Series 4, Year 24

• Strings.

Using dimensional analysis determine the dependance of the frequency of oscillatons of a string if you know that it depends on its length $l$, on the tension $F$ in the string and on its linear density $ρ_{l}$.

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• Downward.

You have a dumbell which consists of a short rod and two heavy discs. You wrap a string around the rod and let the dumbell fall while holding the string. What is the velocity of the dumbell? The discs have mass $M$ and radius $R$. The radius of the rod is $r$ and you can neglect its mass.

Karel has decided to throw his notes into the Sun. Help: him calculate the necessary initial velocity of his notebook so that it reaches the Sun. You can neglect the frictional froces acting on his notebook and you can also assume that the trajectory of the Earth around the Sun is circular. You know the masses of the Earth and the Sun as well as the distance of the Earth from the Sun.

Design a shape for a sandglass so that the dependance of the height of sand on time is linear. If you do not neglect friction etc. you can gain some extra points.

Terka was playing around and spilled five liters of liquid nitrogen in her room. Couple days later she bought five liters of gasoline, brought it to her room and burned it. Could this playing around result in her being sick? To be more concrete describe the change in the temperature, pressure and oxygen concentration in her room (in both cases) if it is perfectly isolated and has dimensions 3$x3x4\;\mathrm{m}$.

To display a cyan blue on your monitor it has to light up both blue and red segment. If you however mix blue and red temperas you see that the resulting color is purple. Imagine that the temperas consist of small pieces and describe how does the color of a mixture of blue and red temperas depends on the size of these pieces.

Determine the maximum height a typical egg can fall from without breaking itself. How does the result change if you wrap the egg in some protective material which is not thicker than 5$\;\mathrm{mm}?$ Try couple of different materials.