KöMaL Problems in Physics, September 2024
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Problems with sign 'M'Deadline expired on October 15, 2024. |
M. 433. Try to throw pebbles or stones of different mass as far as possible. Plot the average distance of the throws as a function of the mass. What is the mass of that stone which you can throw the furthest?
(6 pont)
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Problems with sign 'G'Deadline expired on October 15, 2024. |
G. 857. A small body is at rest on the top of a frictionless hill. If it is gently pushed, it will reach the bottom of the hill at a speed of \(\displaystyle 4~\mathrm{m}/\mathrm{s}\). At what speed would it reach the bottom of the slope if it were not started at rest but at an initial speed of \(\displaystyle 3~\mathrm{m}/\mathrm{s}\)?
(3 pont)
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G. 858. Three students (Ann, Bob and Cecily) are discussing where the Moon rises and sets in Hungary.
Ann: It rises on the western horizon and sets on the eastern horizon, just the opposite of the Sun.
Bob: It rises on the eastern horizon and sets on the western horizon, like the Sun.
Cecily: Depending on the lunar cycle, it sometimes rises on the eastern horizon and sometimes on the western horizon.
Who is right?
Note: Students use the term eastern (western) horizon to refer to the part of the horizon towards the east (towards the west) of the north-south line in the horizon.
(3 pont)
solution (in Hungarian), statistics
G. 859. From a circular plate of radius \(\displaystyle R=2r\), which has uniform mass distribution, another circular plate of radius \(\displaystyle r\) was cut off along the diameter \(\displaystyle AOB\) as shown in the figure. This smaller plate was laid on the bigger one on the other side of the diameter \(\displaystyle AOB\). Where is the centre of mass of the resulting object?
(4 pont)
solution (in Hungarian), statistics
G. 860. We have four identical \(\displaystyle 1.5~\mathrm{V}\) AA batteries. Two and two are connected in series, and then these series pairs are connected in parallel and a load of resistance \(\displaystyle R=10~\Omega\) is connected to the battery system. The internal resistance of each battery is \(\displaystyle r=1~\Omega\).
\(\displaystyle a)\) Draw the schematic figure of the circuit.
\(\displaystyle b)\) What is the current through the load?
\(\displaystyle c)\) Investigate the changes that occur if first two and two batteries are connected in parallel an then these parallel battery systems are connected in series.
(3 pont)
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Problems with sign 'P'Deadline expired on October 15, 2024. |
P. 5580. Two siblings, Anne and Brian are ``fighting'' with sock balls in the stairwell of their house, from a distance of \(\displaystyle 4.5~\mathrm{m}\) (measured horizontally) as shown in the figure. Anne throws the sock ball from the landing at a height of \(\displaystyle 4~\mathrm{m}\) with an initial horizontal velocity of \(\displaystyle 6~\mathrm{m}/\mathrm{s}\) and Brian throws the sock ball from a height of \(\displaystyle 1.5~\mathrm{m}\) with an initial velocity of \(\displaystyle 8~\mathrm{m}/\mathrm{s}\), making an angle of \(\displaystyle 45^\circ\) with the horizontal. Find the minimum distance between the two sock balls if the children threw them at the same time. (Neglect air resistance.)
(4 pont)
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P. 5581. Two rings of different radii are made from thin, flexible steel strips. Each ring slides on a horizontal tabletop, and there is a retarding (viscous-type) force exerted on each, which is proportional to the radius of the ring and to the instantaneous speed of the ring. If the smaller ring is started at \(\displaystyle v_0\), it travels a distance of \(\displaystyle L_0\) until it stops. Push one of the rings so that it collides with the initially stationary other ring such that its velocity before the collision is \(\displaystyle v\). What is the distance between the rings when both stop, if the collision is elastic and
\(\displaystyle a)\) along a straight line,
\(\displaystyle b)\) oblique?
The rings do not rotate either before or after the collisions, and their size is negligibly small with respect to the distances they travel after the collisions.
(5 pont)
solution (in Hungarian), statistics
P. 5582. Using the data from the Cassini spacecraft, a spectacular video (https://www.flickr.com/photos/kevinmgill/44583965185/) has been produced, which shows how Jupiter's moon Europa ``overtook'' the moon Io. This appears to contradict Kepler's laws, since Io, which is closer to Jupiter, has a higher orbital speed than the more distant moon Europa. The resolution to the paradox is the fact that the Cassini probe was also moving when the image was taken. At most how far away from Jupiter can a spacecraft orbiting the planet be, and in what direction should it move, to make such a strange ``role reversal''? Consider the moons and the spacecraft orbiting in nearly the same planes along circular paths.
(5 pont)
solution (in Hungarian), statistics
P. 5583. A thin, uniform-density rod of mass \(\displaystyle 3m\) and of length \(\displaystyle 3L\), shown in the figure, can rotate frictionlessly in a vertical plane about a horizontal axle, which is at a distance of \(\displaystyle L\) from one of the ends of the rod. The rod is held horizontal by a vertical thread attached to the other end.
\(\displaystyle a)\) What is the tension in the thread, and the force exerted on the rod by the axle, in this position?
\(\displaystyle b)\) After the thread is cut, what will the velocity of the lower endpoint of the rod be as the rod passes the vertical position?
\(\displaystyle c)\) What is the force exerted by the axle at this moment?
(4 pont)
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P. 5584. Place a closed loop of thread on a soap-film formed on a frame, which was dipped in soapy water. Then pierce the centre of the loop with a pin. The thread loop is stretched into a circle. Determine the tension in the circular thread as a function of the radius of the circle and the surface tension.
(4 pont)
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P. 5585. In a well-insulated cylinder sealed with a piston, a sample of initially 2 litres of neon gas at a temperature of \(\displaystyle 20~{}^\circ\mathrm{C}\) and at a pressure of \(\displaystyle 10^5~\mathrm{Pa}\) is compressed with a quick movement. What will be the temperature of the gas if 40 J of work is done during the compression?
(3 pont)
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P. 5586. A glass rod, which has a square cross section, is bent to the shape shown in the figure. A parallel beam of light is incident perpendicular to surface \(\displaystyle A\). What is the least ratio of \(\displaystyle R/d\), if all the light incident on surface \(\displaystyle A\) leaves the glass rod through surface \(\displaystyle B\)? The refractive index of the glass is \(\displaystyle {n=1.5}\).
(4 pont)
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P. 5587. Shine a laser beam on the Moon from the surface of the Earth. Rotate the laser source perpendicular to its axis with a motor at an angular speed of \(\displaystyle 100~\mathrm{min}^{-1}\). What is the speed at which the laser spot moves on the surface of the Moon? Is the result consistent with what we have learned in relativity? The effect of the Earth's atmosphere is negligible.
(4 pont)
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P. 5588. A thin insulating ring of radius \(\displaystyle R\), uniformly charged with \(\displaystyle {+Q}\), is placed in a horizontal plane. Along the diameter of the fixed ring (e.g. along a stretched fishing line), a point-like body with a charge \(\displaystyle {+q}\) and mass \(\displaystyle m\) can move frictionlessly. The point-like body is slightly displaced from its equilibrium position. What is the period of the small oscillations that occur?
(6 pont)
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