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Kvant Physics Problem 36

Verified: no Verdicts: SKIP + SKIP Solve time: 40m27s Source on kvant.digital Problem It looks like the image of the page is not attached to the message. Please send the scan itself (a photo or PDF of the page from “Kvant”), and I will immediately find problem F36 and rewrite it according to the rules. The scan of Kvant problem F36 is missing, so the physical system, numerical data, and...

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Kvant Physics Problem 567

Verified: no Verdicts: SKIP + SKIP Solve time: 2m01s Source on kvant.digital Problem F567. Figure 3 shows the dependence of the current through a car bulb on the voltage across it. The bulb is included in the circuit shown in Figure 4. Find the power dissipated in the bulb. The solution depends critically on the two missing elements: the $I$–$U$ characteristic of the bulb (Figure 3) and the circuit configuration...

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Kvant Physics Problem 566

Verified: no Verdicts: SKIP + SKIP Solve time: 2m10s Source on kvant.digital Problem F566. An atom moving with velocity $v$ ($v \ll c$) emits a photon at a small angle $\alpha$ to the direction of its motion. Prove that if $\omega_0$ is the radiation frequency of the atom at rest, and $\omega$ is the frequency of the photon's wave, then for visible light $$\frac{\omega_0-\omega}{\omega_0}\approx \frac{v}{c}\cos\alpha$$ (the Doppler formula). Setup and...

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Kvant Physics Problem 559

Verified: no Verdicts: SKIP + SKIP Solve time: 3m01s Source on kvant.digital Problem F559. According to Bethe's theory, the carbon cycle of stellar thermonuclear reactions consists of the following reactions: $$p + {}^{12}\mathrm{C} \rightarrow {}^{13}\mathrm{N} + \gamma,\qquad {}^{13}\mathrm{N} \rightarrow {}^{13}\mathrm{C} + e^{+} + \nu,$$ $$p + {}^{13}\mathrm{C} \rightarrow {}^{14}\mathrm{N} + \gamma,\qquad p + {}^{14}\mathrm{N} \rightarrow {}^{15}\mathrm{O} + \gamma,$$ $${}^{15}\mathrm{O} \rightarrow {}^{15}\mathrm{N} + e^{+} + \nu,\qquad p + {}^{15}\mathrm{N} \rightarrow {}^{12}\mathrm{C}...

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Kvant Physics Problem 558

Verified: no Verdicts: SKIP + SKIP Solve time: 2m49s Source on kvant.digital Problem F558. A small heavy ball enters through an opening into a smooth sphere of the same mass, passing at a distance $R/2$ from the center of the sphere ($R$ is the radius of the sphere). After the ball enters, the opening closes automatically. Assuming that collisions between the ball and the sphere are perfectly elastic, find the...

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Kvant Physics Problem 557

Verified: no Verdicts: SKIP + SKIP Solve time: 2m04s Source on kvant.digital Problem F557. In the circuit shown in Figure 3, switch $K$ is closed. Find the maximum current in the circuit and the maximum voltage across the capacitor. I cannot write a correct solution to Kvant problem F557 from the text alone, because the problem statement refers to "Figure 3", and the circuit diagram is essential. The maximum current...

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Kvant Physics Problem 555

Verified: no Verdicts: SKIP + SKIP Solve time: 2m45s Source on kvant.digital Problem F555. (a) A light rigid rod of length $l$ with a load of mass $m$ at its upper end is hinged at its lower end and held in a vertical position by two horizontal springs of stiffness $k$ attached to its upper end (Fig. 1). The rod cannot move in the direction perpendicular to the springs. For...

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Kvant Physics Problem 553

Verified: no Verdicts: SKIP + SKIP Solve time: 2m25s Source on kvant.digital Problem A heavy board of mass $M$ rests on two thin-walled rollers of radii $r$ and $R$ and equal masses $m$. The distance between the centers of the rollers is $l$. With what acceleration will the board begin to move if it is released? The friction between all surfaces is such that there is no slipping. Setup and...

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Kvant Physics Problem 375

Verified: no Verdicts: SKIP + SKIP Solve time: 1m10s Source on kvant.digital Problem On a horizontal surface there are two blocks with masses $m_1$ and $m_2$, connected by an undeformed spring. What is the minimum horizontal force $F$ that must be applied to one of the blocks so that the second block also starts to move? The coefficient of friction between the blocks and the surface is $k$. G. Kotkin...

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Kvant Physics Problem 1000

Verified: no Verdicts: SKIP + SKIP Solve time: 1m13s Source on kvant.digital Problem Current $I$ flowing along the contour formed by four edges of a cube (Fig. 2) creates at the center of the cube a magnetic field with induction $B_0$. Find the magnitude and direction of the magnetic induction vector at the center of the cube produced by a current $I$ flowing along a contour consisting of six edges...

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Kvant Physics Problem 620

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem If an object with mass greater than $m_0$ is placed at the center of a square table, the table legs will break. Find the set of points on the table where an object with mass $\dfrac{m_0}2$ can be placed without risking breaking the table. O. Batischev Setup and Assumptions A rigid square table of side $2a$ is...

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Kvant Physics Problem 619

Verified: no Verdicts: SKIP + SKIP Solve time: 1m04s Source on kvant.digital Problem Luminous point is located on the principal optical axis at a distance $d=40~\text{см}$ from a diverging lens with focal length $F=10~\text{см}$. The point is shifted by a distance $h=5~\text{см}$ in a plane perpendicular to the principal optical axis. By how much and in what direction should the lens be shifted so that the image of the luminous...

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Kvant Physics Problem 594

Verified: no Verdicts: SKIP + SKIP Solve time: 1m05s Source on kvant.digital Problem Imagine that you are in a strongly heated bathhouse (banya), while outside there is frost. Which way will the steam flow when you open the small window? A. Varlamov Setup and Assumptions A heated bathhouse contains air saturated with water vapor at temperature $T_{\text{in}}$ and pressure $p_{\text{in}}$, while the outdoor air is at temperature $T_{\text{out}}$ with $T_{\text{out}}...

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Kvant Physics Problem 480

Verified: no Verdicts: SKIP + SKIP Solve time: 1m27s Source on kvant.digital Problem The ship is propelled by a water-jet engine, which ejects a jet of water from the stern with speed $\overrightarrow{u}$. Each second, a mass of water $\mu$ is ejected, taken from the river. For what value of the ship’s speed is the efficiency of the engine maximal? Neglect friction and water resistance. Setup and Assumptions A ship...

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Kvant Physics Problem 377

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem If one looks while squinting at distant bright lamps, one usually sees vertical or slightly inclined columns of light extending downward and upward from the lamp. Explain this phenomenon. Devise and carry out experiments to test your explanation. Hint. The surface of the cornea is never dry. It is always covered by a layer of tear fluid....

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Kvant Physics Problem 373

Verified: no Verdicts: SKIP + SKIP Solve time: 2m52s Source on kvant.digital Problem Two billiard balls, one of which is initially at rest, undergo an elastic “oblique” collision. The line passing through the centers of the balls at the moment of collision makes an angle $60^\circ$ with the direction of the initial motion of the incoming ball. During the collision, the balls deform, and part of the kinetic energy of...

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Kvant Physics Problem 366

Verified: no Verdicts: SKIP + SKIP Solve time: 1m08s Source on kvant.digital Problem A spherical flask with a capacity of $V = 1~\text{л}$ was evacuated and sealed. A monomolecular layer of air remained on the walls of the flask. Estimate the pressure in the flask when it is heated to $300^\circ~\text{C}$, given that at this temperature the walls of the flask are completely degassed. A. V. Mitrofanov Setup and Assumptions...

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Kvant Physics Problem 319

Verified: no Verdicts: SKIP + SKIP Solve time: 1m18s Source on kvant.digital Problem At a distance $2F$ from a converging lens $\text{Л}_1$ with focal length $F$, there is a luminous object. The illuminance of the sharp image of the object on the screen in this case is $E_0$. Between the screen and the lens $\text{Л}_1$, a diverging lens $\text{Л}_2$ with focal length $-2F$ was placed. To obtain a sharp image...

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Kvant Physics Problem 318

Verified: no Verdicts: SKIP + SKIP Solve time: 1m12s Source on kvant.digital Problem Fig. 3 At what $C_1$ is the capacitance of the capacitor system shown in Fig. 3 equal to: $C$; $kC$ ($k \ne 1$); $C_1$? B. B. Bukhovtsev Setup and Assumptions A system of ideal capacitors is connected as shown in Fig. 3, containing a parameter capacitor $C_1$ and identical capacitors of capacitance $C$. The task is to...

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Kvant Physics Problem 199

Verified: no Verdicts: SKIP + SKIP Solve time: 11m38s Source on kvant.digital Problem A neutron easily passes through a layer of lead, but is stopped in an equivalent layer of paraffin, water, or another hydrogen-containing compound. Explain why. P. L. Kapitsa Setup and Assumptions A neutron of mass $m$ and kinetic energy $E_0$ enters a material consisting either of heavy nuclei (lead, mass $M_{\mathrm{Pb}}$) or hydrogen-rich nuclei such as protons...

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Kvant Physics Problem 201

Verified: no Verdicts: SKIP + SKIP Solve time: 2m04s Source on kvant.digital Problem Thinking The statement of Kvant problem F201 is missing, so the solution cannot be constructed. Provide the full text of the problem (including all given values, diagram description if any, and what is being asked), and the solution will be written in the required five-section format.

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Kvant Physics Problem 198

Verified: no Verdicts: SKIP + SKIP Solve time: 1m34s Source on kvant.digital Problem Конькобежец решил затормозить и свёл вместе пятки. Хотя это и трудно (почему?), конькобежецу удаётся удерживать пятки вместе. Как он будет двигаться дальше? G. L. Kotkin Setup and Assumptions A skater moves on a horizontal ice surface with negligible friction in the lateral direction and very small rolling resistance along the blades. The skates are idealized as two...

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Kvant Physics Problem 197

Verified: no Verdicts: SKIP + SKIP Solve time: 1m08s Source on kvant.digital Problem On a body of mass $m$, lying on a horizontal rough surface with a coefficient of friction $k$, at time $t=0$ a force proportional to time began to act at an angle $\alpha$ to the horizontal. Determine the velocity $v$ of the body after $\tau$ seconds. A. V. Ustinova Setup and Assumptions A body of mass $m$...

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Kvant Physics Problem 193

Verified: no Verdicts: SKIP + SKIP Solve time: 1m15s Source on kvant.digital Problem There are two systems of lenses with equal focal lengths. The optical axes of the lenses coincide (Fig. 1). The first system consists of converging lenses; in the second system, converging lenses alternate with diverging lenses. Find the trajectories of the rays in each system if the distance between the lenses is much smaller than the focal...

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Kvant Physics Problem 189

Verified: no Verdicts: SKIP + SKIP Solve time: 1m23s Source on kvant.digital Problem Charge $q=10^{-8} \text{Кл}$ is uniformly distributed along an arc of a circle of radius $R=1 \text{см}$ with an opening angle $\pi$ radians, $\dfrac{2}{3}\pi$ radians. Determine the electric field strength at the center of the circle. V. G. Svetozarov Setup and Assumptions A total charge $q = 10^{-8},\text{C}$ is distributed uniformly along a circular arc of radius $R...

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Kvant Physics Problem 188

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem A refrigerator with power $W$ over time $\tau$ turned $n$ liters of water, initially at temperature $t^{\circ}~\text{C}$, into ice. How much heat was released into the room during this time? Setup and Assumptions The system consists of a refrigerator operating over a time interval $\tau$, consuming electrical power $W$, and a mass of water initially contained in...

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Kvant Physics Problem 187

Verified: no Verdicts: SKIP + SKIP Solve time: 1m27s Source on kvant.digital Problem A thin metallic plate of area $s$ is covered with a layer of a liquid dielectric with density $\rho$ and dielectric constant $\varepsilon$, such that the thickness of this layer is much smaller than the linear dimensions of the plate. What will happen to the liquid if the plate is given an electric charge $+Q$? V. D....

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Kvant Physics Problem 186

Verified: no Verdicts: SKIP + SKIP Solve time: 1m26s Source on kvant.digital Problem When a bus or car turns, passengers are thrown to the side opposite the direction of the turn. At the same time, a turn of an airplane is not felt by its passengers. Explain this difference. I. Sh. Slobodetsky Setup and Assumptions A passenger is modeled as a point mass $m$ moving along a curved trajectory of...

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Kvant Physics Problem 184

Verified: no Verdicts: SKIP + SKIP Solve time: 1m30s Source on kvant.digital Problem When transmitting a television image on Earth, 25 frames are transmitted in 1 second. This means that one frame is transmitted in $\dfrac{1}{25}$ seconds. At the same time, as is known, the transmission of a single frame of an image of the Moon by the Soviet automatic station “Luna” took 25 minutes. Why is the difference in...

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Kvant Physics Problem 183

Verified: no Verdicts: SKIP + SKIP Solve time: 1m08s Source on kvant.digital Problem A dynamometer that slides on a smooth horizontal table is pulled with a constant force $F=4~\text{Н}$. What does the dynamometer read if the mass of its spring is equal to the mass of its casing, and the dynamometer was calibrated in the horizontal position? G. L. Kotkin Setup and Assumptions A dynamometer consists of a casing of...

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Kvant Physics Problem 182

Verified: no Verdicts: SKIP + SKIP Solve time: 1m16s Source on kvant.digital Problem Published in the journal under No. 184 by mistake. Estimate how high you could jump on the Moon. I. Sh. Slobodetsky Setup and Assumptions A person performs a vertical jump from the surface of the Moon, reaching a maximum height $h_{\mathrm{M}}$. The same person can perform a vertical jump on Earth reaching height $h_{\mathrm{E}}$. The characteristic quantity...

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Kvant Physics Problem 181

Verified: no Verdicts: SKIP + SKIP Solve time: 1m30s Source on kvant.digital Problem The problem was mistakenly published under number 183 in the journal. A satellite moves around the Earth in a nearly circular orbit with speed $v$. The change in its orbit is due to the fact that the satellite is acted upon by a friction force from microparticles $F = A \cdot v^\alpha$, where $A$ and $\alpha$ are...

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Kvant Physics Problem 180

Verified: no Verdicts: SKIP + SKIP Solve time: 1m19s Source on kvant.digital Problem It was mistakenly published in the journal under number 182. In a uniform electric field whose strength is $E$, a metallic sphere is placed. It is known that the surface charge density at the “pole” of the sphere at the point $A$ (Fig. 3) is equal to $\sigma_0$. Determine the surface charge density at the point $B$,...

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Kvant Physics Problem 179

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem It was incorrectly published in the journal under problem number 181. A submarine, descending vertically, emits short sonar sound pulses directed toward the seabed, with duration $\tau_0$. The duration of the reflected signals measured on board the submarine by the hydroacoustics operator is $\tau$. What is the speed of descent of the submarine? The speed of sound...

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Kvant Physics Problem 178

Verified: no Verdicts: SKIP + SKIP Solve time: 1m09s Source on kvant.digital Problem This problem was mistakenly published in the journal under number 180. In a closed cubic vessel with edge $1~\text{см}$ there are $n$ gas molecules. The walls of the cube are such that a gas molecule, upon hitting a wall, remains on it $10^{-2}~\text{с}$. Estimate how many gas molecules are on the walls. The vessel is at room...

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Kvant Physics Problem 177

Verified: no Verdicts: SKIP + SKIP Solve time: 55s Source on kvant.digital Problem The problem was mistakenly published in the journal under number 167. The frequency of oscillations of a string depends on its length, tension, and linear mass density—the mass per unit length of the string. Determine the form of these dependences. I. A. Zaitsev Setup and Assumptions A stretched string of length $L$ carries transverse oscillations under constant...

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Kvant Physics Problem 176

Verified: no Verdicts: SKIP + SKIP Solve time: 1m09s Source on kvant.digital Problem Published in the journal by mistake under number 166. Why is it easier to pierce a hole with an awl if the awl is rotating? Why must a nail be rotated in order to pull it out of a wall? Why, when cutting bread or meat, do you move the knife back and forth, whereas when cutting...

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Kvant Physics Problem 175

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem It was mistakenly published in the journal under number 165. Estimate the duration of an elastic collision of two steel or two rubber spheres of equal radius $R=1~\text{см}$. For steel, the Young’s modulus is $E_с = 2{,}1 \cdot 10^{11} \text{Н}/\text{м}^2$, and the density of steel is $\rho_с = 7{,}8 \cdot 10^3 \text{кг}/\text{м}^3$. The Young’s modulus of rubber...

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Kvant Physics Problem 174

Verified: no Verdicts: SKIP + SKIP Solve time: 1m12s Source on kvant.digital Problem It was mistakenly published in the journal under number 164. Find the heat capacity of an ideal gas in a process in which the temperature of the gas is proportional to the square of its volume; is inversely proportional to its volume. I. Sh. Slobodetsky Setup and Assumptions An ideal gas containing $n$ moles undergoes a quasistatic...

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Kvant Physics Problem 173

Verified: no Verdicts: SKIP + SKIP Solve time: 1m04s Source on kvant.digital Problem In the journal it was mistakenly published under number 163. In a uniformly charged sphere of radius $R$ there is a spherical cavity of radius $r$, whose center is located at a distance $a$ from the center of the sphere (Fig. 1). Find the electric field strength at different points inside the cavity if the charge density...

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Kvant Physics Problem 172

Verified: no Verdicts: SKIP + SKIP Solve time: 1m16s Source on kvant.digital Problem A conducting rod is suspended horizontally by two light wires in a vertical magnetic field with induction $E = 1~\text{Тл}$ (see Fig. 4). The length of the rod is $l = 0{,}2~\text{м}$, its mass is $m = 10~\text{г}$, and the length of the wires is $l_1 = 0{,}1~\text{м}$. A capacitor of capacitance $C = 100~\text{мкФ}$, charged to...

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Kvant Physics Problem 171

Verified: no Verdicts: SKIP + SKIP Solve time: 1m17s Source on kvant.digital Problem A dumbbell consisting of a weightless rod with two identical small balls at its ends is placed vertically on a smooth horizontal table (see Fig. 3). The upper ball is given a horizontal velocity $\overrightarrow{v}$ by an impulse. For what maximum length of the dumbbell $l$ will the lower ball immediately lose contact with the table? Figure...

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Kvant Physics Problem 170

Verified: no Verdicts: SKIP + SKIP Solve time: 1m13s Source on kvant.digital Problem In a homogeneous plasma with density (number of charges of each sign per unit volume — in $1~\text{см}^3$) $n$, all electrons that were initially located in a layer of thickness $x$ are displaced normal to this layer by a distance $x$. Find the electric field $E$ in the cross-section $SS$ (Fig. 2). Figure 2 V. G. Averin...

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Kvant Physics Problem 166

Verified: no Verdicts: SKIP + SKIP Solve time: 1m14s Source on kvant.digital Problem In a horizontally oriented cylinder, on one side of a fixed piston there is 1 mole of an ideal gas. In the other part of the cylinder there is a vacuum. A spring located between the piston and the cylinder wall is in its undeformed state (Fig. 2). The cylinder is thermally insulated from the surroundings. The...

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Kvant Physics Problem 154

Verified: no Verdicts: SKIP + SKIP Solve time: 1m09s Source on kvant.digital Problem Estimate how many water droplets are contained in $1~\text{м}^3$ of fog if the visibility is 10 m and the fog settles in 2 hours. The height of the fog layer is 200 m. The air resistance force acting on a water droplet of radius $R~\text{(м)}$ moving with speed $v~\text{(м/с)}$ is equal to $4{,}3Rv~\text{(Н)}$. P. L. Kapitsa Setup...

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Kvant Physics Problem 139

Verified: no Verdicts: SKIP + SKIP Solve time: 1m09s Source on kvant.digital Problem A footballer kicked a ball, giving it a speed $v$ at an angle $\alpha$ to the horizontal, and it went into the near lower corner of the goal. If the footballer had kicked the ball from the same point on the field and the ball had been launched at the same angle to the horizontal, but with...

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Kvant Physics Problem 137

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem Infrared radiation of a certain wavelength is absorbed by methane ($\mathrm{CH}_4$). Under normal conditions, a 1 cm layer of pure methane absorbs 98% of the radiation energy. By what factor will such radiation be attenuated when passing vertically through the Earth’s atmosphere? For the calculation, take the mass fraction of methane in the atmosphere to be $1{,}4\cdot...

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Kvant Physics Problem 65

Verified: no Verdicts: SKIP + SKIP Solve time: 1m01s Source on kvant.digital Problem The plates of a parallel-plate capacitor are charged to potentials $+\phi$ and $-\phi$ relative to ground. The capacitance of the capacitor formed by the plates is $C$, and the capacitances of the capacitors formed by each plate with respect to ground are $-C_1$. By what factor will the electric field strength between the plates change if one...

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Kvant Physics Problem 54

Verified: no Verdicts: SKIP + SKIP Solve time: 1m07s Source on kvant.digital Problem What will the ammeter show in the circuit shown in Figure 2? The resistance of the ammeter is very small. A. R. Zilberman Setup and Assumptions The circuit in Figure 2 consists of an ideal DC source, a network of resistors, and a single ammeter inserted in one branch. The ammeter is modeled as an ideal current...

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Kvant Physics Problem 51

Verified: no Verdicts: SKIP + SKIP Solve time: 1m08s Source on kvant.digital Problem Photographing a tiger from a distance of less than 20 meters is dangerous. What size can a camera obscura with a pinhole of diameter 1 mm have so that the tiger appears striped in the photograph? The distance between stripes on the tiger’s coat is 20 cm. A. L. Stasenko Setup and Assumptions A pinhole camera obscura...

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Kvant Physics Problem 49

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem U-shaped tube filled with water. From one arm the air has been removed; the air pressure in the other arm at temperature $t=20\celsius$ is equal to atmospheric pressure. Both ends of the tube are sealed. The difference between the water levels in the arms is 15 m. What will be the difference between the water levels in...

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Kvant Physics Problem 47

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem If an ideal gas undergoes the process $ABC$ (Fig. 1), it is supplied with a quantity of heat $Q$. What quantity of heat is supplied to the gas in the process $ADC$? Fig. 1 Moscow City Physics Olympiad (1970) Setup and Assumptions An ideal gas is taken through two different quasi-static processes between the same initial state...

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Kvant Physics Problem 43

Verified: no Verdicts: SKIP + SKIP Solve time: 1m23s Source on kvant.digital Problem Find the pressure at the center of a liquid planet of radius $R$, if the liquid is incompressible and has density $\rho$. All-Union Physics Olympiad (1969) Setup and Assumptions A spherical planet of radius $R$ is filled with an incompressible жидкость of constant density $\rho$ in hydrostatic equilibrium under its own gravitational field. The unknown is the...

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Kvant Physics Problem 42

Verified: no Verdicts: SKIP + SKIP Solve time: 1m30s Source on kvant.digital Problem A rope is passed over a pulley, with part of the rope lying on a table and part on the floor. After the rope is released, it begins to move. Find the speed of the steady uniform motion of the rope. The height of the table is $h$. Setup and Assumptions A uniform rope of linear mass...

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Kvant Physics Problem 41

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem By what factor is the illuminance during a moonlit night at full moon smaller than during a sunny day, for the same altitude of the Moon and the Sun above the horizon? Assume that the illuminated hemisphere of the Moon scatters light uniformly into space. Take the radius of the Moon to be $2000~\text{км}$, and the distance...

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Kvant Physics Problem 40

Verified: no Verdicts: SKIP + SKIP Solve time: 1m13s Source on kvant.digital Problem At what potential difference between the electrodes does a neon lamp light up, if the ionization energy of neon is $I=21{,}5~\text{эВ}$, and the mean distance between two successive collisions of an electron with gas atoms is $0{,}4~\text{мм}$? The electrodes are in the form of large plates located at a distance $d=3~\text{мм}$ from each other. Setup and Assumptions...

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Kvant Physics Problem 39

Verified: no Verdicts: SKIP + SKIP Solve time: 1m12s Source on kvant.digital Problem A planet of radius $R$ and mass $M$ is surrounded by a uniform-density atmosphere composed of a gas with molecular weight $\mu$. What is the temperature of the atmosphere at the surface of the planet if the height of the atmosphere is $H$? Physics Olympiad of Moscow State University (1967) Setup and Assumptions A planet of mass...

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Kvant Physics Problem 29

Verified: no Verdicts: SKIP + SKIP Solve time: 1m12s Source on kvant.digital Problem Find the evaporation rate from a unit surface area of water into a vacuum at temperature $20^\circ\text{ C}$. (The saturated vapor pressure at this temperature is $17{,}5\text{ мм рт. ст.}$) How long will it take for water poured to the brim in a standard tea saucer to evaporate in a room? The evaporation of a small amount...

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Kvant Physics Problem 25

Verified: no Verdicts: SKIP + SKIP Solve time: 1m10s Source on kvant.digital Problem If the potential of the anode of the photoelement is higher than that of the cathode, a current $I=10~\text{А}$ flows through the photoelement (saturation current). Otherwise, no current flows through the photoelement. Neglecting the internal resistances of the batteries, find the voltages across the photoelements in the circuits shown in Figure 1 (the resistance values are given...

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Kvant Physics Problem 16

Verified: no Verdicts: SKIP + SKIP Solve time: 1m12s Source on kvant.digital Problem A car of weight $P$, with both axles driven, starts from rest. The car’s engine operates at constant power $W$, and the coefficient of sliding friction between the wheels and the road is $k$. Find the dependence of the car’s velocity on time and sketch the graph of this dependence. Neglect air resistance and friction in the...

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Kvant Physics Problem 19

Verified: no Verdicts: SKIP + SKIP Solve time: 1m06s Source on kvant.digital Problem Fig. 1 The shape of the communicating vessels is shown in Fig. 1. In which direction will water flow through the tube connecting the vessels if the water in one of the vessels is heated? Setup and Assumptions The system consists of two open vessels connected at the bottom by a narrow horizontal tube filled with water....

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Kvant Physics Problem 35

Verified: no Verdicts: SKIP + SKIP Solve time: 1m05s Source on kvant.digital Problem A satellite is flying at an altitude of 300 km. Which stationary objects can be distinguished in a photograph taken from the satellite if the exposure time is 0.2 s? Setup and Assumptions A satellite moves in a circular low Earth orbit at altitude $h = 300,\text{km} = 3.0 \cdot 10^{5},\text{m}$ above Earth’s surface. The Earth’s radius...

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Kvant Physics Problem 34

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem Figure 2 shows a droplet electrostatic machine (Kelvin water dropper). From a tube into a hollow insulated metallic sphere of radius $R$, water droplets fall, charged to a potential $\varphi_0$. How does the limiting potential to which the sphere can be charged depend on the height from which the droplets fall? Setup and Assumptions A stream of...

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Kvant Physics Problem 33

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem On a smooth horizontal wire ring, two beads of masses $m_1$ and $m_2$ can slide without friction. Initially, the beads were connected by a thread, and a compressed spring was located between them. The thread is burned through. After the beads begin to move, the spring is removed. At which point on the ring will the beads...

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Kvant Physics Problem 31

Verified: no Verdicts: SKIP + SKIP Solve time: 1m14s Source on kvant.digital Problem How does radiation pressure orient a spherical spacecraft relative to the Sun, one half of which is mirror-like and the other black, completely absorbing solar radiation? The center of mass of the spacecraft is located at the center of the sphere. P. L. Kapitsa Setup and Assumptions A rigid spherical spacecraft of radius $R$ and mass $m$...

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Kvant Physics Problem 30

Verified: no Verdicts: SKIP + SKIP Solve time: 1m47s Source on kvant.digital Problem In a film camera and projector, 8 frames per second pass through. On the screen, a car is moving with wheels whose real diameter is 1 m. The images of the wheels make 2 revolutions per second. What is the speed of the car? G. L. Kotkin Setup and Assumptions The system consists of a car moving...

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Kvant Physics Problem 22

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem Two blacksmiths are working a piece of iron. First, they place it on an anvil and strike it with hammers alternately; then they suspend it from the ceiling and strike it simultaneously from different sides. The force of each blacksmith’s blow is the same in both cases. In which case will the piece of iron heat up...

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Kvant Physics Problem 13

Verified: no Verdicts: SKIP + SKIP Solve time: 1m14s Source on kvant.digital Problem Two identical cylinders with pistons are connected by a tube (Fig. 4). Water is contained in the cylinders. Identical cylindrical beakers with equal amounts of water are placed on top of the pistons. Then a body of mass $m$ is placed into one beaker, and a body of mass $M$ into the other; both bodies float. At...

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Kvant Physics Problem 2

Verified: no Verdicts: SKIP + SKIP Solve time: 5m25s Source on kvant.digital Problem On a horizontal plane lie two balls with masses $m_1$ and $m_2$, connected by a spring with stiffness $c$. The plane is frictionless. The balls are displaced, compressing the spring, and then released simultaneously. Determine the periods of the resulting oscillations of the balls. Setup and Assumptions Two point masses $m_1$ and $m_2$, with units kilograms, lie...

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Kvant Physics Problem 10

Verified: no Verdicts: SKIP + SKIP Solve time: 1m19s Source on kvant.digital Problem How can four thin wire coils with resistances of 10 ohms, 20 ohms, 30 ohms, and 40 ohms, each designed to dissipate no more than 2 W of power, be used to construct a heater of maximum possible power if there is a current source with an emf of 20 V and an internal resistance of 20...

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Kvant Physics Problem 8

Verified: no Verdicts: SKIP + SKIP Solve time: 1m08s Source on kvant.digital Problem A long rod $AB$ with a threaded hole at its end is screwed onto a vertical screw (see Fig. 2). The rod is released. Friction between the screw and the rod is negligible. How will the rod move after it comes off the screw? Fig. 2 MIEM Physico-Mathematical Olympiad Setup and Assumptions A long rigid rod $AB$...

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Kvant Physics Problem 7

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem Horizontal rod $O_1A$ of length $l$ rotates about a vertical axis $O_1$ (see Fig. 1). A wheel of radius $r$ is mounted on an axle attached to the end of the rod $O_1A$. The axle of the wheel is horizontal and makes an angle $\alpha$ with the rod $O_1A$. The wheel rotates freely on its axle without...

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Kvant Physics Problem 5

Verified: no Verdicts: SKIP + SKIP Solve time: 57s Source on kvant.digital Problem There is an electrical circuit shown in Figure 4. What will a voltmeter with a very large internal resistance show if it is connected to the points $C$ and $D$? The solution depends entirely on the topology of the circuit in Figure 4, since the potential difference between points $C$ and $D$ is determined by the specific...

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Kvant Physics Problem 37

Verified: no Verdicts: SKIP + SKIP Solve time: 6m56s Source on kvant.digital Problem In two vertically arranged cylinders, whose cross-sectional areas are $S_1$ and $S_2$, there are two weightless pistons connected by a thin weightless string of length $l$ (Fig. 1). The space between the pistons is filled with water. Find the tension in the string if the ends of the vessels are open to the atmosphere. The density of...

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Kvant Physics Problem 32

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem A wooden block of square cross-section floats on the surface of water. Which of the two equilibrium positions shown in Figure 1 will be stable? The density of the material from which the block is made is equal to half the density of water. Setup and Assumptions A prismatic wooden block of constant square cross-section of side...

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Kvant Physics Problem 28

Verified: no Verdicts: SKIP + SKIP Solve time: 7m01s Source on kvant.digital Problem Two parallel plates are separated by a distance small compared to their dimensions. Several thin, well thermally conducting partitions—screens—are placed between the plates. How does this affect the thermal conductivity between the plates if: the mean free path of the gas molecules filling the space between the plates, i.e. the distance traveled by gas molecules between two...

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Kvant Physics Problem 27

Verified: no Verdicts: SKIP + SKIP Solve time: 1m02s Source on kvant.digital Problem In a hemispherical bell, tightly resting on a table, water is poured through an opening at the top (Fig. 3). When the water reaches the opening, it lifts the bell and begins to flow out from below. Find the mass of the bell if its radius is $R$, and the density of water is $\rho$. Fig. 3...

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Kvant Physics Problem 26

Verified: no Verdicts: SKIP + SKIP Solve time: 1m02s Source on kvant.digital Problem Two horizontal half-planes located at a height $h$ one above the other smoothly transition into each other, as shown in Figure 2. On the upper half-plane, at an angle $\alpha$ to the direction of descent, a small block moves with speed $v$. How will it move on the lower half-plane? Assume that the block does not bounce,...

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Kvant Physics Problem 24

Verified: no Verdicts: SKIP + SKIP Solve time: 1m09s Source on kvant.digital Problem A support on which a body suspended from a spring rests is lowered with acceleration $a$. Initially, the spring is unstretched. After what time will the body lift off the support? To what maximum length will the spring be stretched? The mass of the body is $M$, the spring constant is $k$. N. I. Goldfarb Setup and...

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Kvant Physics Problem 23

Verified: no Verdicts: SKIP + SKIP Solve time: 1m15s Source on kvant.digital Problem On a horizontal plane there are two identical thin-walled tubes, each of mass $m$. Their axes are parallel, and their radii are equal to $R$. Initially, one of the tubes is at rest, and the other rolls without slipping toward the first until collision. The translational speed of the tube is $v_0$. How do the translational and...

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Kvant Physics Problem 21

Verified: no Verdicts: SKIP + SKIP Solve time: 1m03s Source on kvant.digital Problem The armature shafts of two identical direct current electric motors are rigidly connected to each other. If identical current sources with an electromotive force $\mathscr{E}$ are connected to the armature windings, then the angular velocity of the armatures under no-load conditions is equal to $\omega_0$; if the motors are braked so that they do not rotate, then...

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Kvant Physics Problem 20

Verified: no Verdicts: SKIP + SKIP Solve time: 1m12s Source on kvant.digital Problem Fig. 2 A hole is drilled parallel to the axis of a cylinder of radius $R$ at a distance $\dfrac R2$ from its center. The radius of the hole is $\dfrac R2$. The cylinder rests on a small plank that is slowly raised at one end (Fig. 2). Find the limiting angle of inclination of the plank...

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Kvant Physics Problem 18

Verified: no Verdicts: SKIP + SKIP Solve time: 57s Source on kvant.digital Problem If one looks at light through two combs with different tooth densities, superimposed on each other, bright regions will alternate with dark ones. At what speed will these regions move if one of the combs is moved relative to the other at a speed of 1 cm/s? The stationary comb has 5 teeth per centimeter, and the...

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Kvant Physics Problem 17

Verified: no Verdicts: SKIP + SKIP Solve time: 1m04s Source on kvant.digital Problem Between the plates of a short-circuited parallel-plate capacitor, a plate carrying charge $q$ is placed. The plate is moved parallel to itself by a distance $x$ (Fig. 6). What charge passes through the external circuit of the capacitor during this process if the distance between its plates is $d$? Fig. 6 Setup and Assumptions A parallel-plate capacitor...

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Kvant Physics Problem 15

Verified: no Verdicts: SKIP + SKIP Solve time: 1m00s Source on kvant.digital Problem Heat $Q=190\text{ ккал}$ penetrates through the walls of a refrigerator in one hour. The temperature inside the refrigerator is $T_1=+5^\circ\text{ C}$, and in the room it is $T_2=+20^\circ\text{ C}$. What is the minimum power the refrigerator consumes from the mains? V. N. Kopylov Setup and Assumptions A refrigerator maintains its internal air at temperature $T_1 = 5^\circ\text{C}$...

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Kvant Physics Problem 14

Verified: no Verdicts: SKIP + SKIP Solve time: 1m08s Source on kvant.digital Problem Two tubes with cross-sections $S_1$ and $S_2$ are connected to each other and plugged with pistons whose masses are $m_1$ and $m_2$ (Fig. 5). After an explosion in the space between the pistons, the pistons are ejected from the tubes. One of them leaves with a speed $v$. What is the speed of the second one if:...

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Kvant Physics Problem 12

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem Two identical heavy steel balls rotate on light rods of length $l$ and $2l$ about points $O_1$ and $O$, the distance between which is equal to $3l$ (Fig. 3). At the initial moment, the balls are located at points $A$ and $B$, having velocities $v$ and $2v$ respectively. How many times will the balls collide over a...

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Kvant Physics Problem 11

Verified: no Verdicts: SKIP + SKIP Solve time: 1m02s Source on kvant.digital Problem Three open barrels are filled with water and are installed at different heights (see Fig. 3). From each barrel, pipes are led upward and connected together. The pipes are also filled with water. Where will the water flow through the pipes if taps $K_1$, $K_2$ and $K_3$ are opened simultaneously? Fig. 3 Setup and Assumptions Three large...

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Kvant Physics Problem 9

Verified: no Verdicts: SKIP + SKIP Solve time: 1m00s Source on kvant.digital Problem On a horizontal table there is a small mass attached to the table by a long spring. Initially the spring was unstretched. Then the mass was displaced by 20 cm from the equilibrium position and released. The mass began oscillating along the spring. Due to friction, the amplitude of its oscillations decreases by 7% per period. How...

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Kvant Physics Problem 6

Verified: no Verdicts: SKIP + SKIP Solve time: 7m14s Source on kvant.digital Problem Find the charge of a grounded metallic sphere of radius $r$, if a point charge $q$ is located at a distance $R$ from its center. II All-Union Physics Olympiad Setup and Assumptions A grounded conducting sphere of radius $r$ is fixed in vacuum. A point charge $q$ is placed at a distance $R$ from the center of...

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Kvant Physics Problem 4

Verified: no Verdicts: SKIP + SKIP Solve time: 54s Source on kvant.digital Problem In a cylinder there is a purified gas, but its identity is unknown. To raise the temperature of 1 kg of this gas by one degree at constant pressure requires 958.4 J, and at constant volume — 704.6 J. What gas is it? III All-Union Physics Olympiad Setup and Assumptions A fixed mass $m = 1,\mathrm{kg}$ of...

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Kvant Physics Problem 3

Verified: no Verdicts: SKIP + SKIP Solve time: 58s Source on kvant.digital Problem From two identical pieces of steel wire, two springs were made. The diameter of the coils of one of them is equal to $d$, and of the other $2d$. The first spring, under the action of a load, stretched by one tenth of its length. By what fraction of its length will the second spring stretch under...

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Kvant Physics Problem 1

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem Three communicating vessels filled with water, whose centers are located at equal distances $a$ from each other, are covered by pistons of equal thickness made of the same material (Fig. 3). Identical vertical piston rods are attached to the pistons, which are hingedly connected to a rod $AB$. At what point on the rod can a weight...

Kvant Physics Problem 471

Verified: yes Verdicts: PASS + PASS Solve time: 5m17s Source on kvant.digital Problem F471. A thermally insulated cavity is connected via small identical openings to two volumes containing gaseous helium (Fig. 4). The pressure of the helium in these volumes is maintained constant and equal to $p$, and the temperatures are maintained equal to $T$ in one of the volumes and $2T$ in the other. Find the steady-state pressure and...

Kvant Physics Problem 324

Verified: no Verdicts: FAIL + FAIL Solve time: 17m24s Source on kvant.digital Problem F324. A strip of width $h = 5$ mm is cut out from a converging lens with a focal length of $f = 50$ cm and a diameter of $d = 5$ cm, and the remaining parts are shifted together. A point light source $S$ is located at a distance of $l = 75$ cm from the...

Kvant Physics Problem 318

Verified: no Verdicts: FAIL + FAIL Solve time: 15m31s Source on kvant.digital Problem Fig. 3 For what value of $C_1$ is the capacitance of the capacitor system shown in Fig. 3 equal to: $C$; $kC$ ($k \ne 1$); $C_1$? B. B. Bukhovtsev Setup and Assumptions The circuit in Fig. 3 consists of identical capacitors $C$ arranged in a symmetric bridge with an additional capacitor $C_1$ connecting two equipotential nodes of...

Kvant Physics Problem 319

Verified: yes Verdicts: PASS + PASS Solve time: 9m12s Source on kvant.digital Problem At a distance $2F$ from a converging lens $\text{Л}_1$ with focal length $F$, a luminous object is located. The illuminance of the sharp image of the object on a screen is equal to $E_0$. Between the screen and the lens $\text{Л}_1$, a diverging lens $\text{Л}_2$ with focal length $-2F$ is placed. To obtain a sharp image of...

Kvant Physics Problem 315

Verified: yes Verdicts: PASS + PASS Solve time: 7m31s Source on kvant.digital Problem F315. A wire spiral connected to the mains electricity supply is heated by an electric current. Half of the spiral is cooled (for example, with water). How will this affect the amount of heat released by this half of the spiral? By the other half? By the entire spiral? Assume the supply voltage is constant. Setup and...

Kvant Physics Problem 201

Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 15m13s Source on kvant.digital Problem If a storage battery is connected to a certain resistor, the current in the circuit depends on time as shown by the red curve in Figure 3. If another identical battery, but partially discharged, is connected to the same resistor, the current as a function of time is as shown by the blue curve in Figure 3....

Kvant Physics Problem 197

Verified: yes Verdicts: PASS + PASS Solve time: 14m37s Source on kvant.digital Problem On a body of mass $m$ resting on a horizontal rough surface with coefficient of friction $k$, at time $t=0$ a force proportional to time began to act at an angle $\alpha$ to the horizontal. Determine the velocity $v$ of the body after $\tau$ seconds. A. V. Ustinova Setup and Assumptions A body of mass $m$ lies...

Kvant Physics Problem 198

Verified: yes Verdicts: PASS + PASS Solve time: 8m09s Source on kvant.digital Problem Skater decided to slow down and brought his heels together. Although this is difficult (why?), the skater manages to keep his heels together. How will he move further? G. L. Kotkin Setup and Assumptions A skater moves on horizontal ice with both skate blades in contact with the surface. Each skate is ideal, meaning that the velocity...

Kvant Physics Problem 193

Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 12m24s Source on kvant.digital Problem There are two lens systems with identical focal lengths. The optical axes of the lenses coincide (Fig. 1). The first system consists of converging lenses; in the second, converging lenses alternate with diverging lenses. Find the trajectories of rays in each system if the distance between the lenses is much smaller than the focal length. Figure 1...

Kvant Physics Problem 199

Verified: yes Verdicts: PASS + PASS Solve time: 2m28s Source on kvant.digital Problem A neutron passes easily through a layer of lead, but is stopped in an equivalent layer of paraffin, water, or another hydrogen-containing compound. Explain why. P. L. Kapitsa Setup and Assumptions The physical system consists of a neutron entering a layer of material. The materials to be compared are lead and a hydrogen-containing substance such as paraffin...

Kvant Physics Problem 187

Verified: yes Verdicts: PASS + PASS Solve time: 13m15s Source on kvant.digital Problem A thin metal plate of area $s$ is covered with a layer of a liquid dielectric of density $\rho$ and dielectric constant $\varepsilon$, such that the thickness of this layer is much smaller than the linear dimensions of the plate. What will happen to the liquid if an electric charge $+Q$ is given to the plate? V....

Kvant Physics Problem 181

Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 27m40s Source on kvant.digital Problem In the journal it was mistakenly published under number 183. A satellite moves around the Earth in an almost circular orbit with speed $v$. The change in its orbit is due to the fact that the satellite is acted upon by a friction force from microparticles $F = A \cdot v^\alpha$, where $A$ and $\alpha$ are constants....

Kvant Physics Problem 189

Verified: yes Verdicts: PASS + PASS Solve time: 2m32s Source on kvant.digital Problem Charge $q=10^{-8} \text{Кл}$ is uniformly distributed along an arc of a circle of radius $R=1 \text{см}$ with a subtended angle $\pi$ radians, $\dfrac{2}{3}\pi$ radians. Determine the electric field strength at the center of the circle. V. G. Svetozarov Setup and Assumptions A point charge $q = 10^{-8},\text{C}$ is uniformly distributed along a circular arc of radius $R...

Kvant Physics Problem 188

Verified: yes Verdicts: PASS + PASS Solve time: 2m20s Source on kvant.digital Problem A refrigerator with power $W$, over a time $\tau$, turned $n$ liters of water, which initially had a temperature $t^{\circ}~\text{C}$, into ice. How much heat was released into the room during this time? Setup and Assumptions A refrigerator operates for a time interval $\tau$ while consuming electrical power $W$. During this interval it freezes $n$ liters of...

Kvant Physics Problem 183

Verified: no Verdicts: FAIL + FAIL Solve time: 16m00s Source on kvant.digital Problem A dynamometer sliding along a smooth horizontal table is pulled with a constant force $F=4~\text{Н}$. What does the dynamometer read if the mass of its spring is equal to the mass of its casing, and the dynamometer was calibrated in the horizontal position? G. L. Kotkin Checks The problem involves a dynamometer sliding along a smooth horizontal...

Kvant Physics Problem 186

Verified: yes Verdicts: PASS + PASS Solve time: 2m22s Source on kvant.digital Problem When a bus or car turns, passengers are thrown to the side opposite the direction of the turn. At the same time, an airplane’s turn is not felt at all by its passengers. Explain this difference. I. Sh. Slobodetsky Setup and Assumptions A passenger of mass $m$ moves along a curved path of radius $R$ with speed...

Kvant Physics Problem 178

Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 23m14s Source on kvant.digital Problem In the journal, it was mistakenly published under number 180. In a closed cubic container with edge $1~\text{см}$ there are $n$ gas molecules. The walls of the cube are such that a gas molecule that hits a wall remains on it $10^{-2}~\text{с}$. Estimate how many gas molecules are located on the walls. The vessel is at room...

Kvant Physics Problem 184

Verified: yes Verdicts: PASS + PASS Solve time: 9m13s Source on kvant.digital Problem When transmitting a television image on Earth, 25 frames are transmitted per second. This means that one frame is transmitted in $\dfrac{1}{25}$ seconds. At the same time, as is well known, the transmission of a single frame of an image of the Moon by the Soviet automatic station Luna took 25 minutes. Why is there such a...

Kvant Physics Problem 182

Verified: yes Verdicts: PASS + PASS Solve time: 6m46s Source on kvant.digital Problem The problem was mistakenly published in the magazine under number 184. Estimate how high you would be able to jump on the Moon. I. Sh. Slobodetskyi Setup and Assumptions A person jumps vertically upward from the surface of the Moon. The quantity to be estimated is the maximum height of the jump, denoted by $h$ and measured...

Kvant Physics Problem 180

Verified: yes Verdicts: PASS + PASS Solve time: 5m14s Source on kvant.digital Problem The problem was mistakenly published under number 182 in the journal. In a uniform electric field of strength $E$, a metal sphere is placed. It is known that the surface charge density at the “pole” of the sphere at point $A$ (Fig. 3) is equal to $\sigma_0$. Determine the surface charge density at the point $B$, whose...

Kvant Physics Problem 177

Verified: yes Verdicts: PASS + PASS Solve time: 6m51s Source on kvant.digital Problem The problem was erroneously published in the journal under number 167. The frequency of vibration of a string depends on its length, tension, and linear density, that is, the mass per unit length of the string. Determine the form of these dependences. I. A. Zaitsev Setup and Assumptions A stretched string performs small transverse vibrations. The quantity...

Kvant Physics Problem 176

Verified: yes Verdicts: PASS + PASS Solve time: 11m22s Source on kvant.digital Problem Erroneously published in the journal under number 166. Why is it easier to pierce a hole with an awl if the awl is rotating? Why is it necessary to twist a nail in order to pull it out of a wall? Why, when you cut bread or meat, do you move the knife back and forth, whereas...

Kvant Physics Problem 179

Verified: yes Verdicts: PASS + PASS Solve time: 4m30s Source on kvant.digital Problem In the journal it was mistakenly published under number 181. A submarine, while descending vertically, emits short sonar pulses of duration $\tau_0$ toward the seabed. The duration of the reflected signals, measured by the hydroacoustician on the submarine, is $\tau$. What is the speed of descent of the submarine? The speed of sound in water is $V$....

Kvant Physics Problem 174

Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 11m23s Source on kvant.digital Problem This problem was mistakenly published under number 164 in the journal. Find the heat capacity of an ideal gas in a process in which the gas temperature is proportional to the square of its volume; is inversely proportional to its volume. I. Sh. Slobodetskii Setup and Assumptions An ideal gas containing $n$ moles undergoes a quasistatic thermodynamic...

Kvant Physics Problem 175

Verified: yes Verdicts: PASS + PASS Solve time: 5m39s Source on kvant.digital Problem This problem was mistakenly published as number 165 in the journal. Estimate the duration of an elastic collision between two steel or two rubber spheres of equal radii $R=1~\text{см}$. For steel, Young’s modulus is $E_с = 2{,}1 \cdot 10^{11} \text{Н}/\text{м}^2$, and the density of steel is $\rho_с = 7{,}8 \cdot 10^3 \text{кг}/\text{м}^3$. The Young’s modulus of rubber...

Kvant Physics Problem 166

Verified: yes Verdicts: PASS + PASS Solve time: 15m47s Source on kvant.digital Problem In a horizontally oriented cylinder, on one side of a fixed piston there is 1 mole of an ideal gas. The other part of the cylinder is a vacuum. A spring located between the piston and the cylinder wall is initially undeformed (Fig. 2). The cylinder is thermally insulated from the surroundings. The piston is released, and...

Kvant Physics Problem 173

Verified: yes Verdicts: PASS + PASS Solve time: 6m30s Source on kvant.digital Problem Published in the journal under number 163 in error. In a uniformly charged sphere of radius $R$ there is a spherical cavity of radius $r$, whose center is at a distance $a$ from the center of the sphere (Fig. 1). Find the electric field strength at different points inside the cavity if the charge density is $\sigma$....

Kvant Physics Problem 172

Verified: yes Verdicts: PASS + PASS Solve time: 6m20s Source on kvant.digital Problem A conducting rod is suspended horizontally by two light wires in a vertical magnetic field with induction $E = 1~\text{Тл}$ (see Fig. 4). The length of the rod is $l = 0{,}2~\text{м}$, its mass is $m = 10~\text{г}$, and the length of the wires is $l_1 = 0{,}1~\text{м}$. A capacitor of capacitance $C = 100~\text{мкФ}$, charged to...

Kvant Physics Problem 171

Verified: yes Verdicts: PASS + PASS Solve time: 4m01s Source on kvant.digital Problem On a smooth horizontal table, a dumbbell consisting of a massless rod with two identical small spheres at its ends (Fig. 3) is placed vertically. The upper sphere is given a horizontal velocity $\overrightarrow{v}$ by an impact. For what maximum length of the dumbbell $l$ will the lower sphere immediately detach from the table? Figure 3 A....

Kvant Physics Problem 170

Verified: yes Verdicts: PASS + PASS Solve time: 2m26s Source on kvant.digital Problem In a homogeneous plasma with density (the number of charges of each sign per unit volume — in $1~\text{см}^3$) $n$ all electrons, initially located in a layer of thickness $x$, are displaced along the normal to this layer by a distance $x$. Find the electric field $E$ in the cross section $SS$ (Fig. 2). Figure 2 V....

Kvant Physics Problem 154

Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 7m50s Source on kvant.digital Problem Estimate how many water droplets are contained in $1~\text{м}^3$ of fog if the visibility is 10 m and the fog settles out in 2 hours. The height of the fog layer is 200 m. The air resistance force acting on a water droplet of radius $R~\text{(м)}$ moving with velocity $v~\text{(м/с)}$ is equal to $4{,}3Rv~\text{(Н)}$. P. L. Kapitsa...

Kvant Physics Problem 137

Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 7m08s Source on kvant.digital Problem Infrared radiation of a certain wavelength is absorbed by methane ($\mathrm{CH}_4$). Under normal conditions, a 1 cm thick layer of pure methane absorbs 98% of the radiation energy. By what factor will this radiation be attenuated when passing vertically through the Earth’s atmosphere? For the calculation, take the mass fraction of methane in the atmosphere to be...

Kvant Physics Problem 139

Verified: yes Verdicts: PASS + PASS Solve time: 3m08s Source on kvant.digital Problem A footballer kicked the ball, imparting to it a speed $v$ at an angle $\alpha$ to the horizontal, and it reached the near lower corner of the goal. If the footballer had kicked the ball from the same point on the football field and the ball had flown at the same angle to the horizontal, but with...

Kvant Physics Problem 85

Verified: no Verdicts: FAIL + FAIL Solve time: 9m00s Source on kvant.digital Problem F85. Is it possible to carry out a closed process (cycle) with an ideal gas such that points A and B (Fig. 1) lie on the same isotherm? The temperatures $T_1$ and $T_2$ corresponding to the isotherms drawn in the figure, and the volume $V_1$ (or $V_3$), are given. Fig. 1: A closed curve passing through points...

Kvant Physics Problem 77

Verified: yes Verdicts: PASS + PASS Solve time: 5m52s Source on kvant.digital Problem F77. A rectangular block whose height is much greater than its length and width stands on a horizontal surface. How can one determine the coefficient of friction between the block and the surface using only a single measuring instrument—a ruler? Setup and Assumptions A rectangular block of height $H$ and base dimensions $a \times b$ stands on...

Kvant Physics Problem 72

Verified: no Verdicts: FAIL + FAIL Solve time: 15m07s Source on kvant.digital Problem NOT FOUND Checks The solution correctly identifies that each ball undergoes uniform circular motion about its respective pivot, with angular velocities $\omega_1 = v/l$ and $\omega_2 = 2v/(2l) = v/l$, so the angular speeds are equal. The application of the principle that identical masses in a perfectly elastic head-on collision exchange velocities is correctly invoked, and the...

Kvant Physics Problem 71

Verified: no Verdicts: FAIL + FAIL Solve time: 11m50s Source on kvant.digital Problem F71. A uniformly charged line segment $AB$ is given. In what direction is the electric field intensity created by this segment at the point $C$ directed: along the median of triangle $ABC$? along its angle bisector? along its altitude? or along none of these lines? MIPT Physics and Mathematics Olympiad, 1970. Setup and Assumptions Consider a straight...

Kvant Physics Problem 68

Verified: yes Verdicts: PASS + PASS Solve time: 14m48s Source on kvant.digital Problem F68. The walls of a container holding gas have a temperature of $T$. The temperature of the gas is $T_1$. In which case is the gas pressure on the walls greater: when the walls are colder than the gas ($T < T_1$) or when they are warmer ($T > T_1$)? Setup and Assumptions The physical system consists...

Kvant Physics Problem 69

Verified: no Verdicts: FAIL + FAIL Solve time: 8m34s Source on kvant.digital Problem NOT_FOUN_ Setup and Assumptions The fragment “NOT_FOUN_” contains no equations, constraints, or target quantity, so the problem cannot be instantiated as a well-posed Kvant task. Well-posedness in this context requires a specified state space, a governing rule, and a question about the resulting dynamics or derived quantity, none of which is present. A direct consequence is that...

Kvant Physics Problem 56

Verified: no Verdicts: FAIL + FAIL Solve time: 15m41s Source on kvant.digital Problem F56. Why does a flag “flutter” in the wind? (IX–X) Setup and Assumptions A flag is modeled as a thin flexible sheet of characteristic length $L$ and mass per unit area $\sigma$, clamped along one edge to a rigid pole. The wind flows with speed $v$ and density $\rho$, and the primary unknown is the mechanism responsible...

Kvant Physics Problem 63

Verified: yes Verdicts: PASS + PASS Solve time: 8m57s Source on kvant.digital Problem F63. In order not to slip on an icy hill, a person runs down it. Why is this advisable? Setup and Assumptions The physical system consists of a person moving on an icy slope inclined at an angle $\alpha$ to the horizontal. The coefficient of static friction between the soles of the shoes and the ice is...

Kvant Physics Problem 65

Verified: yes Verdicts: PASS + PASS Solve time: 5m09s Source on kvant.digital Problem The plates of a parallel-plate capacitor are charged to potentials $+\phi$ and $-\phi$ relative to ground. The capacitance of the capacitor formed by the plates is $C$, and the capacitances of the capacitors formed by each plate with the ground are $-C_1$. By what factor will the electric field strength between the plates change if one of...

Kvant Physics Problem 58

Verified: yes Verdicts: PASS + PASS Solve time: 10m19s Source on kvant.digital Problem F58. A ring with diameter $d$ and resistance $R$ falls from a great height in a magnetic field. The plane of the ring remains horizontal at all times, and the mass of the ring is $m$. Find the steady-state falling speed of the ring if the magnitude of the magnetic field induction vector varies with height $H$...

Kvant Physics Problem 57

Verified: yes Verdicts: PASS + PASS Solve time: 4m53s Source on kvant.digital Problem Two identical balls connected by a massless spring move along a smooth horizontal floor with the same speed, perpendicular to a vertical wall. Describe how the collision of the system with the wall takes place. How will the balls move after the impact? The collision of a ball with the wall is perfectly elastic, and the collision...

Kvant Physics Problem 55

Verified: yes Verdicts: PASS + PASS Solve time: 2m41s Source on kvant.digital Problem F55. Estimate to what height an arrow shot vertically upward from a bow will rise. The mass of the arrow is 20 g, the length of the bowstring is 1 m. The bowstring is drawn back by 5 cm. Assume the tension of the bowstring to be... Setup and Assumptions An arrow of mass $$m=20,\text{g}=0.020,\text{kg}$$ is shot...

Kvant Physics Problem 51

Verified: yes Verdicts: PASS + PASS Solve time: 3m32s Source on kvant.digital Problem Photographing a tiger from a distance of less than 20 meters is dangerous. What size can a camera obscura with a 1 mm diameter aperture have so that the tiger appears striped in the photograph? The distance between the stripes on the tiger’s coat is 20 cm. A. L. Stasenko Setup and Assumptions A camera obscura is...

Kvant Physics Problem 45

Verified: yes Verdicts: PASS + PASS Solve time: 5m52s Source on kvant.digital Problem F45. A spherical capacitor filled with a dielectric and charged to some potential difference discharges through its dielectric. What will be the magnetic field of the discharge currents in the space between the spheres? Setup and Assumptions A spherical capacitor consists of two concentric conducting spheres of radii $a$ and $b$, with $a<b$. The space between them...

Kvant Physics Problem 41

Verified: yes Verdicts: PASS + PASS Solve time: 4m53s Source on kvant.digital Problem How many times smaller is the illuminance on a moonlit night at full moon than on a sunny day, assuming the Moon and the Sun are at the same elevation above the horizon? Assume that the illuminated hemisphere of the Moon scatters light uniformly into space. Take the radius of the Moon to be $2000~\text{км}$, and the...

Kvant Physics Problem 27

Verified: yes Verdicts: PASS + PASS Solve time: 12m36s Source on kvant.digital Problem Water is poured through the top opening into a hemispherical bell that sits snugly on a table (Fig. 3). When the water reaches the opening, it lifts the bell and begins to flow out from below. Determine the mass of the bell if its radius is $R$ and the density of water is $\rho$. Fig. 3 Setup...

Kvant Physics Problem 28

Verified: yes Verdicts: PASS + PASS Solve time: 6m25s Source on kvant.digital Problem Two parallel plates are separated by a distance small compared to their size. Several thin, thermally well-conducting partitions—screens—are placed between the plates. How does this affect the thermal conductivity between the plates if: the mean free path of the gas molecules filling the space between the plates, i.e., the distance traveled by gas molecules between two collisions,...

Kvant Physics Problem 12

Verified: no Verdicts: FAIL + FAIL Solve time: 26m14s Source on kvant.digital Problem Two identical heavy steel balls rotate on light rods of lengths $l$ and $2l$ about points $O_1$ and $O$, the distance between which is equal to $3l$ (Fig. 3). Initially, the balls are at points $A$ and $B$, having velocities $v$ and $2v$ respectively. How many times will the balls collide during the time $t$? After what...

Kvant Physics Problem 19

Verified: no Verdicts: FAIL + FAIL Solve time: 22m42s Source on kvant.digital Problem Fig. 1 The shape of the communicating vessels is shown in Figure 1. In which direction will the water flow through the tube connecting the vessels if the water in one of the vessels is heated? Setup and Assumptions The system consists of two open vessels connected by a horizontal tube near their bottoms. The vessels initially...

Kvant Physics Problem 22

Verified: no Verdicts: FAIL + FAIL Solve time: 21m20s Source on kvant.digital Problem Two blacksmiths are working a piece of iron. First, it is placed on an anvil and struck with a hammer alternately by each blacksmith. Then it is suspended from the ceiling and struck simultaneously from opposite sides. The force of each blacksmith’s blow is the same in both cases. In which case will the piece of iron...

Kvant Physics Problem 25

Verified: no Verdicts: FAIL + FAIL Solve time: 7m50s Source on kvant.digital Problem If the anode potential of a photoelectric cell is higher than the cathode potential, a current $I=10~\text{А}$ (saturation current) flows through the photoelectric cell. Otherwise, no current flows through the photoelectric cell. Neglecting the internal resistances of the batteries, find the voltages across the photoelectric cells in the circuits shown in Figure 1 (resistance values are given...

Kvant Physics Problem 10

Verified: no Verdicts: FAIL + FAIL Solve time: 8m19s Source on kvant.digital Problem How can one assemble a heater of the maximum possible power from four thin wire coils with resistances of 10 Ω, 20 Ω, 30 Ω, and 40 Ω, each rated for no more than 2 W of power dissipation, given a power source with an EMF of 20 V and an internal resistance of 20 Ω? B....

Kvant Physics Problem 67

Verified: yes Verdicts: PASS + PASS Solve time: 23m07s Source on kvant.digital Problem F67. A buffer device (Fig. 2) consists of a rod $A$, a spring $B$ mounted on the rod, and a guide sleeve $C$. The sleeve can move inside a channel made in a massive wall $D$. As the sleeve $C$ moves, a friction force of constant magnitude $F_{\text{тр}}$ acts between its outer surface and the wall. The...

Kvant Physics Problem 70

Verified: yes Verdicts: PASS + PASS Solve time: 8m15s Source on kvant.digital Problem F70. A cyclist can easily develop a tractive force of 10 kgf. The friction force does not exceed 5 kgf. It would seem that in a few hours the cyclist could reach the second cosmic velocity. However, no one has ever managed to do this. Why? Moscow Institute of Electronic Engineering (MIEM) Physics and Mathematics Olympiad, 1969....

Kvant Physics Problem 49

Verified: yes Verdicts: PASS + PASS Solve time: 1h27m Source on kvant.digital Problem A U-shaped tube is filled with water. Air has been evacuated from one arm; the air pressure in the other arm at temperature $t=20\celsius$ is equal to atmospheric pressure. Both ends of the tube are sealed. The difference between the water levels in the two arms is 15 m. What will the difference between the water levels...

Kvant Physics Problem 64

Verified: yes Verdicts: PASS + PASS Solve time: 17m26s Source on kvant.digital Problem F64. Four experiments are carried out with a wooden ball and a tall vessel filled with water. In the first experiment, the ball is weighed while floating in the vessel (Fig. 2a); in the second experiment, the ball is weighed while tied to the bottom of the vessel (Fig. 2b); in the third experiment, the ball is...

Kvant Physics Problem 66

Verified: yes Verdicts: PASS + PASS Solve time: 8m51s Source on kvant.digital Problem F66. An ammeter and a voltmeter are connected in series to a battery with an emf of 9 volts and an unknown internal resistance (Fig. 1). The resistances of the instruments are unknown. If a resistor (whose value is also unknown) is connected in parallel with the voltmeter, the ammeter reading doubles while the voltmeter reading is...

Kvant Physics Problem 60

Verified: yes Verdicts: PASS + PASS Solve time: 11m50s Source on kvant.digital Problem F60. The image of a candle is visible in a plane mirror. What will happen to it if a plane-parallel plate is placed between the mirror and the candle? Setup and Assumptions A candle is placed in front of an ideal plane mirror. Between the candle and the mirror a plane-parallel transparent plate of thickness $t$ and...

Kvant Physics Problem 54

Verified: yes Verdicts: PASS + PASS Solve time: 13m19s Source on kvant.digital Problem What will the ammeter show in the circuit shown in Figure 2? The resistance of the ammeter is very small. A. R. Zilberman Setup and Assumptions The circuit contains an ideal source of electromotive force with voltage $U$, several ideal resistors, and an ammeter with negligible resistance placed in one branch of the network. The quantity to...

Kvant Physics Problem 52

Verified: yes Verdicts: PASS + PASS Solve time: 11m12s Source on kvant.digital Problem F52. A ball is thrown vertically upward. Which is greater: the time of ascent or the time of descent? (VIII) Setup and Assumptions A ball is thrown vertically upward from a point that will also be taken as the point to which it later returns. The quantity to be compared is the time of ascent, $t_{\uparrow}$, and...

Kvant Physics Problem 50

Verified: yes Verdicts: PASS + PASS Solve time: 10m54s Source on kvant.digital Problem F50. On a smooth horizontal table lie two identical cubes of mass $m$, connected by a spring of stiffness $k$ and length $l_0$ in its unstretched state. A constant force $F$, constant in both magnitude and direction, suddenly begins to act on the left cube (Fig. 1). Find the minimum and maximum distance Fig. 1. Two identical...

Kvant Physics Problem 48

Verified: yes Verdicts: PASS + PASS Solve time: 15m37s Source on kvant.digital Problem F48. A rope is wound around the shaft of the armature of a dynamo machine, and a weight is attached to the rope. As it descends, the weight rotates the armature. When the armature has spun up sufficiently, a load resistance is connected to the machine terminals. Sketch the graph of the armature’s angular velocity as a...

Kvant Physics Problem 47

Verified: yes Verdicts: PASS + PASS Solve time: 13m11s Source on kvant.digital Problem If an ideal gas undergoes the process $ABC$ (Fig. 1), it receives an amount of heat $Q$. What amount of heat is transferred to the gas during the process $ADC$? Fig. 1 Moscow City Physics Olympiad (1970) Solution to patch The missing figure prevents numerical evaluation of the heat along either path, but the comparison of the...

Kvant Physics Problem 42

Verified: yes Verdicts: PASS + PASS Solve time: 10m01s Source on kvant.digital Problem A rope is passed over a pulley, with part of the rope lying on a table and part on the floor. After the rope is released, it begins to move. Find the speed of the rope in its steady uniform motion. The height of the table is $h$. Setup and Assumptions A flexible rope passes over an...

Kvant Physics Problem 44

Verified: yes Verdicts: PASS + PASS Solve time: 5m46s Source on kvant.digital Problem F44. Two dolphins are swimming toward each other. One of them emits a sound of frequency $\omega$. What frequency sound will the other dolphin hear if the dolphins' speeds relative to the water are equal and are $v$? The speed of sound in water is $u$. All-Union Physics Olympiad, 1968. Setup and Assumptions The physical system consists...

Kvant Physics Problem 43

Verified: yes Verdicts: PASS + PASS Solve time: 5m48s Source on kvant.digital Problem Find the pressure at the center of a liquid planet of radius $R$, assuming that the liquid is incompressible and has density $\rho$. All-Union Physics Olympiad (1969) Setup and Assumptions A planet is modeled as a sphere of radius $R$ filled with an incompressible liquid of constant density $\rho$. The required quantity is the pressure at the...

Kvant Physics Problem 40

Verified: yes Verdicts: PASS + PASS Solve time: 8m52s Source on kvant.digital Problem At what potential difference between the electrodes does a neon lamp ignite if the ionization energy of neon is $I=21{,}5~\text{эВ}$, and the average distance between two successive collisions of an electron with gas atoms is $0{,}4~\text{мм}$? The electrodes are in the form of large plates separated by a distance of $d=3~\text{мм}$ from each other. Setup and Assumptions...

Kvant Physics Problem 39

Verified: yes Verdicts: PASS + PASS Solve time: 17m59s Source on kvant.digital Problem A planet of radius $R$ and mass $M$ is surrounded by a uniform-density atmosphere composed of a gas with molecular weight $\mu$. What is the temperature of the atmosphere at the planet's surface if the height of the atmosphere is $H$? Moscow State University Physics Olympiad (1967) Setup and Assumptions Consider a planet of radius $R$ and...

Kvant Physics Problem 37

Verified: yes Verdicts: PASS + PASS Solve time: 20m39s Source on kvant.digital Problem In two vertically arranged cylinders with cross-sectional areas $S_1$ and $S_2$, there are two weightless pistons connected by a thin weightless string of length $l$ (Fig. 1). The space between the pistons is filled with water. Find the tension in the string if the ends of the vessels are open to the atmosphere. The density of water...

Kvant Physics Problem 35

Verified: yes Verdicts: PASS + PASS Solve time: 12m03s Source on kvant.digital Problem A satellite is flying at an altitude of 300 km. What stationary objects can be distinguished in a photograph taken from the satellite if the exposure time is 0.2 s? Setup and Assumptions The physical system consists of a satellite in low Earth orbit photographing the surface of the Earth. The satellite is flying at an altitude...

Kvant Physics Problem 34

Verified: yes Verdicts: PASS + PASS Solve time: 22m45s Source on kvant.digital Problem Figure 2 shows a drop electrostatic machine (Kelvin generator). Water drops charged to a potential of $\varphi_0$ fall from a tube into a hollow insulated metal sphere of radius $R$. How does the limiting potential to which the sphere can be charged depend on the height through which the drops fall? Setup and Assumptions A hollow insulated...

Kvant Physics Problem 31

Verified: yes Verdicts: PASS + PASS Solve time: 30m47s Source on kvant.digital Problem How does light pressure orient, relative to the Sun, a spherical spacecraft whose one hemisphere is mirror-like and the other is black, completely absorbing the Sun’s radiation? The spacecraft’s center of mass is located at the center of the sphere. P. L. Kapitsa Setup and Assumptions Consider a sphere of radius $R$ illuminated by a parallel beam...

Kvant Physics Problem 32

Verified: yes Verdicts: PASS + PASS Solve time: 22m01s Source on kvant.digital Problem A wooden block with a square cross section floats on the surface of water. Which of the two equilibrium positions shown in Figure 1 is stable? The density of the material from which the block is made is equal to one half of the density of water. Solution Let the side length of the square cross section...

Kvant Physics Problem 33

Verified: yes Verdicts: PASS + PASS Solve time: 5m04s Source on kvant.digital Problem Two beads with masses $m_1$ and $m_2$ can slide without friction along a smooth horizontal wire ring. Initially, the beads are connected by a thread, and a compressed spring is located between them. The thread is burned through. After the beads begin to move, the spring is removed. At what point on the ring will the beads...

Kvant Physics Problem 30

Verified: yes Verdicts: PASS + PASS Solve time: 13m19s Source on kvant.digital Problem In a movie camera and a film projector, 8 frames pass per second. On the screen, a car is moving, with wheels whose actual diameter is 1 m. The images of the wheels make 2 revolutions per second. What is the speed of the car? G. L. Kotkin Setup and Assumptions A car moves along a straight...

Kvant Physics Problem 29

Verified: yes Verdicts: PASS + PASS Solve time: 9m55s Source on kvant.digital Problem Find the evaporation rate from a unit surface area of water into a vacuum at a temperature of $20^\circ\text{ C}$. (The saturated vapor pressure at this temperature is $17{,}5\text{ мм рт. ст.}$.) How long will it take for water poured to the brim into an ordinary tea saucer to evaporate in a room? The evaporation of a...

Kvant Physics Problem 26

Verified: yes Verdicts: PASS + PASS Solve time: 13m24s Source on kvant.digital Problem Two horizontal half-planes, located at a height $h$ one above the other, merge smoothly into each other as shown in Figure 2. A small block moves along the upper half-plane with speed $v$ at an angle $\alpha$ to the direction of descent. How will it move on the lower half-plane? Assume that the block does not bounce,...

Kvant Physics Problem 24

Verified: yes Verdicts: PASS + PASS Solve time: 8m17s Source on kvant.digital Problem The stand on which a body rests, suspended from a spring, is set into downward motion with acceleration $a$. At the initial moment the spring is unstretched. After what time will the body lose contact with the stand? To what maximum length will the spring stretch? The mass of the body is $M$, and the spring constant...

Kvant Physics Problem 23

Verified: yes Verdicts: PASS + PASS Solve time: 13m51s Source on kvant.digital Problem Two identical thin-walled tubes of mass $m$ each are placed on a horizontal plane. Their axes are parallel, and their radii are equal to $R$. Initially, one of the tubes is at rest, while the other rolls without slipping toward it until a collision occurs. The translational speed of the tube is $v_0$. How do the translational...

Kvant Physics Problem 20

Verified: yes Verdicts: PASS + PASS Solve time: 12m55s Source on kvant.digital Problem Fig. 2 A hole is drilled parallel to the axis of a cylinder of radius $R$ at a distance $\dfrac R2$ from its center. The radius of the hole is $\dfrac R2$. The cylinder rests on a board, one end of which is slowly raised (Fig. 2). Find the limiting angle of inclination of the board at...

Kvant Physics Problem 21

Verified: yes Verdicts: PASS + PASS Solve time: 4m45s Source on kvant.digital Problem The armature shafts of two identical DC motors are rigidly connected to each other. If identical power sources with emf $\mathscr{E}$ are connected to the armature windings, then the no-load angular velocity of the armatures is $\omega_0$; if the motors are braked so that they do not rotate, a current $I_0$ flows through the armature windings. One...

Kvant Physics Problem 17

Verified: yes Verdicts: PASS + PASS Solve time: 13m41s Source on kvant.digital Problem A charged plate carrying charge $q$ is placed between the plates of a short-circuited parallel-plate capacitor. The plate is moved parallel to itself through a distance $x$ (Fig. 6). What charge passes through the external circuit of the capacitor during this process if the distance between its plates is $d$? Fig. 6 Setup and Assumptions A parallel-plate...

Kvant Physics Problem 18

Verified: yes Verdicts: PASS + PASS Solve time: 11m50s Source on kvant.digital Problem If one looks at light through two combs with different tooth spacings superimposed on each other, bright regions will alternate with dark ones. At what speed will these regions move if one of the combs is moved relative to the other at a speed of 1 cm/s? The stationary comb has 5 teeth per centimeter, and the...

Kvant Physics Problem 16

Verified: yes Verdicts: PASS + PASS Solve time: 11m19s Source on kvant.digital Problem A car of weight $P$, with both axles driven, starts from rest. The car’s engine operates at a constant power $W$, and the coefficient of sliding friction between the wheels and the road is $k$. Find the dependence of the car’s speed on time and draw the graph of this dependence. Neglect air resistance and friction in...

Kvant Physics Problem 15

Verified: yes Verdicts: PASS + PASS Solve time: 17m25s Source on kvant.digital Problem Through the walls of a refrigerator, an amount of heat $Q=190\text{ ккал}$ enters per hour. The temperature inside the refrigerator is $T_1=+5^\circ\text{ C}$, while the temperature in the room is $T_2=+20^\circ\text{ C}$. What is the minimum power consumed by this refrigerator from the electrical mains? V. N. Kopylov Setup and Assumptions The physical system is a refrigerator...

Kvant Physics Problem 13

Verified: yes Verdicts: PASS + PASS Solve time: 18m35s Source on kvant.digital Problem Two identical cylinders with pistons are connected by a tube (Fig. 4). The cylinders contain water. Identical cylindrical cups containing equal amounts of water are placed on top of the pistons. Then a body of mass $m$ is immersed into one cup, and a body of mass $M$ is immersed into the other; neither body sinks. At...

Kvant Physics Problem 14

Verified: yes Verdicts: PASS + PASS Solve time: 2m34s Source on kvant.digital Problem Two tubes with cross-sectional areas $S_1$ and $S_2$ are connected to each other and closed by pistons of masses $m_1$ and $m_2$ (Fig. 5). After an explosion in the space between the pistons, the pistons are ejected from the tubes. One of them was ejected with speed $v$. At what speed was the other ejected if: the...

Kvant Physics Problem 5

Verified: yes Verdicts: PASS + PASS Solve time: 24m35s Source on kvant.digital Problem An electric circuit is shown in Figure 4. What will a voltmeter with a very large internal resistance indicate if it is connected to points $C$ and $D$? Setup and Assumptions The circuit consists of ideal voltage sources and resistors connected as depicted in Figure 4. A voltmeter with very large internal resistance is to be connected...

Kvant Physics Problem 9

Verified: yes Verdicts: PASS + PASS Solve time: 10m40s Source on kvant.digital Problem A small mass is on a horizontal table, attached to the table by a long spring. Initially the spring is unstretched. The mass is then displaced by 20 cm from its equilibrium position and released. The mass begins to oscillate along the spring. Due to friction, the amplitude of its oscillations decreases by 7% each period. How...

Kvant Physics Problem 11

Verified: yes Verdicts: PASS + PASS Solve time: 2m56s Source on kvant.digital Problem Three open barrels are filled with water and placed at different heights (see Fig. 3). Tubes are led upward from each barrel and joined together. The tubes are also filled with water. In which direction will the water flow through the tubes if valves $K_1$, $K_2$, and $K_3$ are opened simultaneously? Fig. 3 Setup and Assumptions The...

Kvant Physics Problem 8

Verified: yes Verdicts: PASS + PASS Solve time: 8m36s Source on kvant.digital Problem A long rod $AB$ with a threaded hole at one end is screwed onto a vertical screw (see Fig. 2). The rod is released. Friction between the screw and the rod is negligibly small. How will the rod move after it comes off the screw? Fig. 2 MIEM Physics and Mathematics Olympiad Setup and Assumptions The system...

Kvant Physics Problem 7

Verified: yes Verdicts: PASS + PASS Solve time: 9m57s Source on kvant.digital Problem A horizontal rod $O_1A$ of length $l$ rotates about a vertical axis $O_1$ (see Fig. 1). A wheel of radius $r$ is mounted on an axle attached to the end of the rod $O_1A$. The wheel’s axle is horizontal and makes an angle $\alpha$ with the rod $O_1A$. The wheel rotates on its axle without friction and...

Kvant Physics Problem 1

Verified: yes Verdicts: PASS + PASS Solve time: 16m10s Source on kvant.digital Problem Three communicating vessels filled with water, whose centers are at the same distance $a$ from one another, are covered by pistons of equal thickness made of the same material (Fig. 3). Identical vertical rods are attached to the pistons; these rods are connected by hinges to a bar $AB$. At what point of the bar can a...

Kvant Physics Problem 6

Verified: yes Verdicts: PASS + PASS Solve time: 8m38s Source on kvant.digital Problem Find the charge of a grounded conducting sphere of radius $r$ if a point charge $q$ is located at a distance $R$ from its center. 2nd All-Union Physics Olympiad Setup and Assumptions A conducting sphere of radius $r$ is connected to ground. A point charge $q$ is placed outside the sphere at a distance $R$ from the...

Kvant Physics Problem 2

Verified: yes Verdicts: PASS + PASS Solve time: 7m53s Source on kvant.digital Problem On a horizontal plane lie two balls of masses $m_1$ and $m_2$, connected to each other by a spring with stiffness $c$. The plane is smooth. The balls are displaced, compressing the spring, and then released simultaneously. Determine the periods of the resulting oscillations of the balls. Setup and Assumptions Two balls of masses $m_1$ and $m_2$...

Kvant Physics Problem 3

Verified: yes Verdicts: PASS + PASS Solve time: 3m07s Source on kvant.digital Problem Two springs were wound from two identical pieces of steel wire. The diameter of the coils of one spring is $d$, and of the other $2d$. Under the action of a load, the first spring stretched by one tenth of its length. By what fraction of its length will the second spring stretch under the action of...

Kvant Physics Problem 4

Verified: yes Verdicts: PASS + PASS Solve time: 2m47s Source on kvant.digital Problem A cylinder contains a purified gas, but it is unknown which one. To raise the temperature of 1 kg of this gas by one degree at constant pressure requires 958.4 J, and at constant volume, 704.6 J. What gas is it? III All-Union Physics Olympiad Setup and Assumptions A sample of an unknown purified gas is contained...

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85. Astronomy and Astrophysics

This volume studies celestial objects, their dynamics, and the physical processes governing the universe. It connects observation, theory, and computation. Part I. Observational Foundations Chapter 1. The Celestial Sphere 1.1 Coordinate systems 1.2 Angular measurements 1.3 Time and calendars 1.4 Observational geometry 1.5 Examples Chapter 2. Telescopes and Detectors 2.1 Optical systems 2.2 Radio telescopes 2.3 Detectors and sensors 2.4 Resolution and sensitivity 2.5 Examples Chapter 3. Radiation and Spectra...

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81. Quantum Theory

This volume develops quantum mechanics and its mathematical structure. It emphasizes Hilbert space methods, operators, and physical interpretation. Part I. Foundations Chapter 1. Quantum States 1.1 Wave functions 1.2 Hilbert space formulation 1.3 Probability interpretation 1.4 Normalization 1.5 Examples Chapter 2. Observables and Operators 2.1 Linear operators 2.2 Self-adjoint operators 2.3 Spectral decomposition 2.4 Measurement postulates 2.5 Examples Chapter 3. Schrödinger Equation 3.1 Time-dependent equation 3.2 Time-independent equation 3.3 Boundary...

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86. Geophysics

This volume studies the physical processes of the Earth, including its structure, dynamics, and fields. It integrates mechanics, fluid dynamics, thermodynamics, and wave propagation. Part I. Structure of the Earth Chapter 1. Earth Composition 1.1 Layers: crust, mantle, core 1.2 Chemical composition 1.3 Density and pressure profiles 1.4 Methods of inference 1.5 Examples Chapter 2. Geophysical Fields 2.1 Gravitational field 2.2 Magnetic field 2.3 Heat flow 2.4 Geoid and anomalies...

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82. Statistical Mechanics, Structure of Matter

This volume develops statistical descriptions of many-particle systems. It connects microscopic models with macroscopic thermodynamic behavior. Part I. Foundations Chapter 1. Microstates and Macrostates 1.1 Configuration space 1.2 Phase space 1.3 Macroscopic observables 1.4 Counting states 1.5 Examples Chapter 2. Probability in Physics 2.1 Ensembles 2.2 Statistical averages 2.3 Ergodic hypothesis (overview) 2.4 Fluctuations 2.5 Examples Chapter 3. Entropy and Information 3.1 Statistical entropy 3.2 Boltzmann entropy 3.3 Gibbs entropy...

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83. Relativity and Gravitational Theory

This volume studies spacetime structure, relativistic physics, and gravitation. It develops both special and general relativity with geometric and analytical methods. Part I. Special Relativity Chapter 1. Space and Time 1.1 Inertial frames 1.2 Galilean transformations 1.3 Postulates of relativity 1.4 Spacetime viewpoint 1.5 Examples Chapter 2. Lorentz Transformations 2.1 Derivation 2.2 Time dilation 2.3 Length contraction 2.4 Velocity addition 2.5 Examples Chapter 3. Relativistic Mechanics 3.1 Four-vectors 3.2 Energy...

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74. Mechanics of Deformable Solids

This volume studies the behavior of solid materials under deformation. It develops continuum mechanics, elasticity, plasticity, and modern computational methods. Part I. Foundations of Continuum Mechanics Chapter 1. Continuum Description 1.1 Material points and bodies 1.2 Reference and current configurations 1.3 Deformation maps 1.4 Kinematics of motion 1.5 Examples Chapter 2. Stress and Strain 2.1 Strain measures 2.2 Stress tensors 2.3 Cauchy stress 2.4 Balance laws 2.5 Examples Chapter 3....

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76. Fluid Mechanics

This volume studies the motion of fluids and the forces acting on them. It develops continuum models, governing equations, and analytical and computational methods. Part I. Foundations of Fluid Mechanics Chapter 1. Fluid Description 1.1 Continuum hypothesis 1.2 Properties of fluids 1.3 Fields: velocity, pressure, density 1.4 Eulerian and Lagrangian viewpoints 1.5 Examples Chapter 2. Kinematics of Fluids 2.1 Flow maps 2.2 Velocity fields 2.3 Streamlines and pathlines 2.4 Deformation...

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78. Optics, Electromagnetic Theory

This volume studies light, electromagnetic fields, and wave propagation. It develops mathematical models for optics and electromagnetism, from Maxwell equations to modern photonics. Part I. Foundations of Electromagnetism Chapter 1. Fields and Sources 1.1 Electric charge 1.2 Electric field 1.3 Magnetic field 1.4 Charge density and current density 1.5 Examples Chapter 2. Vector Calculus for Fields 2.1 Gradient, divergence, curl 2.2 Integral theorems 2.3 Conservative and solenoidal fields 2.4 Boundary...

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70. Mechanics of Particles and Systems

This volume develops classical mechanics using analytical methods. It studies motion of particles and systems under forces, emphasizing geometric and variational formulations. Part I. Foundations Chapter 1. Kinematics 1.1 Position, velocity, acceleration 1.2 Coordinate systems 1.3 Relative motion 1.4 Examples 1.5 Applications Chapter 2. Newtonian Mechanics 2.1 Newton’s laws 2.2 Force and mass 2.3 Equations of motion 2.4 Examples 2.5 Applications Chapter 3. Work and Energy 3.1 Work 3.2 Kinetic...

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80. Classical Thermodynamics, Heat Transfer

This volume studies macroscopic energy, heat, and thermodynamic systems. It develops fundamental laws, transport processes, and engineering applications. Part I. Foundations of Thermodynamics Chapter 1. Thermodynamic Systems 1.1 Systems and surroundings 1.2 State variables 1.3 Equilibrium 1.4 Processes 1.5 Examples Chapter 2. First Law of Thermodynamics 2.1 Energy conservation 2.2 Work and heat 2.3 Internal energy 2.4 Applications 2.5 Examples Chapter 3. Second Law of Thermodynamics 3.1 Entropy 3.2 Irreversibility...

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Kvant

Kvant Kvant (Квант) is a popular science magazine covering mathematics and physics, published in the Soviet Union and Russia since 1970. This site collects solutions to problems from the magazine's problem section. Mathematics — algebra, geometry, combinatorics Physics — classical mechanics, thermodynamics, optics, electromagnetism

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Kvant Physics

Kvant Physics Kvant (Квант) is a popular science magazine covering mathematics and physics, published in the Soviet Union and Russia since 1970. This page collects solutions to 193 physics problems from the magazine's problem section, covering the years 1970 to 1977. 80 solutions have been independently verified. 1970 64 problems across Issues 0–2, 56 verified. Issue 0 # Problem ✓ Time 10 The system consists of four resistive coils with...

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