KöMaL Problems in Physics, March 2025
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Problems with sign 'M'Deadline expired on April 15, 2025. |
M. 439. Measure the emf and the internal resistance of an AA battery. How do these values change during the discharge of the battery?
(6 pont)
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Problems with sign 'G'Deadline expired on April 15, 2025. |
G. 881. A long 2 m wide conveyor belt moves at a constant speed of \(\displaystyle 0.5~\mathrm{m/s}\). A remote-controlled toy car travels from one end of the conveyor to the other such that it starts from rest, relative to the conveyor belt, moves perpendicular to the conveyor belt. It accelerates uniformly until it reaches the middle of the conveyor belt, where it has a speed of 1 m/s relative to the belt, then decelerates uniformly in the same way, and finally its speed relative to the conveyor belt decreases to zero.
a) How much does the conveyor belt carry the car forward as it crosses?
b) Draw a sketch of the path of the toy car relative to the ground.
(4 pont)
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G. 882. The system shown in the figure is in equilibrium, the rope in the middle is horizontal.
a) On graph paper construct the forces with a ruler and a protractor to determine the tension exerted in the ropes and the value of the unknown angle \(\displaystyle \vartheta\).
b) Estimate the percentage error of the constructed quantities.
(4 pont)
solution (in Hungarian), statistics
G. 883. The circuit shown in the figure contains a battery of internal resistance \(\displaystyle R_\mathrm{b}\) and of electromotive force of \(\displaystyle U_0\), and four resistors each of resistance \(\displaystyle R\). \(\displaystyle U_0=4.5~\mathrm{V}\), \(\displaystyle R_\mathrm{b}=3\,\Omega\), \(\displaystyle R=10\,\Omega\).
a) What is the power of the resistor marked with the arrow?
b) What is the voltage \(\displaystyle U_{AB}\)?
(3 pont)
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G. 884. A glass hemisphere of radius \(\displaystyle r\) and of refractive index \(\displaystyle 1.5\) is illuminated by a light ray at point \(\displaystyle A\) as shown in the figure. The refracted light ray reaches the plane part of the hemisphere at a distance \(\displaystyle y\) from the centre.
a) For which angle of incidence will \(\displaystyle y\) be equal to half of the radius?
b) What is the colour of the light ray if the wavelength of the light in the glass is \(\displaystyle 400~\mathrm{nm}\)?
(3 pont)
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Problems with sign 'P'Deadline expired on April 15, 2025. |
P. 5634. The coefficient of friction between a drawing board and a book resting on it is \(\displaystyle \mu\). One edge of the drawing board is slowly raised.
a) At which angle of inclination \(\displaystyle \alpha\) will the book begin to slip?
b) What is the acceleration of the sliding book when the angle of inclination of the board is \(\displaystyle 2\alpha\)?
c) What is the minimum horizontal acceleration at which the board with an angle of inclination of \(\displaystyle 2\alpha\) should be pushed in order to prevent the book from sliding?
Consider the coefficients of dynamic and static friction to be equal. Give the results in terms of \(\displaystyle \mu\) and \(\displaystyle g\).
(5 pont)
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P. 5635. A gymnast, performing a giant on a high bar, is just tipping over the vertical position at the top. When he reaches the bottom, the steel bar is visibly bent. Let us model the gymnast with a thin, heavy, uniform beam rotating around a horizontal axis (the steel bar). When the beam reaches the bottom from its top position, by what factor of its weight does it pull the bar? (Neglect friction, air resistance, and bending of the bar with respect to the length of the beam.)
(4 pont)
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P. 5636. A space probe has moved away from the Earth in the opposite direction to the Earth's orbital velocity \(\displaystyle v\approx 30~\mathrm{km/s}\), at a speed of \(\displaystyle nv\) relative to the Earth (\(\displaystyle n<1\)). Its further motion is only governed – to a good approximation – by the gravitational field of the Sun.
a) What is the major axis and the numerical eccentricity of the orbit of the space probe?
b) What can be \(\displaystyle n\) in order for the remnants of the probe to reach the surface of the Sun?
(See also the article entitled Mesterséges égitestek mozgásával kapcsolatos problémák és feladatok (Problems and exercises related to the motion of artificial celestial bodies).)
(5 pont)
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P. 5637. The heat capacity of the interior of a freezer is \(\displaystyle 4\cdot 10^5\mathrm{\tfrac{J}{K}}\), and we want to keep the temperature inside the freezer at \(\displaystyle (-18\pm 1)~{}^\circ\mathrm{C}\). The freezer is in a room of temperature \(\displaystyle 20~{}^\circ\mathrm{C}\). The refrigerator motor will start when the temperature inside reaches \(\displaystyle -17~{}^\circ\mathrm{C}\). After the motor has been running for 15 minutes, the temperature in the interior of the refrigerator reaches \(\displaystyle -19~{}^\circ\mathrm{C}\) again. At least what is the power of the freezer's motor? (Assume that the refrigerator is operating as an ideal Carnot heat pump, which is a reversed Carnot cycle.)
(4 pont)
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P. 5638. A point charge \(\displaystyle Q\) on the outside of a neutral metal sphere, at a distance of \(\displaystyle d\) from the centre of the sphere, is moved to a distance of \(\displaystyle 2d\). How much does the potential of the metal sphere change during this process?
(5 pont)
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P. 5639. In the circuit assembled according to the attached circuit diagram, the voltage supply has an electromotive force of 6 V and an internal resistance of \(\displaystyle 2\,\Omega\). The inductance of the ideal coil is \(\displaystyle 1.5~\mathrm{H}\) and the resistance of resistor \(\displaystyle R\) is \(\displaystyle 1000\,\Omega\). Initially, the switch is closed.
a) How much charge flows through resistor \(\displaystyle R\) after the switch is opened?
b) How much heat is dissipated in resistor \(\displaystyle R\) during this time?
(5 pont)
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P. 5640. In Las Palmas, the largest city in the Canary Islands, there is a unique exhibition in Europe that shows the world's aquatic life. One of the attractions of the exhibition is a 400 cubic metre vertical cylindrical marine aquarium, which is serviced by divers. Looking horizontally around the aquarium, how much of the aquarium wall is visible to a diver at a distance \(\displaystyle d\) from the axis of symmetry of the cylinder of radius \(\displaystyle R\)? The refractive index of the water is \(\displaystyle n\).
(5 pont)
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P. 5641. Positively charged pions are unstable particles with a (rest) mass of \(\displaystyle M=139.57\tfrac{\mathrm{MeV}}{c^2}\). Their decay most often results in a muon of mass \(\displaystyle m=105.66\,\tfrac{\mathrm{MeV}}{c^2}\) and a neutrino of mass zero (or negligibly small). One of the decays produced a muon with negligibly small momentum, which can be considered stationary. What was the kinetic energy and speed of the pion before the decay?
(5 pont)
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P. 5642. As early as 1679, Newton suggested that the rotation of the Earth could be proved by a mechanical experiment such that the eastward deflection of a small ball in free fall is measured. Hooke, who was secretary of the Royal Society in Newton's time and also an excellent experimenter, made such measurements in 1680. In a sufficiently wealthy and ambitious equatorial country, the experiment, which was proposed almost 350 years ago, was to be repeated by modern equipment. The plan is to drop tiny steel balls onto a wax plate in a 200 m high vertical tube, from which air was sucked out (in a vacuum tower). Determine the magnitude of the deflection by calculating
a) in an inertial frame of reference,
b) in the coordinate system rotating (accelerating) with the Earth.
(6 pont)
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