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 = ii) a 1.52 AU. The mean orbital radius of Earth We can use this model to calculate the is Rg = 1 AU = 149600000 km. Mars has a magnetic properties of materials containing mean radius of ra = 3397 km and surface unpaired electrons that have negligibly weak gravity ga = 3.71 m/s². Earth has a mean interactions with one another. Let us con- radius of rg = 6371 km. sider a sample of material with n such un- paired electrons per unit volume, placed in- i) (1 point) Find the orbital period Ta of side a homogeneous magnetic field B. Due Mars, i.e. find the length of Martian "year" to spin, each unpaired electron acts as a in Earth years. small magnetic dipole. However, owing to the quantum nature the electron, the projection ii) (1.5 points) How long (tt) does a one-way of its magnetic moment along B can only be trip to Mars take? μB or −μB (μB is called the Bohr magneton).
2. CELL PHONE CAMERA (6 points) A pho- tographer focussed his camera to distance L iii) (1.5 points) The spaceship is put into this ii) (4 points) Calculate M, the magnetic mo- and took a photo. On the photo, all farther orbit by using a powerful rocket. It is more ment per unit volume of the sample, if the objects (up to infinity) turned also out to be efficient to burn fuel as a short burst when temperature of the material is T and the ex- sharp. Additionally, all closer objects down the spaceship is still near Earth. How much ternal magnetic field is B. to distance s were sharp. additional speed (Δv1) does the booster have to be able to give to the spaceship to enter the 5. FRICTION OF A STRING (8 points) i) (4 points) What is the minimum pos- transfer orbit, starting from the north pole? Measure the dynamic coefficient of friction sible L? Neglect the air resistance. μ1 between the ballpoint pen and the string. ii) (2 points) Find the corresponding s. iv) (1.5 points) Estimate the Δv2 needed to Estimate the uncertainty. It might help that enter a circular orbit close to Mars. the dynamic coefficient of friction between Background. We consider the image of the pencil and the same string was measured a point like object to be sharp if its image is v) (1.5 points) What is the minimal duration beforehand and μ2 = 0.20 ± 0.01 was obtained. smaller than one pixel on the sensor. Oth- of the trip to Mars and back? Equipment: dynamometer, string, ball- erwise the image is blurry. The lens of the point pen, pencil and weight. camera may be viewed as a convex lens. The 4. MAGNETIC DIPOLES (7 points) Let us camera is focussed by changing the distance consider the following model for a magnetic between the sensor and the lens. dipole. Some wire with no resistance has Parameters. Calculate the answer for a 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 functions of time t? of glass, and the point A. Find the mini- with specific heat capacity of cw = 4.2 J/(K·g) The efficiency of friction between the mum distance d between the strip and the and the mass of water mw = 270 g. Inter- bodies is µ. point A so that a light ray passing from nal heat capacity of the calorimeter is cal = 90 J/K and its mass is mcal = 27 g. The i) (2 points) Which of the bodies comes down A to the strip after refraction will not be specific heat capacity of aluminum is cal = faster? What was the relative lag of the reflected on the flat face of the cylinder but 0.90 J/(K·g). The voltage and current meas- slower body γ = (t2 − t1)/t1? The times t1 and will pass through it. Draw an appropriate urements may vary due to the fact that the voltage t2, respectively, denote the traveling times of diagram. Note that the refractive index of of the batteries will vary. Your set of batteries the faster or the slower body. Assume that glass is n = 1.5. may become depleted, spares are available. the rolling occurs without slipping.
ii) (2.5 points) Find the minimal angle of A Zener diode is a non-linear circuit ele- inclination α0 for which the cylinder starts ment that acts as a bi-directional diode: it to slide in addition to rolling. allows the current to flow in the positive dir- ection when a forward voltage on it exceeds iii) (2.5 points) If α → 90°, the bodies obvi- a certain threshold value, but it also allows ously lose contact with the surface and fall a current to flow in the opposite direction down in free fall with equal times. What when exposed to sufficiently large negative is the minimal angle of inclination αm, for voltage. Normally the two voltage scales are which both the sphere and the cylinder come quite different, but for our purposes we will down with equal times? take a Zener diode with the following volt- ampere characteristics: for forward currents, 7. BURNING WITH A LENS (7 points) Sun the voltage on the diode is Vd, for reverse rays are focused with a lens of diameter currents, the voltage on the diode is −Vd, d = 10 cm and focal length of f = 7 cm to a for zero current the voltage on the diode is black thin plate. Behind the plate is a mirror. −Vd < V < Vd. Angular diameter of the Sun is α = 32′ and its intensity on the surface of the Earth is Now we connect the inductance L, the ca- I = 1000 W/m², Stefan-Boltzmann constant pacitor C all in series with a switch and a σ = 5.670 × 10⁻⁸ W/(m²K⁴). Zener diode. The switch is initially open. The capacitor is again given the charge q0 > CVd i) (4 points) Find the temperature of the and the switch is then closed. heated point of the plate. ii) (2 points) Make a drawing of the phase ii) (3 points) Using thermodynamic argu- diagram for the system. Note the direction of ments, estimate the maximal diameter of the the system's evolution with arrow(s). lens for which this model can be used. iii) (2 points) Does the evolution of the sys- 6. SPHERE AND CYLINDER (7 points) A tem only necessarily stop for q = 0? Find the sphere and a cylinder are lying on an inclined range of values of q on the capacitor for which surface with inclination angle α. Both have the evolution of the system will necessarily mass m and radius r. The bodies are released from equal initial heights H. The moments of inertia of the sphere and the cylinder are, re- spectively, Is = 2/5 mr² and Icyl = 1/2 mr². The coefficient of static friction between the surface and the bodies is μ.