KöMaL Problems in Physics, November 2024
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Problems with sign 'M'Deadline expired on December 16, 2024. |
M. 435. Attach a weight to the bottom of a household candle so that it floats in a vertical position when placed in water. Light the candle and then measure how the length of the candle submerged in the water varies as a function of its total instantaneous length. Using the measured data, determine the density of the material of the candle.
(6 pont)
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Problems with sign 'G'Deadline expired on December 16, 2024. |
G. 865. A tanker truck delivered milk from Hódmezővásárhely to Debrecen, 200 km away, on a weekly basis. On one of the trips, the driver noticed halfway through the trip that milk was dripping through a small hole in the tank to the road. When he had been on the road for 2 hours, \(\displaystyle 1.2\) litre of milk was missing from the tank. He then increased the speed of the truck in order to get the milk to its destination as quickly as possible. On arrival in Debrecen, he was able to deliver a total of 2 litres less milk than he had originally set off with.
a) What was the average speed at which the truck covered the 200 km distance?
b) In metres what was the distance between two adjacent milk drops on the road on the first and on the second part of the trip, if the volume of each drop is \(\displaystyle 0.2\,\mathrm{ml}\)?
The drops were detaching from the rim of the hole in the tank at regular time intervals.
(4 pont)
solution (in Hungarian), statistics
G. 866. A thin, double-bent tube is made from three long enough pieces. The middle section is horizontal, the first section makes an angle of \(\displaystyle 30^\circ\) with the horizontal, and the third section makes an angle of \(\displaystyle 45^\circ\) with the horizontal. The sections lie in the same vertical plane, with short, smooth bends connecting each section. At the left end of the tube, a tap seals off enough air, such that there is a mercury thread, of length \(\displaystyle L\), at rest in the bottom of the left section of the tube (as shown in the figure). The tap is suddenly opened, causing the mercury thread to move without friction. Initially, the bottom of the mercury thread is at the junction of the first and second tube sections.
What is the maximum fraction of the mercury thread that enters the third section of the tube?
(4 pont)
solution (in Hungarian), statistics
G. 867. A calorimeter contains a mixture of ice and water. The calorimeter is heated at a constant power and the temperature of its contents is measured and plotted against time (see the figure).
At the end of the measurement, there was 850 ml of water in the calorimeter. Determine the heating power of the calorimeter and the initial amount of ice in it.
(4 pont)
solution (in Hungarian), statistics
G. 868. The resistance of the edges of the rhombus shown in the figure is \(\displaystyle R\), and the resistance of the diagonal is \(\displaystyle xR\), where \(\displaystyle x\ge 0\) is a variable parameter. A voltage supply \(\displaystyle U\) is connected across two arbitrary chosen vertices. For each possible case, calculate the total dissipated power in the five resistors as a function of the parameter \(\displaystyle x\).
(4 pont)
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Problems with sign 'P'Deadline expired on December 16, 2024. |
P. 5598. According to the rumours, Tom McPoint, the famous acrobat, will perform his legendary stunt in Hódmezővásárhely in the winter of 2024. He will walk along a wooden plank waving to the audience such that one end of the plank lies on the top of a tower block without fastening it to the building, while the other end extends over the street. On the day before the performance, the acrobat's assistants sprinkle water on the horizontal roof of the chosen building to form a flat layer of ice. Then, at the start of the show, the 6 m long plank is pushed out over the edge of the roof so that it just will not fall over when Tom steps on its end which is on the roof (See the figure).
a) What is the maximum length of the part of the plank that extends above the street at the beginning of the stunt, if Tom's mass is 60 kg and that of the plank is 40 kg?
b) At the end of the stunt, how far will Tom be from the edge of the roof if he walks along the plank in 20 s at a constant speed?
The friction between the plank and the ice is negligible and Tom's feet do not slip on the plank during the stunt.
(4 pont)
solution (in Hungarian), statistics
P. 5599. The shape of a thin metal cup is a paraboloid of revolution, and it has a vertical axis of symmetry and is attached to a horizontal supporting plate at its apex \(\displaystyle O\). The distance between point \(\displaystyle P\) and the plate is \(\displaystyle h_0\). A point-like body is dropped into the thin tube placed inside the cup at its end point \(\displaystyle Q\), and it leaves the tube at point \(\displaystyle P\). The difference in height between the points \(\displaystyle Q\) and \(\displaystyle P\) is \(\displaystyle H\). (See the figure.)
The small body flies out of the tube in the direction of a horizontal straight line, which lies in the plane tangent to the surface of the cup. Within what limits will the distance \(\displaystyle h\) of the small body from the cup vary during the rest of the motion? (Friction is negligible everywhere.)
(5 pont)
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P. 5600. A twin ladder is standing on a frictionless, horizontal ground such that initially the angle between the ground and each side rail is \(\displaystyle \varphi_0\). The spreaders between the front and rear rails prevents them from sliding apart. A man of mass \(\displaystyle M\) sits on top of the ladder. The spreaders break and the ladder opens. At what speed and acceleration does the man reach the ground? What is the ratio of these two quantities to the speed and acceleration, respectively, when the man falls freely from the same height? Investigate the two limiting cases when \(\displaystyle M\to 0\) and when \(\displaystyle M\to\infty\). Consider the two sides of the ladder as uniform rods of mass \(\displaystyle m\) and length \(\displaystyle \ell\), and the man as a point. The two sides of the ladder are held together at the top by a frictionless hinge. (See the figure).
(5 pont)
solution (in Hungarian), statistics
P. 5601. A horizontal membrane vibrates vertically, harmonically, at a frequency of 500 Hz. Fine sand is sprinkled on the membrane and the sand particles are seen to rise into the air to a height of 3 mm above the equilibrium position of the diaphragm. What is the amplitude of the vibration of the membrane?
Consider the collision of the sand particles on the membrane as totally inelastic.
(5 pont)
solution (in Hungarian), statistics
P. 5602. A piston in a horizontal cylinder seals off helium gas. The gas is heated by a 10 W heating element. The gas pushes the piston outwards uniformly. The cross section of the cylinder is \(\displaystyle 10~\mathrm{cm}^2\). The external air pressure is \(\displaystyle 100~\mathrm{kPa}\). What is the speed of the piston?
(4 pont)
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P. 5603. We are observing a small starfish lying on a sandy beach, standing on all fours, from exactly above, at a height of \(\displaystyle 50~\mathrm{cm}\). The tide slowly causes the water level to rise, and \(\displaystyle 40~\mathrm{cm}\) of water covers the animal.
a) How many times larger, i.e., at how many times greater angle do we see the starfish than at the beginning if we do not change our body position?
b) How many times larger do we see the starfish if at high tide we wear glasses which have a power of \(\displaystyle 0.5\) dioptres? Where is the image formed by the lens in this case?
Ignore the distance between the glasses and your eyes.
(4 pont)
solution (in Hungarian), statistics
P. 5604. The electromotive force (emf) of a flat-pack lantern battery is \(\displaystyle U_0=4.5~\mathrm{V}\), its internal resistance is \(\displaystyle R_\mathrm{i}=1.5~\Omega\). We have two alike incandescent bulbs, whose \(\displaystyle I(U)\) characteristics is shown in the figure.
a) How does the terminal voltage of the battery depend on the current? Graph the \(\displaystyle I(U)\) characteristics of the battery.
b) A bulb is connected to the battery. Using the \(\displaystyle I(U)\) graph determine the voltage across the bulb and the current flowing through the bulb (the so-called operating point).
c) The two bulbs are connected
i) in parallel
ii) in series
and then they are connected to the battery. Plot the \(\displaystyle I(U)\) characteristics of the two bulbs (together) when they are connected in series and in parallel, and then determine the new operating point in both cases. Then determine in both cases the voltage across one bulb, and the current flowing through one bulb.
(5 pont)
solution (in Hungarian), statistics
P. 5605. A moving argon atom collides elastically with a neon atom, initially at rest. What is the maximum angle of deflection of the argon atom?
(5 pont)
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P. 5606. A spacecraft with a total mass \(\displaystyle m_0\) is travelling in interplanetary space at a speed of \(\displaystyle v\) without the presence of external forces. To change the direction of its motion, it suddenly turns on its propulsion system, from which propellant flows out at a constant (relative) speed \(\displaystyle u\), all the time in a direction perpendicular to the instantaneous speed of the spacecraft. How much does the mass of the spacecraft decrease until its velocity vector turns \(\displaystyle 90^\circ\) relative to the original direction?
(6 pont)
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