Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
Already signed up?
New to KöMaL?

KöMaL Problems in Physics, May 2022

Please read the rules of the competition.


Show/hide problems of signs:


Problems with sign 'M'

Deadline expired on June 15, 2022.


M. 414. Measure the coefficient of kinetic friction between several sheets of sandpaper with different grit sizes and a wooden block.

(6 pont)

statistics


Problems with sign 'G'

Deadline expired on June 15, 2022.


G. 781. Boil water in a large pot on the stove. Put some cool water in a thin-walled glass, then immerse the glass of water into the boiling water so that it does not touch the walls of the pot. Will the water in the glass boil if we wait for a long enough time?

(3 pont)

solution (in Hungarian), statistics


G. 782. A bicycle is moving uniformly along a horizontal path at a speed of 3 m/s. Its wheels have a diameter of 70 cm. Choose an arbitrary point on the circumference of the wheel and at different positions of the wheel draw the velocity vectors and the acceleration vectors of this point starting from one common point for each quantity, that is draw the velocity and acceleration hodographs.

(4 pont)

solution (in Hungarian), statistics


G. 783. There is a point-like light source at the centre of a uniform glass ball of radius \(\displaystyle R\) and of refractive index \(\displaystyle n\). The sphere is observed from the outside. Where do we see the image of the light source?

(3 pont)

solution (in Hungarian), statistics


G. 784. The figure shows a whole range of simple machines. Friction and the masses of pulleys and levels are negligible. Into which direction will the lowermost object start moving?

(4 pont)

solution (in Hungarian), statistics


Problems with sign 'P'

Deadline expired on June 15, 2022.


P. 5409. The figure shows a whole range of simple machines. Friction and the masses of pulleys and levels are negligible. What are the values of the tension in the threads?

(4 pont)

solution (in Hungarian), statistics


P. 5410. The peregrine falcon can travel long distances without flapping its wing. Doing so, its movement has two parts. In the first part, it circles with its wings extended and rises in an upward flowing column of warm air (thermals) at a vertical speed of \(\displaystyle v_1\). In the second part, it leaves the thermal at an angle of \(\displaystyle \alpha\) with respect to the horizontal and glides at a constant speed to the next thermal at a distance of \(\displaystyle L\). The glide speed \(\displaystyle v_2\) is approximately directly proportional to the sine of the angle \(\displaystyle \alpha\) (the direction of glide with the horizontal): \(\displaystyle v_2=k \sin\alpha\), where \(\displaystyle k\) is a known constant.

\(\displaystyle a)\) To what minimum height must a peregrine rise in the thermal so that the time of its rising and gliding motion should be the shortest possible?

\(\displaystyle b)\) At least how much time is needed for the peregrine to move from the bottom of a thermal to the bottom of the next thermal?

\(\displaystyle c)\) Determine the glide angle which belongs to the motion with the optimal flight time.

Data: \(\displaystyle v_1=2~\frac{\mathrm{m}}{\mathrm{s}}\), \(\displaystyle k=10~\frac{\mathrm{m}}{\mathrm{s}}\), \(\displaystyle L=2\) km.

(5 pont)

solution (in Hungarian), statistics


P. 5411. A satellite orbits the Earth in an elliptical orbit of numerical eccentricity \(\displaystyle c/a=e\), with a period of \(\displaystyle T\). How long does it take for the satellite to go from point \(\displaystyle A\) to point \(\displaystyle B\), shown in the figure?

(4 pont)

solution (in Hungarian), statistics


P. 5412. If a gas is cooled (at constant pressure), then at a sufficiently low temperature the gas will usually liquefy (condense). However, this only happens over a certain pressure range. The figure shows the ``phase diagram'' of carbon dioxide. What are the values of the minimum and the maximum pressure at which this condensation can occur as described above? What happens if cooling is carried out at pressures higher or lower than this range?

(3 pont)

solution (in Hungarian), statistics


P. 5413. A converging lens with a focal length of 20 cm is placed on a convex spherical mirror as shown in the figure. What should the radius of curvature of the mirror be in order that a vertical parallel beam of light incident on the lens remain parallel after reflection from the system?

(4 pont)

solution (in Hungarian), statistics


P. 5414. We formed a circle of radius \(\displaystyle R\) from a piece of metal wire and from the same wire we made one of the diameters of the circle as well. What should the length of the arcs \(\displaystyle AB = AC\) be in order that the equivalent resistance between points \(\displaystyle A\) and \(\displaystyle B\) be the same as the equivalent resistance between points \(\displaystyle B\) and \(\displaystyle C\)?

(4 pont)

solution (in Hungarian), statistics


P. 5415. From a piece of wire of negligible resistance a V shaped figure was bent. The wire has no insulation, the angle between the two parts of the V is \(\displaystyle \alpha= 45^\circ\). It is placed horizontally into a magnetic field whose induction \(\displaystyle \boldsymbol B\) is perpendicular to the plane of the wire. The magnitude of this induction \(\displaystyle \boldsymbol B\) changes with time according to \(\displaystyle B(t)=B_0/t_0\cdot t\), where \(\displaystyle B_0\) and \(\displaystyle t_0\) are known constants. A metal rod, also without insulation, is placed onto the V shaped wire, initially it is fixed, as shown in the figure. The resistance of a unit length of the rod is \(\displaystyle r\).

\(\displaystyle a)\) How much heat is produced in the metal rod in a time of \(\displaystyle t_0\)?

\(\displaystyle b)\) At time \(\displaystyle t_0\) from the moment of switching on (time \(\displaystyle t=0\)), the change in magnetic induction ceases. At this instant we begin to move the metal rod (which was fixed till this time) in the horizontal plane and perpendicularly to the rod at a constant speed of \(\displaystyle v_0\). What should this speed be in order that the value of the current in the rod should not change?

\(\displaystyle c)\) By what factor will the heat produced in the moving rod be greater than that produced in a static rod, if the rod is moved for a time of \(\displaystyle 2t_0\)?

(5 pont)

solution (in Hungarian), statistics


P. 5416. There are five electrons in a region of length 1.1 nm with respect to which the width and thickness of the region is negligibly small. The potential energy in this region is zero, and outside it is very big. We can neglect the interaction of the electrons with each other.

\(\displaystyle a)\) What is the minimum energy required to excite the electrons in the system?

\(\displaystyle b)\) What is the wavelength of the electromagnetic wave that can produce this excitation? Where is this electromagnetic wave in the spectrum?

(5 pont)

solution (in Hungarian), statistics


P. 5417. A small body of mass \(\displaystyle m\) is placed (but not fixed) onto the top of a thin cylindrical ring of radius \(\displaystyle R\) and of negligible mass being at rest on the horizontal ground. The system is displaced from its unstable equilibrium position. As the ring rolls faster and faster, the small body flies off somewhere.

\(\displaystyle a)\) What is the least value of the coefficient of static friction between the surfaces in contact if neither the small body on the ring nor the ring on the ground is skidding during the motion?

\(\displaystyle b)\) Where will the small object hit the ground?

(6 pont)

solution (in Hungarian), statistics


Upload your solutions above.