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Problems in Physics, March 2010

Please read the rules of the competition.


Problems with sign 'M'

Deadline expired on 12 April 2010.

M. 304. Measure the maximum force that can be exerted on an egg along its longer axis, and along the direction perpendicular to this axis, without breaking the egg. How much does the the egg become indented until the moment of break?

(6 points)

Statistics


Problems with sign 'P'

Deadline expired on 12 April 2010.

P. 4234. The speed of a cyclist along the horizontal road is 16 km/h, 12 km/h upward and 24 km/h downward. Once the cyclist went from the village to the town, and it took 3 hours to get to the town and to go back to the village. How far is the city from the village?

(3 points)

Solution (in Hungarian)

P. 4235. The distance between the cars in a long queue of cars is 15 m and the average length of a car is 5 m. The speed of the car-queue is 72 km/h. How long does it take after an accident to develop a 10 km-long car-queue consisting of standing cars if the distance of the cars is 3 m?

(4 points)

Solution (in Hungarian)

P. 4236. A vertical rod of mass 4 kg is hanging on a rope and a 3-kg cat is grasping the bottom end of the rod. The rope is cut off and the frightened cat begins to run up on the rod. While the rod is falling vertically, the cat remains at the same height, with respect to the ground. Calculate the acceleration of the rod.

(4 points)

Solution (in Hungarian)

P. 4237. A path is made to a uniform density cube of mass m, and of edge a, such that when the cube rolls along the path without skiding, its centre moves along a straight line (``square wheel''). At the ``top'' of the path a horizontal initial velocity of v0 is given to the centre of the cube. The static frictional force is big enough, so the cube do not slide anywhere. What is the speed of the centre of mass when the cube touches the lowest point of the path? (The moment of inertia of the cube with respect to its centre of mass is \frac{1}{6}ma^2.)

(4 points)

Solution (in Hungarian)

P. 4238. A thin-walled plexy cylinder of mass M and of radius R can rotate freely without friction about its horizontal axis. Inside there is a solid rubber cylinder of mass m and of radius r. What is the period of the motion of the system when the rubber cylinder is displaced a little from its equilibrium position? It can be assumed that the rubber cylinder rolls without sliding inside the plexy one.

(5 points)

Solution (in Hungarian)

P. 4239. A tractor is used to transport hay bales from point A to the stable denoted by B as shown in the figure. The farmer has to be in a hurry in order to finish the transport before an approaching big thunderstorm. Find the best route for the tractor if the speed of the tractor on the arable land is 75% of its speed on the meadow. (Numerical solutions are also acceptable.)

(4 points)

Solution (in Hungarian)

P. 4240. A man in a small dinghy approaches from a great distance the wave-breaker dam, in which there are two holes 8 m apart. Waves of wavelength \lambda=6 m hit the other side of the dam. What must the path of the dingy be if the man in it wants the dingy to be rocked the least?

(4 points)

Solution (in Hungarian)

P. 4241. A sample of ideal gas is taken through the cyclic process shown in the figure.

a) What is the relation between T1, T2 and T3?

b) Express the efficiency of the heat engine which carry out this cyclic process in terms of x=\frac{V_{2}}{V_{1}} and \kappa=\frac{c_{p}}{c_{V}}.

c) In case of air between what values can this efficiency vary?

(4 points)

Solution (in Hungarian)

P. 4242. A candle is burning in a vertical cylinder standing on the table. The inner wall of the cylinder has a reflexive surface. The height of the cylinder is 30 cm, its diameter is 12 cm, and the centre of the candle light is at a height of 14 cm, 2 cm away from the axis of the cylinder. From which direction will the image of the candle light, which becomes three times wider, be on the top of the cylinder?

(5 points)

Solution (in Hungarian)

P. 4243. The number of turns of a solenoid (straight coil) of length L=0,5 m is N=2000, the area of its cross section is A=16 cm2, and it has an iron core of relative permittivity of \mur=2500. At a moment the current in the coil changes at a rate of 100 A/s. What is the induced voltage in that long straight wire of length 2\ell=20 cm (the wire is next to the centre of the coil)

a) which is perpendicular to the axis of the coil, if its two ends are equidistant from the coil, and its centre is at a distance of \ell from the coil?

b) which has the same direction as the previous one, and one of its end is at a distance of \ell from the axis of the coil?

(5 points)

Solution (in Hungarian)


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