KöMaL Problems in Physics, November 2025
Please read the rules of the competition.
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Problems with sign 'M'The deadline is: December 15, 2025 24:00 (UTC+01:00). |
M. 444. Determine the moments of inertia of an AA battery which are calculated about the axis of symmetry and about an axis perpendicular to the axis of symmetry passing through the centre of mass.
(6 pont)
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Problems with sign 'G'The deadline is: December 15, 2025 24:00 (UTC+01:00). |
G. 901. The sailor of a river barge is inspecting the deck of the barge and is walking at a speed of \(\displaystyle 3.6~\mathrm{km/h}\) along the route \(\displaystyle A\)-\(\displaystyle B\)-\(\displaystyle C\)-\(\displaystyle D\)-\(\displaystyle A\). (See the figure; \(\displaystyle AB=CD=75~\mathrm{m}\), \(\displaystyle BC=AD=15~\mathrm{m}\).) The barge is travelling at a speed of \(\displaystyle 3.6~\mathrm{km/h}\) relative to the riverbank.
a) How many minutes does it take for the sailor to complete his inspection route?
b) What is the distance covered by the sailor relative to an observer on the riverbank while the sailor goes around the deck once?
c) Plot the sailor's path relative to the observer on the riverbank.
(4 pont)
This problem is for grades 1–10 students only.
G. 902. A fruit falls from the top of a 12 m tree, hits a small branch at a height of 8 m, and although it loses 30% of its speed, it can continue falling because it breaks the branch. At what speed does it land? (The fruit is heavy and small.)
(3 pont)
This problem is for grades 1–10 students only.
G. 903. Four children want to play on a see-saw. Their masses are: Anne 20 kg, Brian 25 kg, Cecily 30 kg, Dave 35 kg. There are 2-2 seats on each side of the see-saw, 120 and 150 cm from the pivot. Based on their experience, it is best to swing with the smallest possible torque difference between the two sides. Help them choose the two ``best'' sitting arrangements.
(3 pont)
This problem is for grades 1–10 students only.
G. 904. An electric water heater connected to the 230 V mains is used to heat water. The resistance of the heating element of the water heater during operation is \(\displaystyle 46~\Omega\).
a) What is the electrical power of the water heater?
b) By how many degrees does 1 litre of water warm up in 3 minutes? The energy loss during heating is 20%.
(3 pont)
This problem is for grades 1–10 students only.
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Problems with sign 'P'The deadline is: December 15, 2025 24:00 (UTC+01:00). |
P. 5679. An initially stationary trolley of mass \(\displaystyle M\) can move frictionlessly on the horizontal ground. The top of the trolley is flat and at one of its ends there is a small block of mass \(\displaystyle m=M/2\) (as shown in the figure). The length of the trolley is \(\displaystyle \ell=24~\mathrm{cm}\), and the coefficient of friction between the trolley and the block is \(\displaystyle \mu=0.2\).
a) At what maximum speed of \(\displaystyle v_0\) can we push the small block so that it does not fall off the trolley?
b) What will the speeds of the trolley and the block be at the moment when the block flies off the trolley, if the block is given an initial speed of \(\displaystyle v_1=2v_0\)?
(4 pont)
P. 5680. The angle of elevation of a hillside is \(\displaystyle 30^\circ\), and at the foot of the hill the ground is horizontal. When the hillside was covered with snow everywhere, Peter chose an unusual way of sledding: he started at different initial speeds at a distance of 5 m from the bottom of the slope (see the figure).
a) At what initial speed did the sled stop in the shortest time?
b) How much distance up the hill did the sled cover in this case? The path of the sled coincided with the fall line of the hillside. The hillside and the horizontal ground at the bottom of the hillside form a smooth curve. Friction between the sled and the snow is negligible.
(4 pont)
P. 5681. Two pieces of thread, each of length \(\displaystyle L=10~\mathrm{cm}\), have one end fixed to a vertical axis at a common point, and two small beads of equal mass \(\displaystyle m\) and of charge \(\displaystyle Q\) are attached to the other ends. In equilibrium, each thread makes an angle of \(\displaystyle \varphi=30^\circ\) with the vertical axis. When the axis is rotated uniformly, the beads in the steady state undergo uniform circular motion such that the threads make an angle of \(\displaystyle \alpha=45^\circ\) with the axis of rotation. What is the period of the circular motion of the beads?
(4 pont)
P. 5682. A cylinder of radius \(\displaystyle R\) and of height \(\displaystyle H\) contains a liquid. The cylinder is made to rotate about its axis. The angular speed of rotation is slowly increased until the edge of the liquid is drawn up to the rim of the cylinder. The liquid then just ``disappears'' from the centre of the bottom of the cup.
a) What is the greatest angular speed of the cylinder?
b) What is the initial height of the liquid in the cylinder?
(5 pont)
P. 5683. The figure shows the current-voltage characteristics of different colour LEDs. For a given value of voltage the current of the LED can be read from the graph.
a) Using the graph, determine the power consumption of a red, a green and a blue LED when they are connected in parallel to a voltage supply of voltage \(\displaystyle 2.5~\mathrm{V}\).
b) What is the power of the same three LEDs when they are connected in series to a voltage supply of voltage \(\displaystyle 7.5~\mathrm{V}\)?
(4 pont)
P. 5684. The frame shown in the figure is made from a piece of wire, which has uniform thickness. Calculate the ratio of the equivalent resistances between points \(\displaystyle A\) and \(\displaystyle B\) and between points \(\displaystyle A\) and \(\displaystyle C\).
(5 pont)
P. 5685. It is a well-known phenomenon that a breakpoint appears in the image of a thin, straight stick when it is partly immersed into a bowl of water (see Figure 1).
Figure 1
In Figure 2 there is a photograph of a stick that makes an angle \(\displaystyle \varphi\) with the vertical. When the stick is photographed (or viewed) from a suitable direction, the submerged part of the stick is not visible at all, although the stick still reaches the bottom of the bowl. At what angle \(\displaystyle \varphi\) can this phenomenon occur?
Figure 2
(4 pont)
P. 5686. Astronauts set off to explore the distant universe. Their spacecraft is launched from the Earth and travels at a speed of \(\displaystyle 3/5\,c\). The controllers on Earth send a part of the cargo \(\displaystyle T\) time after the launch of the spacecraft in another rocket travelling at a speed of \(\displaystyle 4/5\,c\).
a) What is the speed of the rocket in the coordinate system of the astronauts?
b) How much time elapses between the launch and the arrival of the rocket, carrying the cargo, in the reference frame of the ground controllers and in the reference frame of the astronauts? The time required to accelerate the rocket and the spacecraft is negligible with respect to \(\displaystyle T\).
(5 pont)
P. 5687. Four balls of radius \(\displaystyle r\) and mass \(\displaystyle m\) are held by thin threads of length \(\displaystyle r\). One end of each thread is fixed at a common point so that the centers of the balls form a square. The two balls at the end of one diagonal of the square are red, the other two are blue. (Most of the mass of each ball is at the centre of the ball, so the moment of inertia of the balls is negligibly small. The frictional force between the balls is also negligible.)
In this arrangement the balls are in an unstable equilibrium, from which the system will move away at the slightest external disturbance and will move towards a state of lower potential energy with increasing acceleration.
a) What is the angle between the threads and the vertical in the initial and in lowest potential energy states?
b) What is the maximum value of the total kinetic energy of the four balls?
(6 pont)
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