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J. Perelman
"Entertaining physics". Book 2.
Chapter 3. Circular motion

MATHEMATICS IN THE CIRCUS

I know that the ranks of "soulless" formulas scare away other fans of physics. But, refusing to explore the mathematical side effects, such enemies of mathematics deprive yourself of pleasure to foresee the course of events and to define its terms. In this example, the case of two or three formulas will help us accurately determine under what conditions can the successful implementation of such amazing trick, as the mileage in "the devil's noose".

Let us calculations.

Denote letters those values that need to calculate:


"Damn loop". Bottom left diagram for calculation.

the letter h denote the height at which rolls down a cyclist;

the letter x will denote the part of the height h, which rises above the top point of the loop; from the figure it is obvious that x = h - AB;

the letter r denote the radius of the circle loop;

the letter m is the total mass of the artist along with the bike; the weight will be expressed through mg, and:

the letter g shows the acceleration forces of the earth gravity;

it is, as you know, 9.8 m/s per second;

the letter v denote the speed of the bike at the moment when it reaches the highest point of the circle.

All these values we can link the two equations. First, we know from mechanics that the speed that gets the bike by the time, rolling on the inclined track, he is With at the point In (this situation is depicted in the lower part in the figure)equal to that which it has in the upper part of the loop, at the point In. The first speed is expressed by the formula *

or v2 = 2gx

Consequently, the speed of the cyclist at point b is equal to √2gx, i.e., v2 = 2gx

Further, in order for the cyclist, reaching the highest point of the circular path, fell down, you need to developing this centripetal acceleration was greater than the acceleration of gravity, i.e., it is necessary that v2/r > g, or v2 > gr.

But we already know that v2 = 2gx; therefore, 2gx > gr or x > r/2.

So, we learned that to successfully complete this puzzling focus it is necessary to arrange "damn loop" so that the top of the inclined portion of the path towered above the top point of the loop more than 1/2 of its radius. The steepness of the slope is not important, one need only to point, from which the cyclist begins to descend, towered above the top of the loop more than 1/4 of its diameter. If, for example, the loop has a diameter of 16 m, the artist must begin the descent PE less than 20-meter height. Don't do it this condition, no art will not help him to describe "damn loop": reaching the top, he will inevitably fall.

Calculation this does not include the effect of the friction force in the bike: it is believed that the velocity at point C and point b are the same. Why not lengthen the path and do a very gentle slope. When sloping downhill as a result of friction, the speed of the bike upon reaching point b would be smaller than at point C.

It should be noted that when performing this trick cyclist riding without a chain, giving the machine the action of gravity, neither speed up nor slow down his movements, he can not, and should not. All his art is to hold the middle of the wooden walkway; at the slightest evasion artist runs the risk of running down the track and to be thrown aside. Speed in a circle is very high: in the circle with a diameter of 16 m rider commits a turnover in 3 seconds. This corresponds to a speed of 60 km per hour! To control the bike at that speed, of course, wonder; but this is not necessary; it is possible to rely on the laws of mechanics. "By itself, bike stunt, " we read in the booklet, which is a professional with proper calculation and sturdy construction of the apparatus is not dangerous. The danger of the trick lies in the heart of the artist. If the hand of the artist tremble, if he will be thrilled to lose self-control, if he suddenly becomes ill, you can expect to have it all".

At the same law rests the well-known "dead loop" and other aerobatics. In "dead loop" paramount right "acceleration" of the pilot on the curve and skillful control of the aircraft.

* While we neglect the energy of the spinning rims of Bicycle wheels; the impact of this condition on the result of the calculation slightly.

Entertaining physics J. Perelman

 




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