KöMaL Problems in Physics, November 2023

Problems with sign 'M' Deadline expired on December 15, 2023.

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M. 426. Measure the viscosity of three different materials found in your household without using a viscosity meter. For example: cooking oil, honey, washing-up liquid, motor oil, shower gel, etc.

(6 pont)

solution (in Hungarian), statistics

Problems with sign 'G' Deadline expired on December 15, 2023.

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G. 829. According to Kepler's second law, the ray from the Sun to the planet will sweep out equal areas during equal time intervals. Determine how many \(\displaystyle \mathrm{km}^2\) the ray from the Sun to the Earth sweeps in each second.

(3 pont)

solution (in Hungarian), statistics

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G. 830. The width of the material of the base of a thin-walled cylinder-shaped glass jar with a radius of 5 cm is twice as large as the width of its vertical wall. What is the maximum height of the jar if it does not tip over when placed on its base on a slope with an elevation angle of \(\displaystyle 30^\circ\)?

(The friction is so large that the pot does not slip on the slope.)

(4 pont)

solution (in Hungarian), statistics

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G. 831. On a boat trip we bought vanilla ice cream, which is sold in a thin-walled, roughly cuboid shape ice cream container. The product's label states that it contains 1250 g ice cream, which has a net volume of 2500 ml. The empty box weighs 81 g and when filled to the top with water, it weighs 2738 g.

The sealed, unopened ice cream container is dropped into the water with the top facing up. Estimate what fraction of its height is submerged in the water, and up to what fraction of its height it is filled with ice cream.

(4 pont)

solution (in Hungarian), statistics

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G. 832. Two walls of a room with an equilateral triangle-shaped floor are covered with flat mirrors (see the figure). A lamp stands in the centre of the room. How many images does the lamp produce?

(4 pont)

solution (in Hungarian), statistics

Problems with sign 'P' Deadline expired on December 15, 2023.

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P. 5517. The coefficient of friction on the upper part of a slope of length \(\displaystyle \ell_1\) is \(\displaystyle \mu_1\), whilst on the lower part of the slope, having a length of \(\displaystyle \ell_2\) is \(\displaystyle \mu_2\). A small body with zero initial velocity starts at the bottom of a slope and stops just at the bottom. What is the angle of inclination of the slope?

Data: \(\displaystyle \ell_1=20\) cm, \(\displaystyle \ell_2=40\) cm, \(\displaystyle \mu_1=0.1\) and \(\displaystyle \mu_2=0.2\).

(4 pont)

solution (in Hungarian), statistics

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P. 5518. A cylindrical body of radius \(\displaystyle R=20~\) cm is fixed in a horizontal position. A piece of light and flexible thread of length \(\displaystyle \ell\) was laid on its slippery surface, as shown in the figure.

To one end of the thread a point-like body of mass \(\displaystyle m\), whilst to the other end another point-like body of mass \(\displaystyle 2m\) were attached. What can the greatest value of \(\displaystyle \ell\) be, if the system remains at rest?

(4 pont)

solution (in Hungarian), statistics

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P. 5519. Dionysus placed \(\displaystyle 2\) identical cylinder-shaped barrels, having the same height \(\displaystyle h\), on the horizontal ground, one on the top of the other (see the figure). Both were nearly fully filled with wine. Heracles' thirteenth task was to drill a hole perpendicularly to the wall of each of the barrels at the same \(\displaystyle xh\) height measured from the bottom of each barrel.

How should Heracles choose the value of the dimensionless factor \(\displaystyle x\), in order that the impact points of the wine rays fall as far apart as possible?

(5 pont)

solution (in Hungarian), statistics

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P. 5520. A horizontal cylinder of length \(\displaystyle 2\ell\) is closed at its both ends, and is divided into two equal parts by a thin piston. Each part contains air with a temperature of \(\displaystyle 100\;{}^\circ\)C and a pressure of 100 kPa. Into one part, enough water is ejected to produce a saturated vapour while the temperature is kept at \(\displaystyle 100\;{}^\circ\)C.

How much will the piston move and what will be the pressure in both parts?

(4 pont)

solution (in Hungarian), statistics

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P. 5521. A hob has 12 identical resistance wires, which are connected as shown in the figure.

What is the percentage distribution of the dissipated power in each of the wires of the hotplate when voltage is applied across points \(\displaystyle A\) and \(\displaystyle B\)?

(4 pont)

solution (in Hungarian), statistics

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P. 5522. Two horizontal, parallel, frictionless insulating rods are spaced \(\displaystyle d\) apart. On the lower rod a small insulating bead of charge \(\displaystyle q\) and of mass \(\displaystyle m\) slides, and on the upper rod another small insulating bead of charge \(\displaystyle -q\) and of mass \(\displaystyle m\) slides, as shown in the figure.

Initially, the beads are spaced far apart and move towards each other with velocities of \(\displaystyle u\) and \(\displaystyle v\), respectively. What will the maximum velocity of each bead be during the motion?

(5 pont)

solution (in Hungarian), statistics

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P. 5523. Three adjacent walls of a regular hexagon-shaped room are covered with flat mirrors (see the figure). A lamp is lit in the middle of the room. How many images does the lamp produce?

(5 pont)

solution (in Hungarian), statistics

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P. 5524. In a closed capillary tube, there are 2 mercury columns between which there is a drop of \(\displaystyle \mathrm{HgI}_2\) (mercury iodide) electrolyte in aqueous solution. The inner diameter of the tube is 0.3 mm. The capillary tube is connected in series with a \(\displaystyle R=390\) k\(\displaystyle \Omega\) resistor and a \(\displaystyle U=10\) V battery through electrodes which are in contact with the mercury. How long does it take for the solution drop to move 1 centimetre? In which direction does this displacement occur?

(4 pont)

solution (in Hungarian), statistics

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P. 5525. In a long cylindrical metal wire of radius \(\displaystyle r\) and of resistivity \(\displaystyle \varrho\), a current of strength \(\displaystyle I\) flows in a uniform distribution. The wire has a constant surface temperature \(\displaystyle T_0\). Determine the temperature of the wire on its axis of symmetry if it is known that the metal has a coefficient of thermal conductivity of \(\displaystyle \lambda\).

(6 pont)

solution (in Hungarian), statistics