Assignment of Series 4 of Year 22

Calculate the shape of mirror, in which the Cyclops head will look like a square. Cyclops head is a sphere with the eye in the middle.

It is known, that the ice subjected to higher pressure has lower melting temperature. This effect is used also during ice skating. Is this effect strong enough at extremely low temperatures (e.g. is pressure from skate big enough to melt the ice)? If not, what is causing sliding?

Engineers in NASA want to use gyroscopes as a energy source for space satellites. What is the maximum energy, which can be stored in rotating cylinder of radius $r$. What is the maximum angular velocity $ω$ which we can spin gyroscope, before it breaks into pieces?

You probably know a fairytale about a man, who wanted from a king to put in sequence 1, 2, 4, 8, 16, ..., 2^{63} rice grains from left to right. Calculate, how you need to reinforce the table, so the grains will not fall down? (More explicitly: calculate center of mass of all rice, if assuming each grain to be a mass point, all in the middle of corresponding square on chessboard.

Disconnected electric circuit on the picture contains one uncharged and one charged capacitor (charge $Q)$. Wires are ideal with no resistance, both capacitors are identical. Total energy of electrical charges in circuit is then $Q⁄2C$. If the charges on capacitors become equal, then total energy will be $Q⁄8C+Q⁄8C=Q⁄4C$, which is half comparing to the beginning. Explain what happened and where the half of energy was used up. The switch is made in such way, that you can assume the energy loss has happened somewhere else.

Get a inflatable ball, blow it up, make a knot on the orifice and measure how its volume is changing in time. Try to find out, how many percent of the surface is taken by pores, through which the air is escaping.

• Foucault pendulum has drawn into a sand at two different demonstrations two different diagrams, as on the picture. Find out, what caused different shape and how long the pendulum was to create such diagrams on the floor of Paris cathedral. How many tips do the stars have in reality?

• What shape will have the water level in barrel standing still at horizontal plane?

• Show, that equation

$$δf=f_{+}−f_{−}=4ω\cdot \textbf{S}/(λ_{0}P)$$ for frequency difference (frequency of beats) of two contra-propagating light beams in laser gyroscope, is valid for any shape of gyroscope, not only circle.