KöMaL Problems in Physics, February 2024
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Problems with sign 'M'Deadline expired on March 18, 2024. |
M. 429. Make a hole on the cap of a one-and-a-half litre cylindrical glass bottle. Fill the bottle about half full with water and screw the cap back on. Turn the bottle upside down and measure how much water comes out. Also measure the height of the water in the bottle when the spout stops. Using the results of the measurement, determine the air pressure when the measurement was done.
(6 pont)
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Problems with sign 'G'Deadline expired on March 18, 2024. |
G. 841. If you tie 20-30 wooden skewers tightly together with a rubber band, why does the bundle take on a nearly cylindrical shape?
(3 pont)
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G. 842. Space researchers hope to build a human moonbase on the Moon soon. Imagine the astronauts celebrating the one-year anniversary of the moonbase with a special fireworks display. A projectile is fired at an angle of \(\displaystyle 45^\circ\), which explodes at an altitude of 100 m at the top of its path into tiny fragments that fly apart at \(\displaystyle 10~\text{m}/\text{s}\) relative to the projectile's centre of mass, glowing brightly for a long time. With respect to the time and position of the launch, when and where do the first and last brightly glowing fragments hit the ground?
(4 pont)
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G. 843. \(\displaystyle 12~\text{cm}\) above a plane mirror, inclined at an angle of \(\displaystyle 45^\circ\), there is a luminous horizontal arrow of length \(\displaystyle 3~\text{cm}\). Find the size and position of the image produced by a converging lens of focal length \(\displaystyle 20~\text{cm}\), which is at a distance of \(\displaystyle 18~\text{cm}\) from the mirror.
(4 pont)
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G. 844. In a steam bath, people are essentially sitting in a cloud of temperature \(\displaystyle 42~{}^{\circ}\text{C}\). In contrast, a \(\displaystyle 95~{}^{\circ}\text{C}\) Finnish sauna has a relative humidity of just 12%. In which of them is there a higher absolute humidity?
(4 pont)
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Problems with sign 'P'Deadline expired on March 18, 2024. |
P. 5544. In the case of a special potato cannon, the initial velocity of a potato fired at an angle of \(\displaystyle \alpha\), measured from the horizontal is \(\displaystyle v_0=(20~\text{m}/\text{s})\cdot\cos\alpha\). The drag force exerted on the projectile can be neglected. How far can you shoot with this potato gun on the horizontal ground?
(4 pont)
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P. 5545. There is a mixture of helium and hydrogen gas in a cylinder, sealed by an easily moveable piston. The mass of the mixture is \(\displaystyle 180~\text{g}\). At constant pressure. \(\displaystyle 156~\text{kJ}\) thermal energy is added to the gas. This causes the gas mixture to do \(\displaystyle 56~\text{kJ}\) of work. How many grams of hydrogen were in the cylinder? What is the temperature change of the gas mixture?
(4 pont)
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P. 5546. In the Eskimos' Palace of Wonders, there is an igloo of radius of \(\displaystyle R=3~\text{m}\) which is built on a circular ice ink, rotating at an angular speed of \(\displaystyle \omega=\pi/6~\text{s}^{-1}\) around a vertical axis. Two children are sitting in the rotating igloo in the position shown in the figure, at an angle of \(\displaystyle \varphi=120^\circ\) with respect to each other. The child sitting at \(\displaystyle A\) manages to launch the puck so that it arrives at the hand of the other child sitting at \(\displaystyle B\), in a time of \(\displaystyle t=2~\text{s}\) after the launch.
\(\displaystyle a)\) With respect to the igloo at what speed and in what direction did the child at \(\displaystyle A\) have to start the puck?
\(\displaystyle b)\) What distance does the puck approach the centre of the igloo as it moves?
(5 pont)
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P. 5547. A small wooden ball is attached to one end of a \(\displaystyle 30~\text{cm}\) long thread, and the free end of the thread is fixed to the bottom of a bucket at a distance of \(\displaystyle 20~\text{cm}\) from the centre. The bucket is filled with water and rotated around its axis of symmetry. (The water covers the ball during the motion.) What is the angular velocity at which the bucket must be spun so that after a long time the thread makes an angle of \(\displaystyle 30^\circ\) with the vertical?
(5 pont)
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P. 5548. A small, flat fridge magnet weighs \(\displaystyle G\). The magnet is placed on the vertical side of the refrigerator and pulled in some direction in a vertical plane perpendicular to the plane of the metal side. The minimum force to move the magnet vertically upwards is \(\displaystyle F_1\) and the force to move it vertically downwards is \(\displaystyle F_2\).
\(\displaystyle a)\) What is the coefficient of static friction between the side of the refrigerator and the magnet?
\(\displaystyle b)\) What is the force exerted by the metal side on the magnet, when the magnet is not pulled?
Data: \(\displaystyle G=0.10~\text{N}\), \(\displaystyle F_1=0.20~\text{N}\) and \(\displaystyle F_2=0.05~\text{N}\).
(See also the exercise numbered G. 702. in KöMal, issue 3, 2020.)
(5 pont)
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P. 5549. We bend a closed planar curve from a piece of thin wire of uniform mass distribution and of mass \(\displaystyle M\). The moment of inertia of the resulting frame is \(\displaystyle \Theta_0\) with respect to an axis which passes through the centre of mass and is perpendicular to its plane. The frame is then subjected to an experiment as shown in the figure (the centre of mass is denoted with the letter combination ``TKP"): the frame is suspended at various points and the period of its small amplitude oscillations in its own plane is measured. What is the minimum possible period of the oscillation?
(5 pont)
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P. 5550. A sphere-shaped lens of radius \(\displaystyle R\) and of refractive index \(\displaystyle n\) is illuminated with a beam of laser light. Into which direction should the beam of light be directed from a point on the principal axis at a distance \(\displaystyle d\) from the centre of the lens, in order that after it refracts on the lens, it crosses the principal axis of the lens also at a distance \(\displaystyle d\) from the centre on the other side of the lens? For a given refractive index, for which distance \(\displaystyle d\) is this possible? Calculate the angle of the direction of the beam described above using the data of \(\displaystyle d=2R\), \(\displaystyle n=3/2\).
(4 pont)
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P. 5551. An electron-positron pair is produced from a photon of energy \(\displaystyle 2~\text{MeV}\), when it passes next to a nucleus, which has a high atomic number. (The heavy nucleus gains only momentum, and absorbs almost no energy.) In the Wilson chamber, placed into magnetic field, both particles travel in the same plane, along circular arcs of radius \(\displaystyle 5~\text{cm}\). What is the magnitude of the magnetic induction vector?
(5 pont)
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P. 5552. Imagine that power \(\displaystyle P\) is delivered through the coaxial cable of length \(\displaystyle \ell\) as shown in the figure. The radius of the inner conductor of the cable, which has negligible resistance, is \(\displaystyle a\), and the radius of the thin-walled outer tube, which can be considered to have similarly negligible resistance, is \(\displaystyle b\). There is a vacuum both outside the cable and between the inner conductor and the outer tube, and a direct current flows through the cable.
\(\displaystyle a)\) What is the value of the current if there is no outward and inward force exerted on the outer tube?
\(\displaystyle b)\) At which end of the coaxial cable – left or right – is the generator (voltage supply) and at which end is the resistor (load)?
(6 pont)
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