KöMaL Problems in Physics, April 2025

Problems with sign 'M' Deadline expired on May 15, 2025.

---

M. 440. Fold a paper air-plane and measure the lift-to-drag ratio. Try to make it sink evenly over as long a distance as possible.

(6 pont)

statistics

Problems with sign 'G' Deadline expired on May 15, 2025.

---

G. 885. The refractive index of the core of an optical fibre is 1.6 and that of the cladding is 1.5. What can the maximum angle between a ray of light and the axis of the optical fibre be in order to produce total reflection inside the fibre? A light ray enters from air into the glass fibre at the centre of the cross sectional plane of the fibre, which is perpendicular to the symmetry axis of the fibre. What is the maximum angle of incidence of the light ray?

(3 pont)

solution (in Hungarian), statistics

---

G. 886. Two identical parallel plate capacitors are connected in parallel, as shown in the figure. They are charged to 200 V and then the voltage source is disconnected. Then the distance between the plates of one of the capacitors is doubled and that of the other is halved. (There's air between the plates.)

a) By what factor does the equivalent capacitance of the system change?

b) What will the voltage be?

c) By what percentage does the energy stored in each capacitor change?

(4 pont)

solution (in Hungarian), statistics

---

G. 887. A steel ball has a diameter of \(\displaystyle 4.160\,\mathrm{cm}\) at \(\displaystyle 0\,^\circ\mathrm{C}\), and a circular hole in an aluminium sheet, made by laser cutting technology, has a diameter of \(\displaystyle 4.150\,\mathrm{cm}\) at \(\displaystyle 0\,^\circ\mathrm{C}\). The coefficients of linear expansion are: \(\displaystyle \alpha_{\textrm{Al}}=2.4\cdot 10^{-5}\,\mathrm{\tfrac{1}{K}}\) and \(\displaystyle \alpha_{\textrm{steel}}=1.2\cdot 10^{-5}\,\mathrm{\tfrac{1}{K}}\). (Ignore the temperature dependence of the coefficients of thermal expansion.)

a) What is the temperature of the steel ball when it is just passing through the hole in the \(\displaystyle 0\,^\circ\mathrm{C}\) aluminium plate?

b) What is the temperature of the aluminium sheet when a steel ball of temperature \(\displaystyle 0\,^\circ\mathrm{C}\) can just fit through the hole?

c) What is the common temperature at which the hole and the ball have the same diameter?

(4 pont)

solution (in Hungarian), statistics

---

G. 888. In an amusement park inside a giant rotating cylinder, people cling to a slant surface attached to the cylinder's mantle. The radius of the cylinder is \(\displaystyle R=5\,\mathrm{m}\), the coefficient of static friction is \(\displaystyle \mu=0.25\), and the slant surface makes an angle of \(\displaystyle \vartheta=30^{\circ}\) with the vertical as shown in the figure.

a) What is the minimum and maximum angular speed of the cylinder so that people do not slide up or down?

b) What is the coefficient of friction such that people do not slip off at any small angular velocity?

c) What is the coefficient of friction such that people do not slide upwards at any angular velocity?

(4 pont)

solution (in Hungarian), statistics

Problems with sign 'P' Deadline expired on May 15, 2025.

---

P. 5643. A bucket of water of mass \(\displaystyle m\) is pulled up from a water well of depth \(\displaystyle h\), while the rope is wound evenly on the cylinder of mass \(\displaystyle M\). The bucket is released from the top. With what acceleration does the bucket fall? After how long and at what speed does the bucket hit the water? The cylinder has a uniform mass distribution, the bucket full of water can be considered as a point-like body, the mass of the rope can be neglected.

Data: \(\displaystyle m=11\,\mathrm{kg}\), \(\displaystyle M=8\,\mathrm{kg}\), \(\displaystyle h=5\,\mathrm{m}\).

(4 pont)

solution (in Hungarian), statistics

---

P. 5644. With what single mass \(\displaystyle m\) should the pulley with the two masses on the right in the figure be replaced so that the body with mass \(\displaystyle m_1\) moves in the same way as it did originally? Neglect the mass of the pulleys and friction.

(4 pont)

solution (in Hungarian), statistics

---

P. 5645. A motorcyclist drives at 36 km/h into a semi-circular ``reverse'' curve. The coefficient of static friction between the asphalt and the wheels is \(\displaystyle 0.58\). The motorcyclist keeps his vehicle (more precisely, the centre of gravity of the motorcycle and the biker) on a circular arc of radius 40 m, while increasing its speed uniformly.

a) At most by how many m/s can the motorcyclist increase his speed?

b) To what speed can the rider accelerate at the end of the turn?

c) How does the rider's angle with the vertical vary in the semi-circular turn?

(5 pont)

solution (in Hungarian), statistics

---

P. 5646. A space trip to Mars is planned. The spacecraft leaves the Earth and enters into an elliptical orbit that touches both planets' orbits. The spacecraft is at perihelion when it is launched and it is at aphelion when it arrives. The return journey follows a similar elliptical orbit. In both cases, departure requires waiting until the two planets are in the correct position. How long will it take to get there and back, and at least how much time will they spend on Mars? Consider the orbits as circles in the same plane, the orbital period of Mars is 687.0 Earth days.

(5 pont)

solution (in Hungarian), statistics

---

P. 5647. An electric water boiler heats up water from \(\displaystyle 20\,^\circ\mathrm{C}\) (room temperature) to \(\displaystyle 60\,^\circ\mathrm{C}\) in a few hours. If you then do not use hot water at all and disconnect the appliance from the mains, the temperature of the water will drop to \(\displaystyle 40\,^\circ\mathrm{C}\) in about 7 days. If we keep the water warm in the boiler but do not use it, how many days of electricity consumption will cost the same as heating up the room temperature water once? (It can be assumed that the heat dissipation of the boiler is proportional to the temperature difference between the water and the environment.)

(5 pont)

solution (in Hungarian), statistics

---

P. 5648. Consider \(\displaystyle N\) distinct points in the space. Connect each point to all the others with the same resistors of resistance \(\displaystyle R\). What is the equivalent resistance between any two points?

(5 pont)

solution (in Hungarian), statistics

---

P. 5649. Peter wears glasses with a power of \(\displaystyle -4\) dioptres and can see objects that are at least 25 cm away clearly. How far away can Peter see objects clearly when he takes off his glasses?

(4 pont)

solution (in Hungarian), statistics

---

P. 5650. There is a little demon in a sample of helium gas, which is at standard temperature and pressure. The little demon selects a cube with side length of 1 nm and counts from time to time the number of nuclei of atoms in the cube. What is the probability of finding zero, one or two nuclei in the cubic region?

(5 pont)

solution (in Hungarian), statistics

---

P. 5651. Each side of an insulated equilateral triangle-shaped sheet with uniform surface charge density \(\displaystyle \sigma\) has a length of \(\displaystyle \sqrt{2}a\). What is the value of the electric field strength at the point which is at a distance of \(\displaystyle a\) from each vertex?

(6 pont)

solution (in Hungarian), statistics