Estonian-Finnish Olympiad 2013 The focal length of its camera f = 4.3 mm been bent into the shape of a square with and the diameter of the lens D = 1.8 mm. The side length a. At some point on the wire is
1. PRISM (8 points) sensor is w = 4.6 mm wide corresponding to a small ideal current source that keeps cur- N = 3264 pixels. rent I flowing in the circuit in all situations.
i) (4 points) A right prism that has an equilateral triangular base with length a is 3. MISSION TO MARS (7 points) A crew of The magnetic moment m of a planar circuit is given by the relation m = IA, with vector placed in a horizontal slit between two tables, astronauts is going to be sent to explore the m pointing in the normal direction of the cir- so that one of the side faces is vertical. How polar region of Mars and search for buried cuit according to the right hand rule (A is the small can the width d of the slit be made be- water ice. Their spaceship will travel from area bounded by the circuit). fore the prism falls out of the slit? There is no Earth to Mars along an elliptic transfer orbit friction between the prism and the tables and tangential to the orbits of both planets. Des- i) (3 points) The dipole is placed inside a the prism is made of a homogeneous material. pite its shortcomings, this orbit is commonly homogeneous magnetic field B, so that the The edges of the slit are parallel. used in space travel due to its relatively good angle between m and B is θ. Find the angles fuel economy. Future manned missions to θₛ and θᵤ that correspond to stable and un- ii) (4 points) Now the prism is placed in the Mars are very likely based on this kind of stable equilibria, respectively. Calculate the slit so that one of its side faces is horizontal. transfer. In this problem you will examine amount of work (w) needed to rotate the di- How small can the width l of the slit be made some aspects of this orbit. pole from θₛ to θᵤ. Give your answer in terms before that position becomes unstable? of m and B.
The mean orbital radius of Mars is Rₐ = We can use this model to calculate the i) ii) a 1.52 AU. The mean orbital radius of Earth magnetic properties of materials containing is Rₑ = 1 AU = 149600000 km. Mars has a unpaired electrons that have negligibly weak mean radius of rₐ = 3397 km and surface interactions with one another. Let us con- gravity gₐ = 3.71 m/s². Earth has a mean sider a sample of material with n such un- radius of rₑ = 6371 km. paired electrons per unit volume, placed in- side a homogeneous magnetic field B. Due i) (1 point) Find the orbital period Tₐ of to spin, each unpaired electron acts as a Mars, i.e. find the length of Martian "year" small magnetic dipole. However, owing to the in Earth years. quantum nature the electron, the projection of its magnetic moment along B can only be 2. CELL PHONE CAMERA (6 points) A pho- ii) (1.5 points) How long (tₜ) does a one-way μ_B or -μ_B (μ_B is called the Bohr magneton). tographer focussed his camera to distance L trip to Mars take? and took a photo. On the photo, all farther ii) (4 points) Calculate M, the magnetic mo- objects (up to infinity) turned also out to be iii) (1.5 points) The spaceship is put into this ment per unit volume of the sample, if the sharp. Additionally, all closer objects down orbit by using a powerful rocket. It is more temperature of the material is T and the ex- to distance s were sharp. efficient to burn fuel as a short burst when ternal magnetic field is B. the spaceship is still near Earth. How much i) (4 points) What is the minimum pos- additional speed (Δv₁) does the booster have 5. FRICTION OF A STRING (8 points) sible L? to be able to give to the spaceship to enter the Measure the dynamic coefficient of friction ii) (2 points) Find the corresponding s. transfer orbit, starting from the north pole? μ₁ between the ballpoint pen and the string. Neglect the air resistance. Estimate the uncertainty. It might help that Background. We consider the image of the dynamic coefficient of friction between a point like object to be sharp if its image is iv) (1.5 points) Estimate the Δv₂ needed to the pencil and the same string was measured smaller than one pixel on the sensor. Oth- enter a circular orbit close to Mars. beforehand and μ₂ = 0.20 ± 0.01 was obtained. erwise the image is blurry. The lens of the v) (1.5 points) What is the minimal duration Equipment: dynamometer, string, ball- camera may be viewed as a convex lens. The of the trip to Mars and back? point pen, pencil and weight. camera is focussed by changing the distance between the sensor and the lens. 4. MAGNETIC DIPOLES (7 points) Let us consider the following model for a magnetic Parameters. Calculate the answer for a dipole. Some wire with no resistance has cell phone made by a well-known company.
Estonian-Finnish Olympiad 2013 8. ZENER DIODE (7 points) An inductance come to a halt. 10. RESISTIVE HEATING (8 points) Meas- L and a capacitor C are connected in series iv) (2 points) Find the decrease Δq in the ure the resistor. You are not asked to estim- with a switch. Initially the switch is open maximum positive value of the capacitor's ate the uncertainty. and the capacitor is given a charge q₀. Now charge q after one full oscillation. How long Equipment: resistor, voltage source (bat- the switch is closed. does it take before oscillation halts? teries), ammeter, calorimeter, thermometer, stopwatch. i) (1 point) What are the charge q on the 9. GLASS CYLINDER (7 points) In the fol- capacitor and the current I in the circuit as lowing figure, there is a half-cylinder, made The calorimeter has V = 0.80 dl of water function of time? Draw a diagram. of glass, that can be rotated around its axis. and initially has temperature T = 20.9°C. The specific heat capacity of water is cᵤ = 4.2 J/(K·g). Internal resistance of the cal- 6. SPHERE AND CYLINDER (7 points) A There is no friction between the surfaces of Find the index of refraction n of the glass. orimeter is negligible. You should take into sphere and a cylinder are lying on an inclined the bodies. The coefficient of friction is μ. account that after some period of time the surface with inclination angle α. Both have ii) (2 points) Which of the bodies comes down temperature of the batteries will vary. Your set of batteries mass m and radius r. The bodies are released faster? What was the relative lag of the may become depleted, spares are available. from equal initial heights H. The moments of slower body γ = (t₂ - t₁)/t₁? The times t₁ and inertia of the sphere and the cylinder are, re- t₂, respectively, denote the traveling times of spectively, I_sphere = 2/5 mr² and I_cyl = 1/2 mr². The the faster or the slower body. Assume that coefficient of friction between the surface and the rolling occurs without slipping. bodies is μ. There is no slipping. iii) (2.5 points) Find the minimal angle of inclinationα₀ for which the cylinder starts i) (2 points) Which of the bodies comes down to slide in addition to rolling. faster? What was the relative lag of the slower body γ = (t₂ - t₁)/t₁? The times t₁ and iv) (2.5 points) If α → 90°, the bodies obvi- t₂, respectively, denote the traveling times of ously lose contact with the surface and fall the faster or the slower body. Assume that down in free fall with equal times. What the rolling occurs without slipping. is the minimal angle of inclination αₘ, for which both the sphere and the cylinder come ii) (2.5 points) Find the minimal angle of down with equal times? inclination α₀ for which the cylinder starts to slide in addition to rolling. 7. BURNING WITH A LENS (7 points) Sun- rays are focused with a lens of diameter iii) (2.5 points) If α → 90°, the bodies obvi- d = 10 cm and focal length of f = 7 cm to a ously lose contact with the surface and fall black thin plate. Behind the plate is a mirror. down in free fall with equal times. What Angular diameter of the Sun is α = 32' and is the minimal angle of inclination αₘ, for its intensity on the surface of the Earth is which both the sphere and the cylinder come I = 1000 W/m², Stefan-Boltzmann constant down with equal times? σ = 5.670 × 10⁻⁸ W/(m²K⁴).
i) (4 points) Find the temperature of the heated point of the plate.
ii) (3 points) Using thermodynamic argu- ments, estimate the maximal diameter of the lens for which this model can be used.
A Zener diode is a non-linear circuit ele- ment that acts as a bi-directional diode: it allows the current to flow in the positive dir- ection when a forward voltage on it exceeds a certain threshold value, but it also allows a current to flow in the opposite direction when exposed to sufficiently large negative voltage. Normally the two voltage scales are quite different, but for our purposes we will take a Zener diode with the following volt- ampere characteristics: for forward currents, the voltage on the diode is Vd, for reverse currents, the voltage on the diode is -Vd, for zero current the voltage on the diode is -Vd < V < Vd.
Now we connect the inductance L, the ca- pacitor C all in series with a switch and a Zener diode. The switch is initially open. The capacitor is again given the charge q₀ > CVd and the switch is then closed.
ii) (2 points) Make a drawing of the phase diagram for the system. Note the direction of the system's evolution with arrow(s).
iii) (2 points) Does the evolution of the sys- tem only necessarily stop for q = 0? Find the range of values of q on the capacitor for which the evolution of the system will necessarily