KöMaL Problems in Physics, May 2025
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Problems with sign 'M'Deadline expired on June 16, 2025. |
M. 441. Place an iron on its inclined support and switch it on. Plot the on-and-off switching of the iron as a function of time. Wait until this on-and-off switching becomes periodic, and use this to estimate the heat that the iron gives off every second, in this steady state. Take the measurements at three different temperature values. Before each measurement, wait for the iron to cool down to room temperature. For the estimation of the power, use the power rating of the iron.
(6 pont)
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Problems with sign 'G'Deadline expired on June 16, 2025. |
G. 889. A toy model crane can lift up to 60 toy concrete beams without breaking the rope. All the linear dimensions of the real crane, as well as the real concrete beams, are 30 times larger than those of the toy model, but the physical properties of the materials are the same. How many real concrete beams can the real crane lift?
(4 pont)
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G. 890. Suppose that there is \(\displaystyle +Q\) charge on the surface of the Earth, and also \(\displaystyle +Q\) charge on the surface of the Moon, in both cases the charges are distributed evenly.
a) Calculate how much charge \(\displaystyle Q\) would be needed so that the electrostatic repulsion balances the gravitational attraction between the Earth and the Moon.
b) Assuming that the distance between the Moon and the Earth is halved, how would the magnitude of the charge \(\displaystyle Q\) balancing gravitational attraction change?
(3 pont)
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G. 891. A domino-shaped block is made from some low-conducting material. The size of the block is: \(\displaystyle 0.5\,\mathrm{cm}\times 3\,\mathrm{cm}\times 6\,\mathrm{cm}\). The resistance of the domino is measured so that current flows between the opposite faces of the domino. The connections are made such that the current density in the domino is uniform. The voltages are chosen so that the magnitude of the current density is the same in all three cases.
a) What is the ratio of the three measured resistances?
b) What is the ratio of the applied three voltage values?
c) What is the ratio of the dissipated energy per second due to Joule heating in the three cases.
(4 pont)
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G. 892. Anyone who has (or has had) a wristwatch has almost certainly experienced that when the Sun shines through the window, the glass of the watch can be used as a mirror to project an amusing, bouncing spot of light onto the wall, the ceiling, the floor, or even into the eyes of your friend. What does the shape of the light spot depend on, the shape of the glass or the shape of the Sun?
(4 pont)
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Problems with sign 'P'Deadline expired on June 16, 2025. |
P. 5652. A stone is thrown from a tower of height \(\displaystyle H\) upwards at an angle of \(\displaystyle \alpha\). Just before impact, the velocity vector of the stone makes an angle of \(\displaystyle \beta\) with the horizontal. How far from the tower did the stone land? Neglect air resistance.
(4 pont)
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P. 5653. Two straight roads cross perpendicularly. On one road a car is approaching the crossroads at a speed of \(\displaystyle 90~\mathrm{km}/\mathrm{h}\), on the other a motorcyclist is approaching it at a speed of \(\displaystyle 72~\mathrm{km}/\mathrm{h}\). At a given (\(\displaystyle t=0\)) instant, the distance between the two vehicles (as the crow flies) is \(\displaystyle 347~\mathrm{m}\). After \(\displaystyle 5\) seconds, their distance decreases to \(\displaystyle 188~\mathrm{m}\).ű
a) How far were the two vehicles from the crossroads initially?
b) What will be the minimum distance between the two vehicles?
For the sake of simplicity, consider both vehicles as point-like objects.
(5 pont)
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P. 5654. Two thin metal rods of the same length, density and cross-section were soldered together to make a straight rod that is twice as long as the original length of a single rod. The Young's modulus of the material of one part (denoted by A) is \(\displaystyle E_\mathrm{A}\) and that of the other part (denoted by B) is \(\displaystyle E_\mathrm{B}>E_\mathrm{A}\). The rod is rotated about an axis perpendicular to it at a given angular speed. The axis of rotation goes through the
a) free end of rod A,
b) free end of rod B,
c) soldered ends.
In which of the three cases will the total elongation of the rod be the greatest and the smallest?
(4 pont)
solution (in Hungarian), statistics
P. 5655. Two identical sound sources at a distance of \(\displaystyle d\) from each other emit sound waves of wavelength \(\displaystyle \lambda\) in the same phase. The waves are detected by a receiver at a point on the circumference of a circle of radius \(\displaystyle R\). The two sound sources are symmetrically positioned with respect to the centre of the circle. Where are the points at which the waves from the two sound sources reinforce each other if \(\displaystyle d=2\lambda\) and \(\displaystyle R\gg d\)?
(4 pont)
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P. 5656. We construct an infinite resistor chain as follows: the resistance of the resistors of the first "step" are \(\displaystyle R\), those of the next are \(\displaystyle 2R\), \(\displaystyle 4R\), \(\displaystyle \ldots\), etc., always twice the value of the previous one (see figure). What is the equivalent resistance of the infinite resistor chain?
(4 pont)
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P. 5657. A cylinder contains air at a pressure that is three times that of its surroundings. A hole is opened in the cylinder. We find that the gas in the cylinder cools rapidly, then slowly warms back to its initial temperature.
a) To what temperature does the gas cool from the initial \(\displaystyle 300~\mathrm{K}\)?
b) What percentage of the initial gas in the cylinder is lost during the rapid cooling,
c) and during the slow warming?
Assume that no air can get into the cylinder from outside.
(5 pont)
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P. 5658. If a spherical body (e.g. a large steel ball) is placed in an empty bowl and a photo is taken, a circle appears in the image (regardless of the direction in which it was photographed) (Figure 1 = 1. ábra). If the bowl is filled with enough water to slightly cover the ball and then a new image is taken, the image will appear as an ellipse (Figure 2 = 2. ábra). From Figure 3 = 3. ábra, which shows the ball enlarged, the scale of the distortion can be easily read. Determine the angle of the camera's principal axis with the horizontal when the picture was taken. (The distance between the camera and the ball is much larger than the diameter of the ball.)
(5 pont)
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P. 5659. Fission reactions often produce the isotope \(\displaystyle {}^{92}\mathrm{Sr}\), which decays into the stable \(\displaystyle ^{92}\mathrm{Zr}\) in two successive \(\displaystyle \beta\) decays: \(\displaystyle {}^{92}\mathrm{Sr}~\xrightarrow{2.66~\mathrm{h}}~{}^{92}\mathrm{Y}~\xrightarrow{3.54~\mathrm{h}}~{}^{92}\mathrm{Zr}\). After preparing a sample of pure \(\displaystyle ^{92}\)Sr, how long will it take for the amount of \(\displaystyle ^{92}\)Y produced to be the most?
Hint: Try to reduce the equations to a single radioactive decay equation. See the article about the addition of oscillations in several different cases: Vigh Máté: Összetett rezgések I–II. (https://www.komal.hu/cikkek/cikklista.h.shtml)
(5 pont)
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P. 5660. A point-like ball of mass \(\displaystyle m\) is threaded to a fixed rigid semicircle-shaped wire, and can move along it frictionlessly. The diameter of the semicircle is horizontal, its radius is \(\displaystyle R\), and the plane of the semicircle is vertical. (See the figure.) A thin thread is attached to the ball and wound through a small pulley at the end of the wire at \(\displaystyle C\). A body of mass \(\displaystyle m\) is attached to the other end of the thread. The ball on the left is released without a jerk from the horizontal position of the thread when the thread makes an angle of \(\displaystyle \alpha=0^\circ\) with the horizontal diameter.
a) What is the speed at which the bodies move when the ball on the left passes the lowest point of the wire?
b) What is the acceleration of the bodies at this moment?
(6 pont)
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