[Problem M26—M30; F29—F35]

F29. Find the rate of evaporation per unit surface of water into a vacuum at a temperature of 20°C.
(The saturated vapor pressure at this temperature is 17.5 mm Hg.)
How long will it take for water in the room to evaporate if it is poured to the top into an ordinary tea saucer?
The evaporation of a small amount of water practically does not change the air humidity in the room, which is equal to 70%.

27

F30. The film camera and film projector produce 8 frames per second. An automatic vehicle with wheels, the actual diameter of which is 1 m, moves on the screen. The images of the wheels make 2 revolutions per second. What is the speed of the automatic?

G. L. Komkin

F31. How does light pressure orient a spherical cosmic corpus relative to the Sun, one half of which is mirror-like, and the other is black, completely absorbing the radiation of the Sun? The center of gravity of the corpus is in the center of the sphere.

P. L. Kapitsa

F32. A wooden block of square cross-section floats on the surface of the water. Which of the two equilibrium positions shown in figure 1 will be stable? The density of the material from which the block is made is equal to half the density of water.

F33. Two beads with masses $m_1$ and $m_2$ can slide along a smooth horizontal conducting ring without friction. At first, the beads were connected by a thread and there was a rolled spring between them. The thread is burned out. After the beads begin to move, the spring is removed. At what point in the rings will the beads collide at the same time? The beads become absolutely elastic.

F34. figure 2 shows a drip electrostatic machine (Kelvin generator). Drops of water charged to potential $\phi$ fall from the tube into the fields of an insulated metal ball of radius $R$. How does the maximum potential to which a ball can be charged depend on the drop height?

F35. The satellite flies at an altitude of 300 km. What stationary objects can be seen in a photograph taken from a satellite if the exposure time is 0.2 seconds?

M26. Let’s assume that in each issue of our magazine there will be five mathematics problems in the “Kvaenta” problem. Let us denote by $f(x, y)$ the number of the first task from the $x$-th issue of the journal for the $y$ year (for example, $f(6,1970)=26$). Write the general formula for $f(x, y)$ for all $x, y$ ($1 \leq x \leq 12$, $y \geq 1970$).

Solve the equation $f(x, y)=y$.

M27. Prove that if $\frac{a}{b-c} + \frac{b}{c-a} + \frac{c}{a-b} = 0$ then $\frac{a}{(b-c)^2} + \frac{b}{(c-a)^2} + \frac{c}{(a-b)^2} = 0$.

M28. a) Of the 19 balls, 2 are radioactive. For any pile of balls, in one check you can find out whether it contains at least one radioactive