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 = i) 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 2. CELL PHONE CAMERA (6 points) A pho- ii) (1.5 points) How long (tt) does a one-way of its magnetic moment along B can only be tographer focussed his camera to distance L trip to Mars take? μB or −μB (μB is called the Bohr magneton). and took a photo. On the photo, all farther objects (up to infinity) turned also out to be iii) (1.5 points) The spaceship is put into this ii) (4 points) Calculate M, the magnetic mo- sharp. Additionally, all closer objects down orbit by using a powerful rocket. It is more ment per unit volume of the sample, if the to distance s were sharp. efficient to burn fuel as a short burst when temperature of the material is T and the ex- the spaceship is still near Earth. How much ternal magnetic field is B. i) (4 points) What is the minimum pos- additional speed (Δv1) does the booster have sible L? to be able to give to the spaceship to enter the 5. FRICTION OF A STRING (8 points) transfer orbit, starting from the north pole? Measure the dynamic coefficient of friction ii) (2 points) Find the corresponding s. Neglect the air resistance. μ1 between the ballpoint pen and the string. Estimate the uncertainty. It might help that Background. We consider the image of iv) (1.5 points) Estimate the Δv2 needed to the dynamic coefficient of friction between a point like object to be sharp if its image is enter a circular orbit close to Mars. the pencil and the same string was measured smaller than one pixel on the sensor. Oth- 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 camera may be viewed as a convex lens. The of the trip to Mars and back? Equipment: dynamometer, string, ball- camera is focussed by changing the distance point pen, pencil and weight. 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 the switch is closed. does it take before oscillation halts? Equipment: resistor, voltage source (bat- teries), ammeter, calorimeter, thermometer, i) (1 point) What are the charge q on the 9. GLASS CYLINDER (7 points) In the fol- stopwatch. capacitor and the current I in the circuit as lowing figure, there is a half-cylinder, made a function of time t after closing the switch of glass. Find the refractive index n for glass The calorimeter has V = 0.80 dl of water — in the time regime when the system os- if the critical angle for total internal reflec- and minimum capacity of water cmin = 270 g. cillates without any dissipation? Draw the tion is θc = 42°. In the figure there is a The specific heat capacity of aluminum is evolution of the system in the phase diagram point-shaped light source S placed at dis- cal = 0.90 J/(K·g) and specific heat capacity (q, I). Note the direction of the system's tance d from the center of the curvature. of water is cw = 4.2 J/(K·g). Internal resist- evolution with arrow(s). Find the distance to the point P where the ance of the calorimeter is rcal = 27 Ω. The light rays, after their first refraction, will internal resistance and the voltage supplies A Zener diode is a non-linear circuit ele- meet. Note that not all light rays will be of the batteries will vary. Your set of batteries ment that acts as a bi-directional diode: it reflected back from the curved surface. may become depleted, spares are available. 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
6. SPHERE AND CYLINDER (7 points) A sphere and a cylinder are lying on an inclined surface with inclination angle α. Both have 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, Isph = (2/5)mr² and Icyl = (1/2)mr². The coefficient of friction between the surface and the objects is μ.
i) (2 points) Which of the bodies comes down faster? What was the relative lag of the slower body γ = (t2 − t1)/t1? The times t1 and t2, respectively, denote the traveling times of the faster or the slower body. Assume that the rolling occurs without slipping.
ii) (2.5 points) Find the minimal angle of inclination α0 for which the cylinder starts to slide in addition to rolling.
iii) (2.5 points) If α → 90°, the bodies obvi- ously lose contact with the surface and fall down in free fall with equal times. What is the minimal angle of inclination αm, for which both the sphere and the cylinder come down with equal times?
7. BURNING WITH A LENS (7 points) Sun- rays are focused with a lens of diameter d = 10 cm and focal length of f = 7 cm to a black thin plate. Behind the plate is a mirror. Angular diameter of the Sun is α = 32' and its intensity on the surface of the Earth is I = 1000 W/m², Stefan-Boltzmann constant 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.