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Deadline expired on 11 May 2006. 
P. 3884. There is a layer of oil on the surface of water and a wooden board is floating in it, such that half of its volume is submerged in the water and onefourth of its volume is in the oil. What is the density of the board if the density of oil is 70% of the density of water?
(3 pont)
statistics
P. 3885. Find the conditions for the system shown in the figure to be in equilibrium.
(4 pont)
statistics
P. 3886. In the end of 2005 when the row broke out over the price of natural gas between Ukraine and Russia, Hungary had a storage of 2.4^{.}10^{9} m^{3} (standard form) natural gas. In Hungary to store the gas, exhausted gas or oil fields are used which are filled with water (e.g. near Hajdúszoboszló) and the gas is compressed into these reservoirs under a pressure of 200 bar.
a) At what maximum depth can these reservoirs be?
b) In Hungary what is the least useful volume of these reservoirs?
c) The stored gas is mainly methane. Estimate the total mass of the compressed gas.
d) Natural gas is cooled down under the temperature of 160 ^{o}C, and condensed. The density of the gas increases to 600 times of the density when the gas is at standard temperature and pressure. How many oiltankers can be filled with the gas described above if the volume of one tanker is 160 000 cubic metre?
(4 pont)
statistics
P. 3887. An object is projected horizontally with a speed of v_{0}=10 m/s.
a) How much time elapses until the velocity and the displacement enclose an angle of 15^{o}?
b) What is the greatest angle enclosed by the velocity and the displacement?
(5 pont)
statistics
P. 3888. A man is walking along a straight road from point A to point B with a uniform speed. Point O is on the perpendicular bisector of the line segment AB, and at O there is a man who is beating a drum. Another day another man is walking from A to B, with the same speed as the first man was walking, and plays the drum, while an observer listens at point O. In which case does the observer count more beats while the man moves from A to B? (The beating starts in both cases well before the man leaves A, and lasts at least until the man reaches B. The beating is uniform, its period is the same in both cases and much smaller than the time needed to walk from A to B.)
(5 pont)
statistics
P. 3889. A syringe is filled with water. Its volume is 20 cm^{3}, and the cross section of its interior part is 4 cm^{2}. The syringe is held vertically such that its nozzle is at its top, and its 100 g piston is pressed and moves with a constant speed. The ejected water has an initial upward velocity of 2 m/s, and the crosssection of the beam of water at the nozzle is 1 mm^{2}. (Neglect the dissipated energy due to friction.)
a) Find the speed of the piston.
b) What is the total work done by us?
c) Express and graph the force exerted on the piston as a function of time.
(4 pont)
statistics
P. 3890. A charged metal sheet is placed into uniform electric field, perpendicularly to the electric field lines. After placing the sheet into the field, the electric field on the left side of the sheet will be E_{1}=5.6^{.}10^{5} V/m and on the right it will be E_{2}=3.1^{.}10^{5} V/m (see the figure).
a) Find the total charge of the sheet if you know that a force of 0.08 N is exerted on it.
b) Find the area of the sheet.
(5 pont)
statistics
P. 3891. The resistance of each resistor of the electric circuit shown in the figure is R. Find the effective resistances between any two junctions of the circuit.
(4 pont)
statistics
P. 3892. A diffraction grating is illuminated perpendicularly to its plane by light of wavelength 680 nm. The grating spacing is 5^{.}10^{6} m.
a) How many diffraction maxima can be observed on a screen which is parallel to the plane of the diffraction grating.
b) What is the angle between the incident beam and the direction of the furthest reinforcement?
(4 pont)
statistics
P. 3893. The electron of a hydrogen atom can be replaced by a muon for a short time. The new atom is called muonium. (The muon is similar to the electron but its mass is 207 times greater than the mass of the electron, and its decay time is very short. Find the radius of the muonium in its ground state. How much energy is needed to ionize the muonium atom? Data: The radius of the hydrogen atom is about 0.05 nm, and its ionization energy is 2.2 aJ.
(4 pont)
statistics