KöMaL Problems in Physics, February 2026
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Problems with sign 'M'Deadline expired on March 16, 2026. |
M. 447. Measure the spring constant of a loose coil spring using weights of different masses comparable to the mass of the spring,
a) using a static method,
b) using a dynamic method (by studying vibrations).
Compare the results obtained using the two methods and try to explain any differences.
(6 pont)
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Problems with sign 'G'Deadline expired on March 16, 2026. |
G. 913. A table tennis player wants to hit back the ball, travelling at a speed of \(\displaystyle v_1\), with a higher speed, \(\displaystyle v_2\), in the opposite direction to the original velocity. How fast should the racket be?
Assume that the collision is perfectly elastic and that the mass of the ping-pong ball is negligible compared to the racket held by the player.
(4 pont)
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G. 914. A tug-of-war competition between two teams of three people each is held on horizontal asphalt. On the left side, the masses of the competitors are 50 kg, 60 kg, and 70 kg, and the coefficient of friction between their shoes and the asphalt is 0.6, 0.5, and 0.4, respectively. On the right side, the masses of the tug-of-war competitors are 45 kg, 55 kg, and 65 kg, and the coefficient of friction between their shoes and the asphalt is 0.6 for all of them. Which team will win the competition if the competitors exert maximum force?
(3 pont)
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G. 915. A triangle-shaped thin plate with sides \(\displaystyle a\), \(\displaystyle b\), and \(\displaystyle c\) has a uniform mass distribution and weighs \(\displaystyle G\). The plate is supported horizontally at the vertices of the triangle. What is the force exerted by the plate on the support points?
(4 pont)
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G. 916. In cartoons, characters often lift off the ground while holding a bunch of balloons. Estimate the minimum number of nearly spherical, helium-filled balloons with a diameter of 20 cm that would be needed to lift a child of mass 25 kg.
a) Ignore the mass of the balloon and string.
b) Assume that the mass of the balloon and string is 2 g.
(4 pont)
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Problems with sign 'P'Deadline expired on March 16, 2026. |
P. 5706. From a thin iron rod with a uniform mass distribution, pieces of lengths \(\displaystyle a\), \(\displaystyle b\), and \(\displaystyle c\) are cut off, and a triangular rigid frame is constructed from them. The total weight of the iron frame is \(\displaystyle G\). The frame is supported at its vertices in a horizontal position. What force is exerted by the iron frame on the support points?
(4 pont)
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P. 5707. Eduard rides his bicycle downhill at a constant speed along a long slope of constant angle of inclination. How does the power dissipated by the brakes depend on the speed?
The total mass of Eduard and his bicycle is \(\displaystyle m\), the slope angle is \(\displaystyle \alpha\), and without braking, Eduard would accelerate to a speed of \(\displaystyle v_\mathrm{max}\).
(4 pont)
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P. 5708. The coefficient of friction between the horizontal table shown in the figure and the body of mass \(\displaystyle m\) on it is \(\displaystyle \mu=0.5\). The pulleys, which can be considered uniform density discs, can rotate without friction, and the ropes do not slip on the rims of the pulleys. What is the magnitude and direction of the force exerted on the ceiling by the fixed pulley attached to it?
(5 pont)
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P. 5709. In May 2025, the Korond stream flooded the salt mine in Parajd. According to reports, approximately five million tons of water flowed into the mine system.
Let us model the mine as an empty sphere of radius \(\displaystyle a\) with a volume equal to the volume of the inflowing water, which just touches the Earth's surface. The contact point is the entrance to the mine. Estimate how much the direction of a plumb line placed at a distance of \(\displaystyle 2a\) from the entrance to the mine on the Earth's surface would have deviated during the flood of the mine.
(4 pont)
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P. 5710. A sample of helium gas is taken through the cyclic process, which consists of an isobaric expansion, an isochoric cooling, and an adiabatic compression process. What is the maximum efficiency of this cycle?
(5 pont)
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P. 5711. The simple circuit shown in the figure, was built from five alike capacitors of capacitance \(\displaystyle C\), and an ideal battery of electromotive force \(\displaystyle U\).
a) How much charge accumulates on the plates of the middle capacitor?
b) How does this value change if we replace one of the capacitors with another capacitor of capacitance \(\displaystyle nC\)?
(4 pont)
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P. 5712. A point-like firefly moves along a straight line and crosses the principal axis of a thin lens of a focal length of 30 cm. The angle between the path of the firefly and the principal axis is 60\(\displaystyle ^\circ\), and the angle between the image of the firefly and the principal axis is 30\(\displaystyle ^\circ\). Measured from the lens, at what distance did the firefly cross the principal axis of the lens?
(5 pont)
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P. 5713. According to an amateur space researcher student, a large, flat aluminium foil could remain in equilibrium, even at a stationary position, somewhere in the solar system (but far from the planets). For the sake of simplicity, he assumed that the aluminium sail would reflect 100% of the sunlight. (It is known that the radiant flux density reaching the top of the Earth's atmosphere is \(\displaystyle 1360\,\mathrm{W/m^2}\).) What is the maximum thickness of the ``light sail" that would make this idea feasible?
(5 pont)
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P. 5714. A small ball with mass \(\displaystyle m\) was suspended on a thin silk thread of length \(\displaystyle \ell\). At the same height, at a distance of \(\displaystyle \ell/2\) from it, a small metal sphere was fixed to an insulating stand. By means of an electrostatic machine the sphere can be given different positive charges of \(\displaystyle +Q\). The ball which is suspended on the insulating thread has a negative charge of \(\displaystyle -q\).
The charge \(\displaystyle Q\) is changed very slowly, we always wait till the pendulum reaches equilibrium position. The angle \(\displaystyle \varphi\) between the silk thread and the vertical is some \(\displaystyle \varphi(Q)\) function of the charge \(\displaystyle Q\).
a) For very small \(\displaystyle Q\), \(\displaystyle \varphi\) is also very small, and with good approximation the angle of deflection varies proportionally to the charge \(\displaystyle Q\): \(\displaystyle \varphi\approx\tfrac{1}{Q_0}\cdot Q\), where \(\displaystyle Q_0\) is a constant having the same dimension as charge. How can \(\displaystyle Q_0\) be expressed with the other parameters of the problem (\(\displaystyle m\), \(\displaystyle g\), \(\displaystyle \ell\) and \(\displaystyle q\))?
b) How many stable equilibrium states belong to very small, very large, and medium \(\displaystyle Q\) charges? Provide a rough (qualitative) description of the \(\displaystyle \varphi(Q/Q_0)\) function when charge \(\displaystyle Q\) is increased from zero to a large value (\(\displaystyle Q\gg Q_0\)) and then decreased back to zero.
c) Verify the qualitative considerations with detailed calculations and determine the numerical coordinates of the characteristic points of the graph of the deviation-angle versus charge function.
(6 pont)
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