KöMaL Problems in Physics, September 2025
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Problems with sign 'M'Deadline expired on October 15, 2025. |
M. 442. Measure the density of a – not yet used – dishwashing sponge.
(6 pont)
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Problems with sign 'G'Deadline expired on October 15, 2025. |
G. 893. A 4 m long carpet lies on the floor of the corridor. Fold the carpet (in four equal layers) by holding the left edge (B) and move it continuously horizontally at a speed of 20 cm/s first to the right, then to the left and then to the right again (see the figure).
a) How long does it take to fold the carpet?
b) After 5 s from the start of folding, what length of the carpet has a rightward velocity? The carpet is made of thin, easy to fold material, and does not slide on the floor. The changes in the direction of the motion are momentary.
(4 pont)
solution (in Hungarian), statistics
G. 894. From a regular hexagonal plate with side \(\displaystyle a\) and a uniform mass distribution, cut out three equilateral triangles with side \(\displaystyle a\), as shown in the figure. Place the three cut-out triangles on one of the remaining triangles. Where is the centre of mass of the resulting shape?
(4 pont)
solution (in Hungarian), statistics
G. 895. The University of Bremen has a very high drop tower. The cylindrical, completely sealed drop tube has an internal cross-sectional area of \(\displaystyle 12~\mathrm{m}^2\), and it takes nearly two hours to pump out essentially all the air from the tube, during which the air pressure at the top and bottom of the tower is continuously measured. Suppose that at a given moment the pressure difference between the top and bottom of the tower is 1000 Pa. How many moles of air were in the cylinder of the drop tower at this moment?
(4 pont)
solution (in Hungarian), statistics
G. 896. Five resistors are connected to the terminals \(\displaystyle A\) and \(\displaystyle B\) of a 24 V voltage source as shown in the figure. The resistances are \(\displaystyle R_1=40~\Omega\), \(\displaystyle R_2=50~\Omega\), \(\displaystyle R_3=R_4=10~\Omega\), and \(\displaystyle R_5=20~\Omega\).
a) Determine the equivalent resistance of the circuit in the closed and open positions of the switch.
b) How much does the power of resistor \(\displaystyle R_4\) change when the closed switch is opened?
(4 pont)
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Problems with sign 'P'Deadline expired on October 15, 2025. |
P. 5661. A trolley with mass \(\displaystyle m_1=4~\mathrm{kg}\) rolls without friction on the horizontal ground at a speed of \(\displaystyle v_1=2~\mathrm{m}/\mathrm{s}\). A brick of mass \(\displaystyle m_2=1~\mathrm{kg}\) is held by a grabber above the subsequent path of the trolley, just at the height of the trolley (as shown in the figure). When the trolley gets underneath the brick, the grabber releases the brick, which is then placed on the trolley. (Its vertical displacement, and hence the speed at which the brick reaches the trolley, is negligible.) There is friction between the trolley and the brick, the coefficient of friction is \(\displaystyle \mu=0.2\). The trolley is long enough to prevent the brick from sliding off of it.
a) What is the common speed at which the brick and the trolley move when the brick is no longer sliding on the trolley?
b) How much distance does the brick cover and for how long does the brick slide on the trolley before this common speed is reached?
c) What distance relative to the ground is covered during this by the brick and by the trolley?
(4 pont)
solution (in Hungarian), statistics
P. 5662. A force of constant magnitude \(\displaystyle F\) is exerted on an initially stationary body of mass \(\displaystyle m\) on a horizontal surface for time \(\displaystyle t\). The body then stops after time \(\displaystyle t'\) due to the frictional force. How much distance does the body cover?
(4 pont)
solution (in Hungarian), statistics
P. 5663. The system shown in the figure is in equilibrium. A load of 225 kg is hung on the end of a 45 kg supporting rod. Determine the magnitude and direction of the tension in the cable and of the force exerted by the hinge on the supporting rod.
(4 pont)
solution (in Hungarian), statistics
P. 5664. We often hear that the melting ice at the poles slows down the Earth's rotation around its axis. Estimate the order of magnitude of this phenomenon. Antarctica can be considered to have an area of \(\displaystyle 14~\mathrm{million~km}^2\), and the Arctic ice sheet has the same size. Investigate that cases when the thickness of ice decreases by 1 m due to melting at the South Pole or at the North Pole.
a) By what amount does sea level change in one case and the other?
b) How much does the length of an Earth day change?
(5 pont)
solution (in Hungarian), statistics
P. 5665. In a cylinder of volume \(\displaystyle 2~\mathrm{dm}^3\), a piston is enclosing air at a pressure of \(\displaystyle 200~\mathrm{kPa}\). Both the cylinder and the piston are made of a material with very poor thermal conductivity (e.g. glass). The piston is suddenly pulled and then locked when the volume of the enclosed air increased to \(\displaystyle 4~\mathrm{dm}^3\).
a) What is the minimum temperature to which the gas can cool from the initial \(\displaystyle 300~\mathrm{K}\)?
b) After a long time of waiting, the enclosed air warms up to the initial temperature. What is the maximum heat absorbed by the enclosed air during this time?
(4 pont)
solution (in Hungarian), statistics
P. 5666. The charge of the plates of a parallel plate capacitor is changed. Initially the voltage between the plates is \(\displaystyle U_0\). The charge of the positive plate is increased by a factor of three and the charge of the negative plate is halved. What will the voltage between the plates be?
(4 pont)
solution (in Hungarian), statistics
P. 5667. The figure shows an incandescent lamp between two concave mirrors. The mirror on the right produces a parallel beam of light, while the small mirror on the left prevents a significant part of the light from the bulb to escape from this mirror arrangement, which is similar to a car's reflector.
a) Explain why this reflector produces a stronger light with the small mirror on the left than without it.
b) Use a ruler to make measurements in the figure and find the ratio of the focal lengths of the two mirrors.
c) Based on your measurements with the ruler, estimate what percentage of the light from the incandescent lamp is reflected in the parallel beam produced by the reflector without the small mirror, or with the small mirror.
(5 pont)
solution (in Hungarian), statistics
P. 5668. Deuterium is an isotope of hydrogen, with a nucleus consisting of a proton and a neutron. Compare the mass of a deuterium atom in kilograms with the sum of the masses of a proton, a neutron and an electron. What is the explanation for the difference? To make their calculations easier, nuclear physicists express this difference in the form of \(\displaystyle 2.2~\mathrm{MeV}/c^2\). Prove that this is the same as the difference you calculated! The appropriate data can be found in the table https://www.komal.hu/cikkek/atomtomegek.pdf.
(3 pont)
solution (in Hungarian), statistics
P. 5669. One end of a thin thread of length \(\displaystyle L=20~\mathrm{cm}\) is attached to an axle of radius \(\displaystyle r=0.5~\mathrm{cm}\), and \(\displaystyle L/2\) length of the thread is wound on the axle. The axle is attached to a disc of radius \(\displaystyle R=5~\mathrm{cm}\), of mass \(\displaystyle m=0.5~\mathrm{kg}\) and of uniform mass distribution (see the figure). Keeping the other end of the vertical thread in a fixed position, the disc is released.
a) What is the tension in the thread during the motion of the uniformly accelerating disc (``yoyo'')?
b) What is the speed of the axle of the disc at the moment the thread gets unwound?
c) When the vertical movement of the disc is reversed, the tension in the thread increases for a short time (the disc ``jerks'' the thread). Estimate the average value of the tension in the thread during the jerk. The mass of the axle, the deviation of the thread from the vertical and air resistance can be neglected. Assume that the angular speed of the disc during the ``turn'' is constant.
(6 pont)
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