KöMaL Problems in Physics, October 2024
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Problems with sign 'M'Deadline expired on November 15, 2024. |
M. 434. A one-and-a-half litre thick-walled PET bottle is placed on a slope. At a certain angle of inclination (if friction is big enough) the bottle will topple over. Measure how this angle depends on the amount of water in the bottle.
(6 pont)
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Problems with sign 'G'Deadline expired on November 15, 2024. |
G. 861. Anna took her dog Buddy for a walk. Anna was walking on the avenue at a speed of \(\displaystyle 0.8~\mathrm{m}/\mathrm{s}\) when she dropped her house key near a linden tree. At this instant Buddy was at the litter bin, 4 m away from the linden tree, and he was trotting at a speed of \(\displaystyle 1.2~\mathrm{m}/\mathrm{s}\). When Buddy reached the lamppost, \(\displaystyle 3.6\) m away from the litter bin, he turned back and began to trot back and forth between the litter bin and the lamppost at the same constant speed of \(\displaystyle 1.2~\mathrm{m}/\mathrm{s}\). When Anna reached the litter bin, she noticed that she had dropped the key, so she turned back to get it. She was moving at \(\displaystyle 0.8~\mathrm{m}/\mathrm{s}\) the whole time. How did the length of the retractable leash, which was kept taut, change from the time she dropped the key to the time she picked it up? Plot a graph.
(4 pont)
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G. 862. A pulley system consists of \(\displaystyle n\) pulleys. Regardless of their size each moving pulley has the same weight of \(\displaystyle G\), and the rope can be considered ideal. How much weight should be hung on the end of the rope so that the system is in equilibrium if the pulley system
\(\displaystyle a)\) is not loaded, and arranged as shown in figure \(\displaystyle a)\);
\(\displaystyle b)\) is loaded with a pulley of weight \(\displaystyle G\) and the pulleys are arranged as shown in figure \(\displaystyle b)\)?
(4 pont)
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G. 863. A solid iron ball is floating in mercury. What percentage of its volume is submerged in the mercury? How does this percentage change if water is added on top of the mercury such that it completely covers the iron ball?
(3 pont)
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G. 864. In the circuit shown in the schematic diagram, a power source supplies \(\displaystyle U=24~\mathrm{V}\) DC voltage, the voltmeter reads \(\displaystyle 10~\mathrm{V}\), ammeter \(\displaystyle \mathrm{A}_1\) reads \(\displaystyle 0.2~\mathrm{A}\), and ammeter \(\displaystyle \mathrm{A}_2\) reads \(\displaystyle 0.7~\mathrm{A}\).
\(\displaystyle a)\) Determine the resistance of each resistor.
\(\displaystyle b)\) How much heat is dissipated in the circuit during 2 minutes?
All three meters can be considered ideal, and the resistance of the connecting wires and the internal resistance of the power source are negligible.
(4 pont)
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Problems with sign 'P'Deadline expired on November 15, 2024. |
P. 5589. A car travelling on a straight track with constant acceleration covers the first part of the path, which has a length of \(\displaystyle s_1=25~\mathrm{m}\) in \(\displaystyle t_1=2~\mathrm{s}\) and then it covers the second, \(\displaystyle s_2=15~\mathrm{m}\) long part of the path in \(\displaystyle t_2=3~\mathrm{s}\).
\(\displaystyle a)\) What is the acceleration of the car?
\(\displaystyle b)\) What is the speed of the car at the beginning of the first part of the path and at the end of the second part of the path?
(4 pont)
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P. 5590. We want to make a ping-pong ball bounce uniformly with the help of a motor in the following way: a piston is moved along a vertical axis with an amplitude of 3 cm, and the ball always hits the piston in the equilibrium position once in a period. What should the frequency of the motor be in order to achieve this process? The coefficient of restitution is \(\displaystyle k=0.8\).
(4 pont)
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P. 5591. We make three different compound pendulums (physical pendulums).
\(\displaystyle a)\) A thin rod with uniform mass distribution is bent into a circle of radius \(\displaystyle R\), and is supported by a wedge at a point on the inner rim of the circle, see the figure. The circular rod is free to rotate about the wedge in its plane.
\(\displaystyle b)\) A semicircle is cut from a similarly bent rod forming a circle of the same radius. This semicircular rod is supported at the midpoint of its circumference.
\(\displaystyle c)\) The same procedure is followed as in the previous case, but only a relatively short arc, an octant of the circular arc is placed at its midpoint to the wedge.
All three pendulums are slightly displaced and the periods of their swings are measured. Which of these periods will be the greatest and which will be the smallest?
(5 pont)
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P. 5592. The suspension points of a weightless inextensible rope are located at a horizontal distance of \(\displaystyle L\) and a vertical distance of \(\displaystyle H\) from each other as shown in the figure. On a cold winter day, snow of density \(\displaystyle \varrho\) accumulated on the rope in width of \(\displaystyle d\). The height of the snow layer varies linearly as a function of the horizontal coordinate, between zero and \(\displaystyle h_\textrm{max}\). At the equilibrium position, the tangent of the rope at its right-hand end is exactly horizontal. Determine the minimum and maximum tension in the rope.
(5 pont)
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P. 5593. A heat engine is operated with a sample of helium, which has a constant amount. The pressure-density graph of the different parts of the cyclical process is shown in the figure. The temperature of the gas in its initial state (1) is 400 K, and during the process between states (2) and (3) the product of the pressure and the density of the gas is constant.
\(\displaystyle a)\) What is the temperature of the gas at state (2) and at state (3)?
\(\displaystyle b)\) What is the efficiency of this heat engine?
(5 pont)
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P. 5594. There are two neutral metal disks of radius \(\displaystyle R\) between the plates of a parallel plate capacitor having a large area. The disks are parallel to the plates of the capacitor, and they are connected with several thin, unstretched metal threads. The disk at the top is fixed to the top plate of the capacitor with thin insulating rods. Each disk is at a distance of \(\displaystyle d\ll R\) from the capacitor plate which is closer to it. (See the figure.)
How much does the total tension in the metal threads change when a voltage of \(\displaystyle U\) is applied to the capacitor?
Data: \(\displaystyle R=10~\mathrm{cm}\), \(\displaystyle d=0.5~\mathrm{cm}\), \(\displaystyle U=5000~\mathrm{V}\).
(5 pont)
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P. 5595. Two concave mirrors with small aperture angles and with focal length of \(\displaystyle f\) are placed opposite to each other so that their principal axes coincide and their distance from each other is \(\displaystyle 2f\) (see figure).
A point-like light source is placed at a point \(\displaystyle T\) on the principal axis. Where is the position of point \(\displaystyle T\) if the emitted light rays meet at point \(\displaystyle T\) after they are reflected from the two mirrors?
(4 pont)
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P. 5596. If the radius of a spherical object, having uniform mass density, is less than a critical value, then gravity is so strong on its surface that even light cannot escape from it, so it behaves like a black hole. Let us estimate this critical radius, knowing that its value depends only on the mass of the object, the universal gravitational constant and the speed of light in vacuum. Estimate how much a bowling ball of mass \(\displaystyle 7.25~\mathrm{kg}\) should be compressed, in order that it behaves like a black hole!
(4 pont)
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P. 5597. Three alike stars of mass \(\displaystyle m\) form an equilateral triangle at any instant. At a given instant of time, the triangle has a side length \(\displaystyle L_0\), the speed of all three stars is \(\displaystyle v_0\), and the direction of each velocity vector is tangent to the circumscribed circle of the triangle formed by the stars. Determine the period of the pulsating ``triple star'' system.
(6 pont)
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