Remember those orange plastic ka-ta-fo-you - light-from-ra-zha-te-li, attached-la-yu-schi-e-sya to the spokes of the ve-lo-si-ped-no-go ko-le-sa? Attach the ka-ta-fot to the very rim of the ko-le-sa and follow its tra-ek-to-ri-ey. The obtained curves are at the top of the family of cycloids.

At the same time, the co-le-so is called a pro-from-a-circle (or circle) of a cycle.

But let's go back to our century and switch to more modern technology. On the way, a ka-mu-shek fell, which got stuck in the flow of the ko-le-sa. Having turned a few circles with the wheel, where does the stone go when you jump out of the flow? Against the right-hand movement of the motor cycle or along the right-hand side?

As you know, the free movement of the body is on the way along the path to that trajectory along which then it moved. The ka-sa-tel-naya to the cycl-o-i-de is always to the right along the direction of movement and passes through the upper point ku about the surrounding area. According to the right-hand direction of movement, our ka-mu-shek is also moving along.

Do you remember how you rode through the puddles in childhood on a bicycle without a rear wing? The wet streak on your back is the confirmation of life's expectation that it has just received a re-zul -ta-ta.

The 17th century is the century of the cycle. The best scientists have studied its amazing properties.

Some kind of tra-ec-to-ria will bring the body, moving under the action of the force of gravity, from one point to another in a short time? This was one of the first tasks of that na-u-ki, which now has the name va-ri-a-tsi-on-noe-use- number.

Mi-ni-mi-zi-ro-vat (or max-si-mi-zi-ro-vat) you can have different things - path length, speed, time. In za-da-che about the bra-hi-sto-khron mi-ni-mi-zi-ru-et-sya it’s time (what the hell-ki-va-et-sya sa-mime on -name: Greek βράχιστος - least, χρόνος - time).

The first thing that comes to mind is straight-line tra-ek-to-ria. Yes, we’ll also look at the re-turn-around cycle with the return point at the top of the given points. check. And, following Ga-li-leo Ga-li-le-em, - a four-vertical circle that connects our points.

Why did Ga-li-leo Ga-li-lei look at the quarter-vertical circle and think that this was the best in terms of le time-me-ni tra-ek-to-ria descent? He wrote broken ones into it and noticed that as the number of links increased, time later decreased. From here, Ga-li-ley naturally moved to the circle, but made the wrong conclusion that this tra-ek -ria is the best among all possible ones. As we see, the best tra-ek-to-ri-ey is a cycl-o-i-da.

Through two given points it is possible to create a single cycle with the condition that at the top point there is point of return of the cycle. And even when the cyclic comes under the motherfucker to pass through the second point, it will still cri -howl of the quickest descent!

Another beautiful za-da-cha, connected with the cycl-lo-i-da, - za-da-cha about the ta-u-to-chron. Translated from Greek, ταύτίς means “the same,” χρόνος, as we already know, “time.”

We’ll make three one-on-one hills with a pro-fi-lem in the form of cycles, so that the ends of the hills are aligned and settled down at the top of the cycle. We set up three bo-bahs for different you-so-yous and let’s move on. It’s a surprising fact that everyone will come down one day!

In winter, you can build a slide of ice in your yard and check this property live.

For-yes-cha-about-that-chrono-it-is-in-the-look-up-of-such-a-curve that, starting from any-bo-go-start- But after all, the time of descent to the given point will be the same.

Christian Huy-gens knows that the only thing that is chronic is the cycl-o-i-da.

Of course, Guy-gen-sa doesn’t in-t-re-so-val the descent along the icy mountains. At that time, scientists didn’t have such a big deal out of love for art. For-yes-that-we-have-been-studied,-is-ho-di-from life and for-pro-s of those times. In the 17th century, long-distance sea voyages were already completed. Shi-ro-tu seas have already been able to determine up to a hundred precisely, but it’s surprising that for a long time they couldn’t determine -deal with everything. And one of the pre-la-gav-shih methods from the shi-ro-you was based on the presence of precise chro-no-meth ditch

The first person who thought of making ma-yat-no-new clocks that would be accurate was Ga-li-leo Ga-li-ley . However, at the moment when he begins to re-create them, he is already old, he is blind, and in the remaining year The scientist does not have time to complete his life. He tells this to his son, but he hesitates and begins to f-------------------------- near death and doesn’t have time to sit down. The next famous figure was Christian Huygens.

He noticed that the period of ko-le-ba-niya usually ma-yat-ni-ka, ras-smat-ri-vav-she-go-sya Ga-li- le-em, za-vis-sit from the beginning of the po-lo-zhe-niya, i.e. from am-pl-tu-dy. Thinking about what the trajectory of the load's movement should be so that time does not depend on it -se-lo from am-pl-tu-dy, he decides for-da-chu about that-u-to-chron. But how can you make the load move in a cyclic manner? Translation of theo-re-ti-che-re-studies into a practically-ti-che-plane, Guy-gens de-la-et “cheeks” , on which on-ma-you-va-et-sya ve-rev-ka ma-yat-no-ka, and decides a few more ma-te-ma-ti-che -skih tasks. He argues that the “cheeks” should have the profile of the same cycle, thereby suggesting that evo-lyu-that cycle-lo-i-dy is a cycle-lo-i-da with the same pa-ra-met-ra-mi.

In addition, the proposed Guy-gen-som construction of the cycl-lo-and-distance-but-no-go pos-vo-la-et on -count the length of cycles. If there is a blue point, the length of which is equal to what you are talking about from the circle, bend the thread as much as possible, then its end will be at the point where the “cheeks” and cyclic-and-dy-tra-cross ek-to-rii, i.e. at the top of the cycle-and-dy-“cheeks”. Since this is half the length of the ar-ki cycl-o-i-dy, then the full length is equal to eight ra-di-u-sam pro-iz-vo- dyad's circle.

Christ-an Huy-gens made a cyclic-and-distant ma-yat-nik, and the hours with him pro-ho-di-li-is-py-ta-niya in the sea Pu-te-she-stvi-yah, but didn’t get used to it. However, the same as the watch with the usual ma-yat-nik for these purposes.

Why, one-on-one, there are still hours of fur-lowness between us and the usually-veined ma-yat-no-one ? If you look, then with small defects, like the red one, “cheeks” cyclic and-far-but-go ma-yat-have almost no influence. Accordingly, movement in a cyclic and circular manner with small deviations is almost identical yes, yes.

5. Parametric equation of cycloid and equation in Cartesian coordinates

Let us assume that we are given a cycloid formed by a circle of radius a with a center at point A.

If we choose as a parameter determining the position of the point the angle t=∟NDM through which the radius, which had a vertical position AO at the beginning of the rolling, managed to rotate, then the x and y coordinates of point M will be expressed as follows:

x= OF = ON - NF = NM - MG = at-a sin t,

y= FM = NG = ND – GD = a – a cos t

So parametric equations cycloids have the form:

When t changes from -∞ to +∞, a curve will be obtained consisting of an infinite number of branches such as those shown in this figure.

Also, in addition to the parametric equation of the cycloid, there is also its equation in Cartesian coordinates:

Where r is the radius of the circle forming the cycloid.

6. Problems on finding parts of a cycloid and figures formed by a cycloid

Task No. 1. Find the area of ​​a figure bounded by one arc of a cycloid whose equation is given parametrically

and the Ox axis.

Solution. To solve this problem, we will use the facts we know from the theory of integrals, namely:

Area of ​​a curved sector.

Consider some function r = r(ϕ) defined on [α, β].

ϕ 0 ∈ [α, β] corresponds to r 0 = r(ϕ 0) and, therefore, the point M 0 (ϕ 0 , r 0), where ϕ 0,

r 0 - polar coordinates of the point. If ϕ changes, “running through” the entire [α, β], then variable point M will describe some curve AB given

equation r = r(ϕ).

Definition 7.4. A curvilinear sector is a figure bounded by two rays ϕ = α, ϕ = β and a curve AB defined in polar

coordinates by the equation r = r(ϕ), α ≤ ϕ ≤ β.

The following is true

Theorem. If the function r(ϕ) > 0 and is continuous on [α, β], then the area

curvilinear sector is calculated by the formula:

This theorem was proven earlier in the topic definite integral.

Based on the above theorem, our problem of finding the area of ​​a figure limited by one arc of a cycloid, the equation of which is given by the parametric parameters x= a (t – sin t), y= a (1 – cos t), and the Ox axis, is reduced to the following solution .

Solution. From the curve equation dx = a(1−cos t) dt. The first arc of the cycloid corresponds to a change in the parameter t from 0 to 2π. Hence,

Task No. 2. Find the length of one arc of the cycloid

The following theorem and its corollary were also studied in integral calculus.

Theorem. If the curve AB is given by the equation y = f(x), where f(x) and f ’ (x) are continuous on , then AB is rectifiable and

Consequence. Let AB be given parametrically

L AB = (1)

Let the functions x(t), y(t) be continuously differentiable on [α, β]. Then

formula (1) can be written as follows

Let’s make a change of variables in this integral x = x(t), then y’(x)= ;

dx= x’(t)dt and therefore:

Now let's get back to solving our problem.

Solution. We have, and therefore

Task No. 3. We need to find the surface area S formed from the rotation of one arc of the cycloid

L=((x,y): x=a(t – sin t), y=a(1 – cost), 0≤ t ≤ 2π)

In integral calculus, there is the following formula for finding the surface area of ​​a body of revolution around the x-axis of a curve defined parametrically on a segment: x=φ(t), y=ψ(t) (t 0 ≤t ≤t 1)

Applying this formula to our cycloid equation we get:

Task No. 4. Find the volume of the body obtained by rotating the cycloid arch

Along the Ox axis.

In integral calculus, when studying volumes, there is the following remark:

If the bounding curve curved trapezoid is given by parametric equations and the functions in these equations satisfy the conditions of the theorem on the change of variable in a certain integral, then the volume of a body of rotation of a trapezoid around the Ox axis will be calculated by the formula

Let's use this formula to find the volume we need.

The problem is solved.

Conclusion

So, in the course of this work, the basic properties of the cycloid were clarified. We also learned how to build a cycloid, I found out geometric meaning cycloids. As it turned out, the cycloid has enormous practical applications not only in mathematics, but also in technological calculations and physics. But the cycloid has other merits. It was used by scientists of the 17th century when developing techniques for studying curved lines - those techniques that ultimately led to the invention of differential and integral calculus. It was also one of the “touchstones” on which Newton, Leibniz and their first researchers tested the power of new powerful mathematical methods. Finally, the problem of the brachistochrone led to the invention of the calculus of variations, which is so necessary for physicists of today. Thus, the cycloid turned out to be inextricably linked with one of the most interesting periods in the history of mathematics.

Literature

1. Berman G.N. Cycloid. – M., 1980

2. Verov S.G. Brachistochrone, or another secret of the cycloid // Quantum. – 1975. - No. 5

3. Verov S.G. Secrets of the cycloid // Quantum. – 1975. - No. 8.

4. Gavrilova R.M., Govorukhina A.A., Kartasheva L.V., Kostetskaya G.S., Radchenko T.N. Applications of a definite integral. Guidelines And individual assignments for 1st year students Faculty of Physics. - Rostov n/a: UPL RSU, 1994.

5. Gindikin S.G. The stellar age of the cycloid // Quantum. – 1985. - No. 6.

6. Fikhtengolts G.M. Course of differential and integral calculus. T.1. – M., 1969

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