KöMaL Problems in Physics, April 2026
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Problems with sign 'M'Deadline expired on May 15, 2026. |
M. 449. Construct a torsion pendulum from a mason jar by suspending it vertically with two parallel threads. (The threads should each be 1 metre long, and their spacing should be equal to the diameter of the jar's opening. They can be secured with the screw cap.) Measure the period of the torsion pendulum as a function of the amount of granulated sugar poured into the jar!
(6 pont)
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Problems with sign 'G'Deadline expired on May 15, 2026. |
G. 921. Children are descending a mountain slope inclined at \(\displaystyle 10^\circ\) on sleds. At the base of the slope, the surface continues horizontally. The icy snow layer is uniform throughout the entire path, so the coefficient of friction is identical on both the inclined and horizontal sections. The sled starts from the top of the slope without initial velocity, travels down the incline, and then proceeds along the horizontal surface until it comes to a complete stop. The time required to travel the slope is one-third of the time spent moving along the horizontal section.
a) How much farther did the sled travel on the level section compared to the slope?
b) Determine the coefficient of kinetic friction.
c) After sliding down, one of the children pulls the sled back to the starting point. How many times more work does he perform on the slope than on the horizontal section? (The child always pulls with a force parallel to the ground and moves at constant speed.)
(4 pont)
G. 922. We have two springs of equal length but different spring constants, one red and one blue. If we stretch the two springs connected ``in series'', the elastic energy of the red spring will be twice that of the blue one. What is the ratio of the elastic energies if the two springs are connected ``in parallel'' and stretched that way?
(4 pont)
G. 923. A one-litre cubic container holds half a litre of mercury and half a litre of water. Another container, with twice the linear dimensions, is likewise filled with mercury and water. The hydrostatic pressure at the bottom of the two containers is the same.
a) What is the hydrostatic pressure at the bottom of the containers?
b) What quantities of mercury and water are there in the second container?
(4 pont)
G. 924. A person wearing \(\displaystyle +6\) dioptre contact lenses borrows a pair of \(\displaystyle +2\) dioptre glasses and puts them on over their contact lenses. Under what circumstances, if any, would this be reasonable? Or is it completely pointless?
(4 pont)
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Problems with sign 'P'Deadline expired on May 15, 2026. |
P. 5724. On a long one-way main street, traffic-light-controlled intersections are very closely spaced. The lights are green half the time and are timed to give drivers a green wave at 50 km/h. When Edward is not stopped at a red light, he rides his bicycle at 25 km/h, braking and accelerating abruptly. What is his average speed? (The intersections are so close together that Edward can pass many of them during one full light cycle.)
(5 pont)
P. 5725. Two identical cylinders with homogeneous mass distribution are placed on a slope so that they touch each other, their axes are horizontal. There is friction both between the cylinders and between each cylinder and the slope. Can this system remain in equilibrium if it is left alone?
(4 pont)
P. 5726. A lightweight but rigid rod is bent at a right angle, with arm lengths \(\displaystyle \ell_1\) and \(\displaystyle \ell_2\). Small bodies of masses \(\displaystyle m_1\) and \(\displaystyle m_2\) are attached to the ends of the arms. The rod can rotate freely in its own plane about a horizontal axis passing through the elbow of the rod. What is the period of the oscillation of the system when it is slightly displaced from its equilibrium position?
(4 pont)
P. 5727. A 30 cm high air column is enclosed by an 8 kg piston in a vertical, thermally insulated cylinder with a cross-sectional area of \(\displaystyle 2\,\mathrm{dm^2}\). The external air pressure is 100 kPa. The enclosed gas is kept in stable equilibrium by a constant downward force acting on the piston. At a certain moment the piston is released and allowed to move freely upward until it reaches a bumper located 10 cm above its initial position. A heating coil ensures that the gas inside the cylinder drives the piston upward with constant acceleration.
a) What is the initial pressure of the gas if the piston moves frictionlessly and the duration of its motion is 0.2 s?
b) Give the power of the heating coil as a function of time.
(4 pont)
P. 5728. In the infinite circuit shown in the figure, each resistor has resistance \(\displaystyle R\), each capacitor has capacitance \(\displaystyle C\), and each inductor has inductance \(\displaystyle L\). Determine how the current in the circuit varies as a function of time if an alternating voltage supply \(\displaystyle U_{A,B}=U_\mathrm{max}\sin\omega t\) is connected across points \(\displaystyle A\) and \(\displaystyle B\).
(4 pont)
P. 5729. Two point-like bodies of equal mass \(\displaystyle m\), carrying charges \(\displaystyle +q\) and \(\displaystyle -q\), are initially held at rest at a distance \(\displaystyle d\) from each other. If they are released simultaneously, they will collide after some time. However, if the experiment is repeated in a uniform magnetic field of appropriate strength, directed perpendicular to the line joining the bodies, the collision does not occur. According to the solution to the problem set for the 2025 Eötvös competition (see the January 2026 issue of our journal), the minimum value of magnetic induction necessary to avoid collision is: \(\displaystyle B_{\mathrm{min}}=4\sqrt{km/d^3}\)
a) Using numerical methods, determine and plot the trajectory of the charges for magnetic inductions \(\displaystyle {B=\tfrac{4}{5}B_{\mathrm{min}}}\), \(\displaystyle B\approx B_{\mathrm{min}}\), and \(\displaystyle B=\tfrac{5}{4}B_{\mathrm{min}}\).
b) For \(\displaystyle B<B_{\mathrm{min}}\), determine and plot the time elapsed until the collision of the charges as a function of \(\displaystyle B/B_{\mathrm{min}}\).
c) When \(\displaystyle B>B_{\mathrm{min}}\), the charges periodically move away from each other a distance of \(\displaystyle d\) again and again. Determine and plot this period as a function of \(\displaystyle B/B_{\mathrm{min}}\). Hint for the numerical solution can be found in Péter Csóka and Barnabás Seprődi, ``Solving Physics Problems Using Numerical Methods'', published in the November 2024 issue of our journal.
(5 pont)
P. 5730. Laser light of wavelength 535 nm falls on a standard (consisting of identical slits) diffraction grating. A diffraction maximum is visible at an angle of \(\displaystyle 35^\circ\), and the highest observable diffraction order is the fifth. Determine the grating constant.
(4 pont)
solution (in Hungarian), statistics
P. 5731. If a very high-energy gamma photon collides with a stationary electron, pair production may occur, and in some cases more than one electron-positron pair can be created.
a) What is the minimum energy of the gamma photon if \(\displaystyle n\) electron-positron pairs are created in the process?
b) What is the least velocity of the particles created after pair production?
(5 pont)
P. 5732. A man of mass \(\displaystyle M\) hangs from the end of a rope ladder. The ladder consists of two parallel ropes of negligible mass which are connected by rigid rungs of mass \(\displaystyle m\) and of length \(\displaystyle \ell\). The spacing between adjacent rungs is \(\displaystyle h\). The mass of the man is much larger than the total mass of the rungs. At what speed do long wave, small angle-amplitude torsional waves propagate along the ladder?
(6 pont)
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