Construction of a regular 17-gon

38

Find the distance from the center of the heavy chain to the axis of rotation.

F134. A heavy plate moves with a speed $v$ in a direction perpendicular to its plane. At an angle $\alpha$ to the direction of motion of the plate, a light ball moves at a speed $u$. Determine the magnitude and direction of the ball’s velocity vector after its elastic collision with the plate.

a) Distance when the ball speed is directed towards and away from the plate.

6. How does the answer change if the plate moves in a direction making an angle $\beta$ with its plane?

F135. A mine lying on the ground explodes from detonation. The mine fragments begin to move symmetrically in all directions with the same speeds $v$. The sizes of all osokols are the same. What fraction of the sedges will fall outside the circle of radius $R$ centered at the point of explosion?

G. L. Kstkin

F136. A parallel plate capacitor has a capacitance of $C$. Charge $+q$ was placed on one of the capacitor plates, and charge $+4q$ was placed on the other. Determine the potential difference between the plates of the capacitor.

V. E. Belonukkin

F137 *. Infrared radiation of the defining wavelength is absorbed by methane ($\text{CH}_4$). Under normal conditions, a 1 cm thick layer of pure methane absorbs 98% of the radiation energy. How many times will such radiation be weakened when passing vertically through the Earth’s atmosphere?

When calculating, take the weight content of methane in the atmosphere equal to $1,4 \cdot 10^{-6}$.

S. M. Kozel

Answers to the crossword (see page 31)

Horizontally: 1. VTSNIIOT-PUSK. 6. Obscurantist. 7. Telephone 8. Joint. 11. Egor. 18. Sax. 14. Akbin. 17. Ataman. 20. Ogly. 21. Stopkop. 22. Presentation.

Constructing a correct 17-gon

Richmond gave a simple construction of the seventeenth triangle $P_0P_1 \dots P_{16}$ (see figure).

Let’s connect point $P_0$ with point $J$, lying on radius $OB$ at a distance $1/4OB$ from the center. On the diameter passing through the point $P_0$, we choose points $E$ and $F$ so that $\Rightarrow OJE$ is equal to four times the angle $OJP_0$, and $\Rightarrow FJE$ is equal to $45^\circ$. Let a circle built on $FP_0$ as a diameter intersect

$OB$ at point $K$, and let a circle with center $E$ and radius $EK$ intersect $OP_0$ at points $N_3$ (between $O$ and $P_0$) and $N_5$. Let us restore the perpendiculars to $OP_0$ at these two points until they intersect with the original circle at points $P_3$ and $P_5$. Then the arc $P_3P_5$ (and the equal arc $P_1P_3$) is equal to the $2/17$ circle. The proof uses several times the fact that the roots of the equation

$x^2+2x\text{ctg}, 2\alpha-1=0$ are equal to tg$\alpha$ and - ctg$\alpha$.

(G. S. M. Coxter, Introduction to Geometry, M., “Hayka”, 1966, p. 49.)