Nordic-Baltic Physics Olympiad 2018 of time. Write down an equality relating these bars is stationary in time (changes very slowly). periodic array of stripes; cross-section of such a
three vectors to each other. sheet is shown in figure. The refraction index
i) (2 points) Sketch the direction of the wind velocity at the points A and B. of the material from which the sheet is made
1. GRAVITATIONAL RACING (11 points) — iii) (2 points) Prove that the total angular momentum of this system is zero. n ≈ 1.47.
Máté Vigh and Jaan Kalda. While in generic
case, the dynamics of three gravitationally interacting bodies is complicated and chaotic, there ii) (2.5 points) Estimate the magnitude of the
are special cases when the dynamics is regular. wind velocity at point A. Make use of the fact
In particular, there are cases when bodies perform periodic motion. The simplest case of such iv) (2 points) Construct the positions of the that at point A, the lines of constant pressure
periodic motion is when all three bodies, positioned at the vertices of an equilateral triangle, points O₂ and O₃ (use the figure on the separate are almost straight. The density of air near the
rotate as a rigid body. In what follows we consider a more complicated periodic motion. sheet). Motivate your construction. ground ρ ≈ 1.23 kg/m³. i) (4 points) Determine the pitch d of the Fresnel
Relatively recently ¹, it was discovered that v) (2 points) Construct the positions of the points prism (see figure for the definition of the pitch).
three equal point masses can move periodically P₂ and P₃ (use the figure on the separate sheet). iii) (2.5 points) Estimate the magnitude of the
along a common 8-shaped trajectory shown in Find two independent ways of construction and wind velocity at point B. ii) (4 points) Determine the angle β of the prism.
the figure (arrow denotes the direction of motion). This figure is based on a computer simulation and has a correct shape. If needed, you can motivate your methods. Hint. When an object of iii) (4 points) Assuming that in the range of visible light, the refraction index n ≈ n(λ) of the
measure distances from the enlarged version of mass m (for example a slab of Fresnel prism material is a linear function of
the figure (on a separate sheet) using a ruler. vi) (2 points) Find the ratio of the speed of the air) is moving with velocity v wavelength λ, determine the chromatic dispersion dn/dλ.
bodies at point O to the speed at point P. in a frame of reference rotating with angular velocity Ω, it
2. SPEED CAMERA (6 points) — Mihkel feels a fictitious force called the
Kree. In this problem, we analyse the working principle of a speed camera. The transmitter of the speed camera emits an electromagnetic Coriolis force given by F_C/m ≈
wave of frequency f₀ ≈ 24 GHz having waveform cos(2πf₀t). The wave gets reflected from 2vΩ sin φ, where the angle φ and the directions
an approaching car moving with speed v. The are indicated in the figure below.
reflected wave is recorded by the receiver of the
speed camera.
Let us enumerate the three bodies with numbers 1, 2, and 3, according to the order in which i) (2 points) Express the frequency f₁ of the reflected wave.
they pass the leftmost point P shown in the figure. Let O₂ and O₃ denote the positions of
the bodies 2 and 3, respectively at that moment ii) (2 points) In the speed camera, the received
when the body 1 is passing the middle point O. waveform is multiplied with the original emitted waveform. Express all the frequency components present in the multiplied signal.
Similarly, let P₂ and P₃ denote the positions of
the bodies 2 and 3, respectively at that moment iii) (2 points) Given the lowest frequency component f_low ≈ 4.8 kHz present in the multiplied 4. FRESNEL PRISM (12 points) — Eero
when the body 1 is passing the leftmost point P. signal, calculate the speed of the car v. Note: the Uustalu and Jaan Kalda. Equipment: a sheet of
Let T denote the full period of motion of each speed of light c ≈ 3.0×10⁸ m/s and the following Fresnel prism, a sheet with purple and magenta
of the bodies along this 8-shaped trajectory. trigonometric identity might turn to be useful: stripes (see separate sheet), a piece of cardboard
i) (2 points) Express the following travel times for cos β cos γ ≈ ½[cos(β + γ) + cos(β - γ)]. paper (can be used as a screen), ruler, measuring tape, stand, green laser (λ₀ ≈ 532 nm). NB!
one of the bodies: (a) from O₂ to O; (b) from O₃ to P₂. Avoid looking into direct or reflected laser light,
ii) (1 point) Let v1, v2, and v3 denote the velocities of the three bodies at a certain moment 3. WEATHER FORECAST (7 points) — Johan it my damage your eyes! The intensity maximum
of time. Runeson. The map shown on a separate page for the scattered light from magenta stripes is
¹ Cristopher Moore, Phys. Rev. Lett. 70, 3675 (1993) shows isobars at a constant height close to sea λ_m ≈ 630 nm, and from cyan stripes — λ_c ≈
level. You may assume that the pattern of iso- 495 nm.
5. MAGNETIC BILLIARD (9 points) — Jaan m, and q)? i) (2 points) Draw a phasor diagram for this circuit showing the voltage vectors between the following nodes: V_AD, V_DB, V_AB, V_AE, V_EB, V_AF Fresnel prism is a transparent sheet with a
Kalda. Consider two absolutely elastic dielectric balls of radius r and mass m one of which 6. CUBE (5 points) — Taavet Kalda. A laser and V_FB.
carries isotropically distributed charge +q, and pointer of power P is directed at a glass cube.
the other — -q. There is so strong homogeneous magnetic field B, parallel to the axis z, that The refractive index of the cube is n. The surface ii) (3 points) The voltages between the nodes D, the submarine creates turbulent water inside the
electrostatic interaction of the two charges can of the cube has anti-reflective coating so that E, and F are known to have the following values: submarine; what is the total kinetic energy of
be neglected; neglect also gravity and friction there is no partial reflection when light travels V_DE ≈ 7V, V_DF ≈ 15V, and V_EF ≈ 20V; what this water turbulence (which later ends up being
forces. The first ball (negatively charged) moves from one medium to the other. The speed of is the value of the input voltage V₀? dissipated as heat), once the inflow has stopped
with speed v and collides with the second ball light is c. due to equalized pressures?
which had been resting at the origin. The collision is central, and immediately before the impact, the velocity of the first ball was parallel to i) (3 points) What is the maximum force by 8. AIR IN A SUBMARINE (6 points) — Johan 9. BLACK BOX (??? points) — Jaan Kalda, Mihkel Heidelberg. The black box has three terminal
the x-axis. which the laser pointer can push the cube if the Runeson, Jaan Kalda. A submarine of unknown wires: "blue", "black" and "white", and contains
laser needs to be kept parallel to one of the faces nationality is travelling near the bottom of the in a star configuration: a battery, a capacitor, an
i) (1 point) What is the speed of the second ball of the cube (this means that the laser beam can Baltic sea, at the depth of h ≈ 300 m. Its interior inductor in series with a diode. You may consider the diode to be "ideal" — it conducts current perfectly one way and not at all the other
immediately after the collision? propagate only in a two-dimensional plane)? is one big room of volume V ≈ 10 m³ filled with
ii) (2 points) Sketch the trajectories of the centres air (M ≈ 29 g/mol) at pressure p₀ ≈ 100 kPa and way. You may neglect internal resistance of the
of the both balls during the subsequent motion. ii) (2 points) What is the maximum force by temperature t₀ ≈ 20°C. Suddenly it hits a rock battery and capacitor, but the inductor has considerable internal resistance. The multimeter's
which the laser pointer can push the cube if the and a large hole of area A ≈ 20 cm² is formed at internal resistance when measuring voltages is
iii) (3 points) What is the average velocity (magnitude and direction) of the balls during their orientation of the laser can be arbitrary? the bottom of the submarine. As a result, the R_m ≈ 10 MΩ and it displays a new reading every
subsequent motion? submarine sinks to the bottom and most of it t ≈ 0.4 s.
7. LCR-CIRCUIT (5 points) — Jaan Kalda. is filled fast with water, leaving a bubble of air
iv) (3 points) Consider the same situation as before, except that the there are three differences Consider the circuit shown in the figure. at increased pressure (no air escapes the submarine). The density of water ρ ≈ 1000 kg/m³ and i) (3 points) Draw the electrical circuit that is inside the black box. Motivate your solution with
in the assumptions: the both balls have now free fall acceleration g ≈ 9.81 m/s². Molar heat measurements.
identical positive charge +q; electrostatic repulsion between the balls is no longer negligible; capacitance of air by constant volume c_V ≈ 5R/2,
the collision is not necessarily central (but the where R ≈ 8.31 J/K mol is the gas constant. ii) (1 point) Determine the electromotive force
balls move still at the same value of z so that V of the battery.
the collision will not induce any motion in the i) (2 points) What is the volume rate (m³/s) at
direction of the z-axis). which the water flows into the submarine immediately after the formation of the hole? iii) (1 point) Determine the internal resistance of
the inductor.
Let P_i denote the point where the surfaces of the two balls are in contact ii) (2 points) The flow rate is so large that the submarine is filled with water so fast that the heat iv) (3 points) Estimate the value of the capacitance C.
during the i-th collision. What is the maximal exchange between the gas and the water can be
distance between P_i and P_j (maximize over all neglected (this applies also to the next question). v) (3 points) Estimate the value of the inductance
the values i, j = 1, ..., ∞, and over all the impact What is the volume of the air bubble once water L.
parameters of the collisions for fixed values of B, flow has stopped?
Equipment: Black box, multimeter, stopwatch.
iii) (2 points) The water stream rushing into