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A rope is placed on two wedges (see figure) in such way, that in center is not touching the base. The situation is symmetric in left-right direction. Calculate maximal part of rope can be free standing in static equilibrium. Bonus: for which angle is the ratio biggest?

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.

• 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

$$δf=f_{+}-f_{-}=4**ω**\cdot \textbf{S}/(λ_{0}P)$$ 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.

Jakub took chain and held it at two places by fingers and started to spin it at the angular speed $ω$, same as at the picture. Marek saw it and asked, what is the shape of the chain 2 dimensions. What is the correct answer, if neglecting gravitation field?

Assume copper hemispherical pot, with diameter of 40 cm. The pot is filled with a soup and is placed in water. It is floating 10cm under water level. Will the water start to fill the pot, if the border of pot is pulled up by a chain by 10 cm?

In this question we investigate influence of temperature on to moment of inertia of a metal body. Lets have an axe going through the body. How will change the moment of inertia $J$ when the temperature is increased by $ΔT$, if the coefficient of thermal expansion of metal is $α$. If you have problem with a arbitrary shape of body, consider sphere of cylinder.

Carefully measure if the rolling resistance of cylinder depends on its radius or not.

Find the properties of material, which you need to make a rope for a lift from geostationary orbit to the Earth surface. Is such material available on Earth?