring new interactive sub-section!
Here, talented friends to bring you brainstorm! Then the magic question can be found in scientific explanation!
Ask your own "whimsy" in the "comments" in it! As long as there are enough problems "creative" and "depth", and can be answered by existing knowledge, know-jun help you find experts to answer!
Monday's fall festival parade again, so I rolled it out for everyone to Bubu, take a glimpse of what mathematical beauty.
This thing and 17th centuries a puzzle about until later after calculus was positive solutions is established. Although the problem is not difficult, but the results are amazing.
I would like to ask a more "two" Question: What is the shortest path between two points?
Well, do not suspicion I was teasing you, or take non-Euclidean geometry shaking clever, really hope Liangshouyitan say a straight line.
Iron beads [line]
Now we look at this program we want to explore the question:?? O If AB two points in space vertically, then the quickest route between two points, what is?
A few maps, if we use the wire connection between two points above wearing a round bead, so what kind of posture path allows the beads to the fastest downhill from point A to point B?
Note that this problem to add gravity (but does not consider friction and air resistance), the piece of iron beads ScS fastest land line to point B, to give you two minutes ......
It will not be the first straight line method? In any case, we all know that this is the shortest path between two points. So the beads need to move the distance is the shortest, and the beads do not need to change the direction of deviation, in strict accordance with the direction of the initial slide buried in the end.
Will it be in the form of a second parabolic path the fastest? Parabola is a kind of horizontal displacement and vertical motion into the motion path of the square, more in line with nature under the influence of gravity fall trajectory of objects (in fact, those falsely make you spit your money in one place "weightless experience" flying, flying is this kinds of paths.)
There is a third type of ski jumping path, it would be one of the fastest it? Walk this path has an advantage, that is, in the beginning will get a higher acceleration, when the maximum acceleration time, to put this advantage into the horizontal displacement after a short time over the second half.
There is not possible, in fact, for the fall, in fact, the path did not matter? You see, because it is energy conservation thing, the same height, the beads have a potential energy is the same, then the kinetic energy finally obtained is the same, then we can say that in fact, choose the path that has no effect on the speed?
Finally, the path will not be the fastest? In fact, there are other possible? For example, a perfect arc?
Eh? Sounds looks like a little truth! what do you think? Do not wash its hands of the mouse, and I read on ......
Newton, Bernoulli, Huygens, Leibniz, Chin Hao Shen, Luo Bida (anyway, are some ancient science Pa)
In the 17th century, appeared to get together a large number of distinguished mathematicians: Newton, Bernoulli, Huygens, Leibniz, Chin Hao Shen, Luo Bida ...... they are solving the problem, the question of who is Jacob Bernoulli his brother, John Bernoulli :
"I, John Bernoulli, trying to find the world's best mathematicians no more difficult for people to cool me fair and equitable than the debut problem, can solve this problem people will be able to name for oneself, Immortal Fame. become with Pascal, Fermat and other large cattle par V. Please allow me, on behalf of the entire mathematical community in particular, can make everyone PubMed mathematical thinking skills and stamina in today's issue. If anyone can submit and answer me, I will disclosure and grant its deserved reward. "
Contains the history of Newton first found the correct solution and answers. Decades earlier than Galileo, Galileo since the hand is not calculus, come to the wrong answer, so we do not from shame, I do not know normal.
[ Steepest curve (Brachistochrone Curve) ]
The problem there is an optimal solution, this curve has a tongue-twisting names, called Brachistonchrone curve (from the Greek etymology, brachistos shortest mean, chronos means time). This is really read up tired tongue, but first, do not frown, Leibniz would like more read awkwardly call it Tachystopote ......
The shape of the steepest curves approaching the "ski jump" (on view of a third), starting near vertical acceleration gain the ability to make beads half after rapid horizontal displacement by an average fastest. The figure animation, the red piece is the "steepest curve." (Galileo concluded that the fault in the perfect arc is the fastest path.)
[On] the calculation of the variable
Calculation of the optimal solution to get in here, is not a function to a variable in the minimization, but rather a function of the other variables is minimized. This is the " variational method ."
The basic idea is to calculate the "conservation of energy." Beads fall into the potential kinetic energy. If we put the curved path length denoted s, each section of wireless small path denoted ds, too:
Different paths will have different functions, where our goal is to find the smallest functional expression of y.
We know the path is continuous (no potholes and sudden ups and downs), and we know there is only one variable is the acceleration, so get a second derivative d2y / dx2, and we know the value of the start and end points.
You copy a shortcut to directly answer it, here is the angle with respect to the tangent of the parametric equation
Equation K is a curve through endpoints (xB, yB) coefficients guarantee.
[ Cycloid (Cycloid) ]
The resulting image on the formula, is the next figure we can see, "cycloid" beautiful ......
Describes a point on the circle, circle the track in a straight line over the movement of time.
Imagine your car to run on such a slope shape wheels is that black spots, that it moves the fastest section is in the range of this cycloid 0, the vertical descent from a horizontal position to return of this path.
[That in the end there is wool? ]
Steepest roller coaster curve for the construction of a huge significance, build a roller coaster engineer who always struggled for a limited distance from rappelling to reach the highest speed to cool you as soon as possible. As we have just permit, "the steepest curve (Brachistochrone Curve)" is the fastest path between two points.
This is also of great use in competitive sports. If you are a skier, the goal is the shortest time to the finish line, you do not care about the shortest path between two points, but the quickest route. If you decline the steepest curve along the path, you'll get more acceleration advantage.
[To see] here are good students
This thing could be more interesting.
In the framework of a uniform force field, "the steepest curve (Brachistochrone Curve)" is also sometimes referred to as "tautochrone (tautochrone)" (still thank the Greeks, taut means "equal").
You can place objects in "isochronous curve" any position, they will fall the same time to the same location.
The higher the position of the object, will be faster, and the lower position of the object together through the lowest point. (Specific duration is multiplied by the radius of the arc divided by the square root of g).
You can hang the mighty sharp fried Wolfram day to play more exciting examples
We recall high school physics teacher talked about the pendulum motion period depends on the length of the arm, but similar results this argument is only an ideal state. When the pendulum really thrown up, in fact, there is a fine arm length changes slightly:
When the arm is long and swing very young, this error is very small, but this error is the escape. First discovered this problem is mathematician Huygens , he used a special method called the "inverted cycloid involute (involute of an inverted cycloid)" corrects the error (mentioned later), to create a perfect pendulum (Huygens pendulum), he was the first in the history of human cycloid pendulum appears in the top of the error.
If the arm is half the length of the circumference of the cycloid, the bell hammer track running along a cycloid movement in a fixed length of time, duration and location of the high point independent swing. Involute description refers to a movable arm on a point along the curve movement, and the intersection of the selected track on tangents. (If you know every word, which is not really my fault ...... The graph blue period is called "involute").
The figure is designed Huygens pendulum, pendulum top two metal reeds, now known as Huygen's Chops.
When the pendulum swings, slings it labeled reed, reed shape is involute cycloid pendulum therefore perfect run along the cycloid.
[Cycloid, the steepest curves and isochronous curve]
Cycloid characteristic in novel "Moby Dick" is also described:
"Refining whale oil vessel" also includes a brilliant mathematics. Whaler Pequod number of the port side pot, the pot when I talc polished walls when noticed this amazing phenomenon, everything according to the rules of the cycloid, regardless of where to start, both with the same time to fall bottom of the pot.
If you're still playing four-wheel drive model cars, then you can tell the kids, if you are on a curved shape steepest slide game, no matter where the car from the start, the game is fair.
(Of course, clever little guys will tell you, the red car will run fastest).
In line with the requirements of a mathematical shape skateboard slip bowl game, both sides should be in line with "isochronous curve". If you are in this game and the rivalry, then you can rest assured that no matter what equipment they are stepping on everyone's time-consuming in its base are the same. If the shape of unhappy, then you'd better not go directly along the slope, preferably a slide steepest curve trajectory.
(Again involute )
I think it is worth to talk about last involute, cycloid it as interesting, but also more practical role to play in the work. Such as gears. Early cycloid gear are in accordance with the outline of production.
This gear teeth generally have a wider cross-section, and therefore stronger and more powerful, but in the modern manufacturing industry has rarely met. As shown above, is composed of two cycloid cycloid gear contour constituted like this gear on the bike now more common. In the last movie, you'll see the tooth root portion has been cut off a piece, which is common practice on the timepiece gear (in order to reduce weight, and more important to reduce the impact and friction.)
Today, more common involute gear based on the profile (imagine a lot of Huygen's Chops composed of gear it wants).
When this gear meshing, when the point of contact between two teeth stable, less friction, more stable operation. No jitter and other noise-shaped gear happen. But also has the advantage that the gear center distance between the two gears can be freely changed without changing the wheels of the transmission ratio (and gerotor gears must be fixed center distance between the two).
Finally, involute gear top and bottom is flat, curved sides only, so relatively easy to process.
Cycloidal gear now still on the bike, watches, clocks common, but beyond that, basically involute gear of the world line.
[Rolling Stone]
Next time, if you see another lonely hillside tumbling boulders, please remember that the 17th century University of tyrants who!
(Compiled: via datagenetics)
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