**P. 4541.** How long is the shadow of a 1 m long rod, which is fixed perpendicularly to the ground at the equator,

*a*) at noon, on the 21-st of June;

*b*) 2 hours later after noon on the 21-st of June?

(5 points)

**P. 4545.** Two cylinder shaped vessels of cross section *A* are fixed vertically to a trolley, which was initially at rest. The two vessels are connected with a thin horizontal tube which is equipped with a tap. The distance between the axes of the vessels is *L*. The vessel in the left hand side is filled with some liquid of density ; and at this time the total mass of the system at rest is *m*.

After opening the tap what is the speed of the cart at the moment when the speed of the water levels in the vessels is *v*? (Rolling friction, friction in the bearings and air-drag are negligible.)

(5 points)

**P. 4546.** The top part of a thermally insulated, vertical cylinder of cross section *A*=0.5dm^{2} is fixed. There is a sample of Helium gas in the cylinder and initially its pressure is *p*_{0}=10^{5} Pa, its length is _{0}=50 cm. The bottom part of the cylinder is closed with an initially fixed thermally insulating piston of mass *m*=10 kg. The external air pressure is also *p*_{0}. Then the piston is released, and it can move frictionlessly in the cylinder.

*a*) What is the length of the helium confined in the cylinder when the speed of the piston is maximum?

*b*) What is the maximum speed of the piston?

*c*) What is the greatest length of the confined gas?

(5 points)

**P. 4549.** A thyristor is an electrical element made of semiconductor, and can be used in many different ways. Its current-voltage characteristics is quite strange. (See *figure 1.*)

The thyristor is connected as shown in *figure 2,* and the voltage *U*_{0} is slowly increased from zero to *U*_{max }, and then it is slowly decreased back to zero.

Graph the current as a function of the voltage *U*_{0}, if

*a*) and ,

*b*) and .

(5 points)