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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.

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?

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.

A ship is floating on pic. 1. Its owner had solved the problem of windless weather by attaching a powerful hair-dryer to the stern of the ship and pointng it against the sail. Consider at what conditions will the ship move forward or backward. What improvements you would suggest to make the ehgine work more efectively.

Two cylinders of radius $R$ with parallel horizontal axis placed at distance $a$ rotate in opposite directions. A plate of mass $m$ and length $2a$ is laid on them little bit asymetrically (see pic. 2). (viz obr. 2). The coeficient of friction between the plate and the cylinders is $μ$. What will happen

• when the angular velocities of the cylinders are the same,

• when the angular velocities of the left one is two times higher than of the second one.

Picture 3 shows an aerial view of three steam engines with clouds of smoke (seen from above). They move steadily on three straight parallel railways. The velocity of the first engine is 50 km/h, of the third one 70 km/h; their directions are shown on the picture. At what speed does the second engine move?