3th Round of 9th year

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1. ... a gnawed prism( points)

A gnawed prism (see pic.) of mass $M$ was put on a smooth horizontal plane. Small cubes of masses $μ$ and $m$ are situated in the lowest part of the pit and on the right slope respectively. There's no friction both between the prism and the surface and the cubes and the prism.

Determine the relations between masses $M$, $m$, $μ$ necessary for the cube $m$ to start to move when the cube $μ$ gets loose.

2. ... remote research( points)

It’s note easy to obtain precise astrometric data about Mercury. Many errors were rooted out just after radioastronomical measurement on the 60’s. Let’s follow this work. Radioastronomers emit a signal towards Mercury at $t$_{0} = 0 s and receives its reflection in an interval between $t$_{1} = 1070,15624 s and $t$_{2} = 1070,18879 s. Next they analyze the red shift of the received wave. The initial frequency 100 MHz was blurred between $f$_{1} = 99,97739700 MHz and $f$_{2} = 99,97740506 MHz. Supposing that the angle between equatorial plane of Mercury and the eclipses is small determine from these data the distance and relative speed of Mercury from the observatory, its radius, angular velocity and the length of one rotation period.

3. ... Pinoccio's hat( points)

Papa Carlo made a hat for Pinoccio from a thin metallic plate in the form of cone of height 20 cm with top angle of 60 degrees. Will this decoration keep itself on his head that could be imaged as a perfectly smooth ball of 15 cm diameter?

4. ... refrigerator( points)

A refrigerator stands in a thermally insulated room, plugged in and switched on. After one hour of work we turn it off and let the room temperature stabilize. Find the differ-ence of this temperature from the original temperature.

P. ... water pendulum( points)

A box vessel of a negligible mass has a square base of side length $a$ and a its height is 2$a$. The water body inside has a form of cube. Find the maximum distance $h$ between the bottom and the place where we hang the vessel so as it won't turn upside down after the water gets frozen (see pic. – the cut is lead in the vertical direction through the center of mass). Discuss two cases:

  • the vessel is absolutely rigid and the water freezes from the bottom,
  • the vessel is elastic enough and the water keeps its cube form after freezing. The ice slides along the sides of the vessel, the distance $h$ remains constant.

E. ... gravitation( points)

Try to measure the free fall acceleration in as many different ways as possible. Carry out 10–20 measurements for each method, compare their results and their errors.

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