KöMaL Problems in Physics, May 2024
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Problems with sign 'M'Deadline expired on June 17, 2024. |
M. 432. Make an ``hourglass" from semolina and two PET bottles with their mouths turned towards each other. Insert various reducers between the bottles to vary the diameter of the circular opening \(\displaystyle d\). Measure the time \(\displaystyle T\) for the semolina to flow down as a function of the diameter \(\displaystyle d\). The theoretical expectation is \(\displaystyle T\sim d^\gamma\). What exponent \(\displaystyle \gamma\) results from the measurement?
(6 pont)
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Problems with sign 'G'Deadline expired on June 17, 2024. |
G. 853. A police car is driving on the motorway. Its speed was recorded by the computer system of the car, and the following graph \(\displaystyle v_\mathrm{r}(t)\) was produced. Suddenly a motorcyclist overtakes the police car, the police car measures the speed of the motorcyclist. The graph \(\displaystyle u_\mathrm{m}(t)\) shows the speed of the motorcyclist with respect to the police car.
\(\displaystyle a)\) Plot the graph \(\displaystyle v_\mathrm{m}(t)\), which is the speed of the motorcyclist relative to the ground.
\(\displaystyle b)\) During the time period shown in the graphs, when were the two vehicles the furthest from each other? What was this greatest distance?
\(\displaystyle c)\) After the 60th second, they continue at constant speed. When will the police car overtake the motorcyclist?
(4 pont)
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G. 854. A spherical balloon is filled with some gas, whose density is smaller than that of the air. In order to make the balloon float at a certain height in the room, paper clips are hung on the free end of the rope, which is tied to the balloon. The floating was not achieved because the balloon rose if 7 paper clips were hung on the end of the rope, but descended if 8 were hung on the end of the rope. What is the density of the gas in the balloon? The density of the air in the room is \(\displaystyle 1.20~\tfrac{\mathrm{kg}}{\mathrm{m}^3}\). The balloon has a diameter of \(\displaystyle 26~\mathrm{cm}\) and a mass of \(\displaystyle 3~\mathrm{g}\) (before it was inflated). The mass of the rope is \(\displaystyle 2~\mathrm{g}\), the mass of one paper clip is \(\displaystyle 0.6~\mathrm{g}\). The thickness of the material of the balloon, the volume of the rope and the paper clips can be ignored.
(4 pont)
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G. 855. Newton did not write the mirror and lens equation in the usual form \(\displaystyle \left(\tfrac{1}{t}+\tfrac{1}{k}=\tfrac{1}{f}\right)\). He measured the object distance \(\displaystyle x_\mathrm{t}\) from the focus on the side of the object, and the image distance \(\displaystyle x_\mathrm{k}\) from the focus on the image side as shown in the figure.
\(\displaystyle a)\) Determine the lens equation with these parameters in its simplest form.
\(\displaystyle b)\) How can this formula be used for a diverging lens?
(4 pont)
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G. 856. In Budapest during a full Moon in December, or during a full Moon in June is the altitude belonging to the culmination of the Moon higher?
(4 pont)
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Problems with sign 'P'Deadline expired on June 17, 2024. |
P. 5571. A small ball is dropped from a height of \(\displaystyle h\) above the corner of a table of height also \(\displaystyle h\). At most how far from the corner of the table can the ball hit the ground? Consider the impact to be perfectly elastic.
(5 pont)
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P. 5572. Alistair, whose mass is 60 kg, attempts a bungee jump from a 50 m high bridge (measured from the water level). The length of the rope is adjusted so that Alistair just touches the surface of the water during the jump. Thus, his very light but sufficiently flexible rope has a spring constant of 72 N/m. Benedict, whose mass 80 kg, will attempt a similar jump. His rope is of the same quality as Alistair's, but shorter.
\(\displaystyle a)\) How long are Alistair's and Benedict's bungee ropes? What is the spring constant of Benedict's rope?
\(\displaystyle b)\) What is the maximum acceleration of Alistair and Benedict?
\(\displaystyle c)\) Which of them reaches the surface of the water faster if they jump off the bridge at the same time?
We can assume that the elongation of the ropes is directly proportional to the stretching force. Ignore the height of the jumpers and air resistance.
(5 pont)
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P. 5573. A uniform-density cylinder was placed on a slope with an angle of inclination of \(\displaystyle 30^\circ\), and tied to the slope at its highest point above the centre of its mass with a horizontal thread as shown in the figure. What should be the least value of the coefficient of friction between the cylinder and the slope in order that the cylinder remain at rest in this position?
(4 pont)
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P. 5574. Pistons encloses samples of nitrogen gas in a vertical, insulated cylinder sealed at its bottom. Initially the samples have the same volume and the same temperature of \(\displaystyle T_1=300~\mathrm{K}\). The upper piston can move frictionlessly, has a mass of \(\displaystyle {m=40}~\mathrm{kg}\), and is thermally insulated, the lower one is made of good thermal conducting material and is fixed. The external air pressure is \(\displaystyle 10^5~\mathrm{Pa}\), the pressure of the gas at the bottom part is \(\displaystyle 1.2\cdot 10^5~\mathrm{Pa}\). The area of the cross section of the cylinder is \(\displaystyle {A=1}~\mathrm{dm^2}\), \(\displaystyle h=0.5~\mathrm{m}\).
\(\displaystyle a)\) What are the masses of the samples of nitrogen in the bottom and in the top parts of the cylinder?
An electric heating element gives \(\displaystyle Q=1580~\mathrm{J}\) thermal energy to the system.
\(\displaystyle b)\) By how much does the internal energy of the gases in the top and bottom parts change?
\(\displaystyle c)\) How much does the volume of the upper gas change?
(4 pont)
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P. 5575. A small electric dipole is placed above a large grounded metal plate at a height of \(\displaystyle h\). This is done in such a way that its dipole moment \(\displaystyle \boldsymbol{p}\) points upwards as shown in the figure.
Determine the positions of the points on the metal sheet where the surface charge density is zero.
(5 pont)
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P. 5576. Suppose that we have measuring instruments that measure not only the RMS value of sinusoidal AC current or AC voltage, but also that of any periodic signal.
\(\displaystyle a)\) Connect an ideal inductor with self-inductance \(\displaystyle L\) to a voltage generator with frequency \(\displaystyle f=1/T\), which provides a symmetrical square-wave signal. Using ideal voltmeter and ammeter, measure the RMS value of the current through the inductor and the RMS value of the voltage across the voltage generator. The first figure shows the square waveform, and we know that at time \(\displaystyle t=0\) the current through the inductor is zero.
\(\displaystyle b)\) Connect an ideal capacitor with capacitance \(\displaystyle C\) to a voltage generator with frequency \(\displaystyle f=1/T\), which provides a symmetrical square-wave signal. Using ideal voltmeter and ammeter, measure the RMS value of the current through the capacitor and the RMS value of the voltage across the voltage generator. The second figure shows the square waveform, and we know that at time \(\displaystyle t=0\) the voltage across the capacitor is zero.
What are the readings of the meters for the connection in part \(\displaystyle a)\) and in part \(\displaystyle b)\)? Give the answers in terms of \(\displaystyle I_\mathrm{max}\), \(\displaystyle U_\mathrm{max}\), \(\displaystyle L\), \(\displaystyle C\) and \(\displaystyle f\).
(5 pont)
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P. 5577. On the left side of a converging lens, a point-like object was placed, from which two rays were plotted after passing through the lens. One of the rays just passes through the focus of the lens. Using the figure as a guide, construct the position of the light source (using a pair of compasses and a ruler). Describe the steps of construction.
(4 pont)
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P. 5578. Due to the increased solar activity expected to peak in 2024, a very spectacular auroral display was observed in Hungary on the evening of 5 November 2023. However, unlike the regular auroral display in the Arctic, it was mainly reddish instead of the dominant green. Assuming visible light, what is the maximum number of collisions of a single electron, arriving with the solar wind at a speed of 2000 km/s, which causes the emission of light?
(3 pont)
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P. 5579. In Helsinki, the sun rises in just a bit more than 6.5 minutes on Christmas Day. At which day of the year will the time of the sunrise be the shortest in the same place? How long is the fastest sunrise in Helsinki?
Helsinki lies on a plain, at 60 degrees north latitude. The apparent diameter of the sun is about half a degree. Ignore the small variations due to the eccentricity of the Earth's orbit and the effect of the atmosphere in this exercise.
(6 pont)
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