Mathematical and Physical Journal
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KöMaL Problems in Physics, December 2017

Please read the rules of the competition.


Show/hide problems of signs:


Problems with sign 'M'

Deadline expired on January 10, 2018.


M. 373. Measure the heat capacity of a great piece of pebble (or a small sized cobble).

(6 pont)

statistics


Problems with sign 'G'

Deadline expired on January 10, 2018.


G. 617. A lorry travels at a speed of 70 km/h along a horizontal, circular path of radius 120 m. What is the minimum value of the coefficient of static friction if the vehicle does not slide?

(3 pont)

solution (in Hungarian), statistics


G. 618. At most what amount of water can be pumped up in a quarter of an hour from a depth of 50 m by means of a submersible pump of power rating 2 kW?

(3 pont)

solution (in Hungarian), statistics


G. 619. Point \(\displaystyle C\) can be connected to either point \(\displaystyle X\) or to point \(\displaystyle Y\) by means of a piece of flexible wire as shown in the figure of the circuit.

\(\displaystyle a)\) What is the value of current \(\displaystyle I\) in the main branch in each case?

\(\displaystyle b)\) What is the value of this current if the flexible wire is disconnected from point \(\displaystyle C\)?

(3 pont)

solution (in Hungarian), statistics


G. 620. When there is a thunderstorm, and a lightning strikes a lake, then although it has negligibly small probability that a fish is just struck by the lightning, there are bodies of dead fish floating on the surface of the water. What is the reason?

(3 pont)

solution (in Hungarian), statistics


Problems with sign 'P'

Deadline expired on January 10, 2018.


P. 4980. When there is a thunderstorm, and a lightning strikes a lake, then although it has negligibly small probability that a fish is just struck by the lightning, there are bodies of dead fish floating on the surface of the water. What is the reason?

(3 pont)

solution (in Hungarian), statistics


P. 4981. An object of mass \(\displaystyle m\) and of speed \(\displaystyle v\) is shot through the suspended object of mass \(\displaystyle M\), as shown in the figure, and leaves it at a speed of \(\displaystyle v/2\). The hung object of mass \(\displaystyle M\) is first suspended on a negligible mass rigid rod of length \(\displaystyle \ell\), and second it is suspended on a thread of length \(\displaystyle \ell\). After the object of mass \(\displaystyle M\) was shot through, it moves along the circular path of radius \(\displaystyle \ell\) in both cases. Determine the values of the necessary speed of \(\displaystyle v\) in both cases. What is the ratio of these speeds?

(4 pont)

solution (in Hungarian), statistics


P. 4982. A solid cylinder of radius 5 cm can be rotated about its own horizontal symmetry axis. A long piece of thin thread is wrapped around the lateral surface of the cylinder, such that an object, which has the same mass as the cylinder, is attached to the free end of the thread.

\(\displaystyle a)\) The system begins to move from rest. How many revolutions does the cylinder turn in 1.2 seconds?

\(\displaystyle b)\) What is the speed of the suspended object after \(\displaystyle N\) complete turns of the cylinder?

(Neglect air resistance.)

(4 pont)

solution (in Hungarian), statistics


P. 4983. A cylinder, whose walls are thermally insulated, is attached to the inside part of the wall of a fixed closed cylinder as shown in the figure. The wall of the container is made of some heat conducting material. The thermally insulated piston of cross sectional area of 1 dm\(\displaystyle {}^2\) in the cylinder can move frictionlessly. Initially both in the container and in the cylinder there is a sample of air of volume 2.6 dm\(\displaystyle ^3\) at a pressure of \(\displaystyle 10^5\)Pa, and at a temperature of 27 \(\displaystyle {}^\circ\)C.

The air in the cylinder is then heated by means of an electric heater, while the temperature of the air in the container remains constant.

\(\displaystyle a)\) How much distance does the piston move if the temperature of the air in the cylinder increases to 77 \(\displaystyle {}^\circ\)C?

\(\displaystyle b)\) Sketch the change of states of the air inside the cylinder on the \(\displaystyle p\)–\(\displaystyle V\) diagram.

\(\displaystyle c)\) Estimate how much heat is absorbed by the air inside the cylinder.

(5 pont)

solution (in Hungarian), statistics


P. 4984. Express, in terms of the ratio of the specific heat capacities, \(\displaystyle \kappa=c_p/c_V\), what portion of the absorbed heat by a sample of gas, expanding at constant pressure, is the work done by the expanding gas.

(3 pont)

solution (in Hungarian), statistics


P. 4985. The focal length of the objective lens of a camera is 3 cm. A photo is taken about a distant object and then the image is enlarged by a scale factor of three. What is observed to be greater, the object or its image on the picture? By what factor is the greater one larger than the other?

(4 pont)

solution (in Hungarian), statistics


P. 4986. At each vertex of an equilateral triangle a point-like charge \(\displaystyle Q\) is fixed. A point-like charge of mass \(\displaystyle m\) and of charge \(\displaystyle q\) is oscillating in the middle of the triangle along one of the medians of the triangle. The amplitude of the oscillation is much smaller than the diameter \(\displaystyle D\) of the circumscribed circle of the triangle.

What is the angular frequency of the oscillation? (Consider only the electric forces.)

(5 pont)

solution (in Hungarian), statistics


P. 4987. The flux linkage of a circular copper loop of radius \(\displaystyle R\) and of cross sectional area \(\displaystyle A\), is changing in time as \(\displaystyle \Phi(t)=\Phi_0-kt\). The loop is in uniform magnetic field and the magnetic induction is perpendicular to the plane of the loop.

\(\displaystyle a)\) What is the tension in the loop at the moment of \(\displaystyle t_0\)?

\(\displaystyle b)\) What is the value of the current in the loop at the moment when the magnetic induction is zero?

Data: \(\displaystyle R=\)10 cm, \(\displaystyle A=0.5~\rm mm^2\), \(\displaystyle \Phi_0=0.04\) Vs, \(\displaystyle k=5\) mV, \(\displaystyle t_0=2\) s.

(5 pont)

solution (in Hungarian), statistics


P. 4988. The Sun has a radius of 1.4 million km and is making one complete rotation in approximately 25.4 days about its axis, which is approximately perpendicular to the plane of the orbit of the Earth. Consequently one part of the Sun is moving away from us whilst the other is approaching us, which leads to the broadening of the spectral lines due to the Doppler effect.

What is the value of this broadening in nm for a spectral line of wavelength approximately 550 nm?

(4 pont)

solution (in Hungarian), statistics


P. 4989. The half-life of the alpha decaying Uranium isotope of \(\displaystyle {}^{235}\)U is 704 million years. Besides the alpha decays spontaneous fissions may occur as well (resulting in daughter nuclides of greater mass). As an average 0.0056 fissions occur in 1 kg Uranium 235 in one second.

\(\displaystyle a)\) What percent of the \(\displaystyle {}^{235}\)U nuclides undergo spontaneous fission?

\(\displaystyle b)\) What would the half-life of \(\displaystyle {}^{235}\)U be if only spontaneous fissions occurred?

(4 pont)

solution (in Hungarian), statistics


P. 4990. The cross section of the wider part of a fixed vertical tube is \(\displaystyle A_1\), whilst that of its narrower part is \(\displaystyle A_2\). In the tube, between two pistons, there is some liquid of density \(\displaystyle \varrho\). The pistons are connected by means of a rigid rod of length \(\displaystyle \ell\). The masses of the rod and the pistons are negligible. The ambient air pressure is \(\displaystyle p_0\).

What is the magnitude and the direction of the force exerted in the rod if

\(\displaystyle a)\) the narrower,

\(\displaystyle b)\) the wider

part of the tube is on the horizontal tabletop?

What strange thing happens if \(\displaystyle \ell\) is ``relatively large''?

(6 pont)

solution (in Hungarian), statistics


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