KöMaL Problems in Physics, January 2026
Please read the rules of the competition.
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Problems with sign 'M'The deadline is: February 16, 2026 24:00 (UTC+01:00). |
M. 446. Some identical books on a shelf are parallel to each other and tilted at a small angle relative to the vertical. Measure the force exerted by the last book on the side wall of the shelf. How does the force depend on the number of books and the angle of tilt?
(6 pont)
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Problems with sign 'G'The deadline is: February 16, 2026 24:00 (UTC+01:00). |
G. 909. A racing car is moving on a straight test track. The relationship between the distance \(\displaystyle s\) measured from the starting point of the track and the time \(\displaystyle t\) elapsed since the start of the time measurement – if the length is measured in metres and the time in seconds, and the units of measurement are not written out – can be expressed as follows: \(\displaystyle s=10+10\,t+2\,t^2\). How does the distance-time function change if we choose kilometres as the unit of length and hours as the unit of time? Calculate the position, speed, and acceleration of the racing car 1/4 minute after the start in both systems.
(3 pont)
This problem is for grades 1–10 students only.
G. 910. A rhombus-shaped plate with a weight of \(\displaystyle G\) and of uniform mass distribution is supported horizontally at its vertices. The force exerted on the support at one of the vertices is \(\displaystyle G/5\). What are the forces exerted on the other supports at the other vertices?
(4 pont)
This problem is for grades 1–10 students only.
G. 911. The virtual image of point \(\displaystyle P\) produced by a thin diverging lens is at point \(\displaystyle P'\) as shown in the figure. The principal axis of the lens is marked by a continuous line, and each division on the grid corresponds to \(\displaystyle 10~\mathrm{cm}\) horizontally and \(\displaystyle 1~\mathrm{cm}\) vertically. What is the focal length of the lens?
(4 pont)
This problem is for grades 1–10 students only.
G. 912. In the circuit shown in the figure, the switch is initially open.
a) How much current flows through the resistors and batteries in the circuit before and after the switch is closed?
b) How much are these currents if the polarity of the voltage source next to the switch is reversed?
(4 pont)
This problem is for grades 1–10 students only.
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Problems with sign 'P'The deadline is: February 16, 2026 24:00 (UTC+01:00). |
P. 5697. During a fireworks display, projectiles are fired from the same location at the same initial speed in all directions, creating a bright flash at the peak of their trajectory. Along what surface are the flash points located? Neglect air resistance.
(4 pont)
P. 5698. On the level ground, a two-seater glider with mass \(\displaystyle M\) is lifted into the air by means of a winch. The winch is connected to the glider by a long tow rope with spring constant \(\displaystyle D\) and mass \(\displaystyle m\). When the towing begins, the acceleration is \(\displaystyle a\), while there is friction between the rope and the grassy ground, where the coefficient of friction is \(\displaystyle \mu\). Calculate how much the horizontal tow rope stretches shortly after the plane starts moving.
Data: \(\displaystyle m=150~\mathrm{kg}\), \(\displaystyle M=400~\mathrm{kg}\), \(\displaystyle a=3~\mathrm{\tfrac{m}{s^2}}\), \(\displaystyle D=2500~\mathrm{\tfrac{N}{m}}\), \(\displaystyle \mu=0.15\).
(5 pont)
P. 5699. An artificial moon orbits the Earth in the same direction as the Earth's rotation, in the plane of the equator, at a constant height above the surface of the Earth, which is of 4 times the radius of the Earth.
a) What is the period of the artificial satellite?
b) How many days does it take for the artificial satellite to pass over a selected point on the equator?
(4 pont)
P. 5700. According to the legend, Dido, the princess of Tyre, after being forced to flee her homeland, arrived in North Africa, where she asked the local ruler for as much land as she could enclose with an ox-hide. The ruler agreed, so Dido cut the skin into a long, narrow strip, made a fence out of it, and then chose the largest possible area of land along the coast, founding the city of Carthage. According to a lesser-known version of the story, Dido's voyages took her to a circular island with a radius of 1 km, somewhere in the Mediterranean Sea. What was the maximum area of land she could have selected if the length of her fence was 1 km? Dido could consider the smaller part of the divided island as her own.
(5 pont)
P. 5701. A sample of nitrogen gas was taken through an isochoric process, during which some heat was transferred to it. Then the gas was immediately compressed adiabatically so that the work done by the environment on the gas was equal to the heat transferred during the isochoric process. Finally, the gas was heated in an isobaric process until the mechanical work done by the gas became equal to the heat transferred in the isochoric process. During the three processes, what was the change in the temperature of the nitrogen if the temperature of the gas increased by \(\displaystyle 80~{}^\circ\mathrm{C}\) in the isochoric process?
(4 pont)
P. 5702. A copper pipe is bent into a circle and its ends are soldered together, that is, a hollow torus with copper walls is created. What is the electric field strength inside the copper tube and in the material of its wall, if the current in a sufficiently long, straight coil is changed at a constant rate in time and the copper tube is positioned as shown in the figure,
a) next to the coil,
b) coaxially with the coil?
c) What is the electric field strength in the wall of the copper tube in the previous situation if the tube is cut somewhere?
Supplying power to the coil was set up in a way that the magnetic effects of both the connecting cables and the resulting current occurring along the symmetry axis of the solenoid were eliminated. (E.g. power is supplied by a coaxial cable, and the solenoid is double-wound.)
ÁBRA
(5 pont)
P. 5703. We observe a 2-millimetre fossil perpendicularly from a distance of 25 cm, which is the distance of normal vision. If we looked any closer, we would not be able to see it clearly with the naked eye.
a) What is the angle subtended by the fossil, if we do not use a magnifying glass?
b) In order to see the details of the fossil better, we look at it through a thin magnifying glass with a focal length of 5 cm. At what angle do we see the fossil if the magnifying glass is 3 cm away and our eyes are 25 cm away from the object?
c) Place the magnifying glass as close to your eye as possible. What is the maximum angle at which you can see the fossil with the magnifying glass if you can freely change the distance between the object and your eye?
(5 pont)
P. 5704. Using a device, charged particles are accelerated from rest, through a voltage of \(\displaystyle U_\mathrm{gy}\), to a speed significantly less than the speed of light. The particles moving in a straight line in vacuum enter to a parallel plate capacitor, where they are deflected and exit the capacitor at some angle relative to the original direction of their motion. The length of the parallel plate capacitor is \(\displaystyle \ell\), the distance between the plates is \(\displaystyle d\), and the deflection voltage is \(\displaystyle U_\mathrm{el}\). The charged particles enter the deflecting field along the symmetry line of the parallel plate capacitor, as shown in the figure, and do not collide with the plates during their motion. (The effect of gravity can be ignored.)
a) How does the angle of deflection \(\displaystyle \alpha\) depend on the given data?
b) How does the type of particle used in the experiment affect the angle of deflection?
(4 pont)
P. 5705. The Little Prince wandered onto a strange planet, where he sat down to play on the shore of a honey lake. He threw pebbles of identical size and mass \(\displaystyle m\) into the very deep lake. He observed that when he dropped a pebble from directly above to the surface of the lake, it accelerated to a maximum speed of \(\displaystyle v_\mathrm{max}\) in the honey. (It can be assumed that the magnitude of the drag force acting on the pebble is directly proportional to the speed of the pebble.) Then, the Little Prince threw a pebble into the lake from the shore so that it entered the honey close to the shore at a velocity of \(\displaystyle v_0\) in a nearly horizontal direction. He calculated that the pebble can reach the bottom of the lake at a maximum distance of \(\displaystyle x_\mathrm{max}\) from the shore (measured horizontally).
a) To what depth \(\displaystyle y\) did the pebble go while it moved horizontally a distance of \(\displaystyle x<x_\mathrm{max}\) measured from the shore?
b) How much work is done by the drag force on the pebble while the pebble sinks in the honey to point \(\displaystyle P\), which is at a distance of \(\displaystyle x\) from the shore and at a depth of \(\displaystyle y\)?
The Little Prince gave the results in terms of the parameters \(\displaystyle v_0\), \(\displaystyle v_\mathrm{max}\) \(\displaystyle x_\mathrm{max}\) and the variable \(\displaystyle x\), using the following integral formulas: \(\displaystyle \int_0^a\frac{1}{1-u}\,\mathrm{d}u=\ln\frac{1}{1-a}\), \(\displaystyle \int_0^a\frac{u}{1-u}\,\mathrm{d}u=\ln\frac{1}{1-a}-a\), \(\displaystyle \int_0^a \frac{u^2}{1-u}\,\mathrm{d}u=\ln\frac{1}{1-a}-a-\frac{a^2}{2}\).
(6 pont)
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