2025年国际奥林匹克竞赛（IPHO）物理试题

Theory 0 01-1 nternational Rgptod English(Official) Hydrogen and galaxies(10 points) This problem aims to study the peculiar physics of galaxies,such as their dynamics and structure.In particular,we explain how to measure the mass distribution of our galaxy from the inside.For this we will focus on hydrogen,its main constituent. Throughout this problem we will only use h,defined ash=h/27. Part A-Introduction Bohr model We assume that the hydrogen atom consists of a non-relativistic electron,with mass me,orbiting a fixed proton.Throughout this part,we assume its motion is on a circular orbit. A.1 Determine the electron's velocity v in a circular orbit of radius r. 0.2pt In the Bohr model,we assume the magnitude of the electron's angular momentum L is quantized,L=nh where n is an integer.We define a7.27x103. A.2 Show that the radius of each orbit is given byr=n2n,where n is called the 0.5pt Bohr radius.Express n in terms of a,me,c and h and calculate its numerical value with 3 digits.Express u,the velocity on the orbit of radius n,in terms of a and c. A.3 Determine the electron's mechanical energy E,on an orbit of radius r in terms 0.5pt of e,Eo.n and n.Determine E in the ground state in terms of a,m and c. Compute its numerical value in eV. Hydrogen fine and hyperfine structures The rare spontaneous inversion of the electron's spin causes a photon to be emitted on average once per 10 million years per hydrogen atom.This emission serves as a hydrogen tracer in the universe and is thus fundamental in astrophysics.We will study the transition responsible for this emission in two steps. First,consider the interaction between the electron spin and the relative motion of the electron and the proton.Working in the electron's frame of reference,the proton orbits the electron at a distance n.This produces a magnetic field B. A.4 Determine the magnitude B of B,at the position of the electron in terms ofup 0.5pt e,a,c and n. Second,the electron spin creates a magnetic moment Its magnitude is roughly=h.The fine (F)structure is related to the energy difference AEg between an electron with a magnetic moment parallel to B and that of an electron with anti-parallel to B.Similarly,the hyperfine (HF)structure is related to the energy difference AEHe,due to the interaction between parallel and anti-parallel magnetic moments of the electron and the proton.It is known to be approximately3.72 where is the proton mass. Theory 屈0 Q1-2 FRANCE 2025 English(Official) A.5 Express AE as a function of a and E. 0.5pt Express the wavelength HF of a photon emitted during a transition between the two states of the hyperfine structure and give its numerical value with two digits. Part B-Rotation curves of galaxies Data ·Kiloparsec:1kpc=3.09×10l9m ·Solar mass:1M。=1.99x1030kg We consider a spherical galaxy centered around a fixed point O.At any point P,let p=p(P)be the volumetric mass density and=(P)the associated gravitational potential(i.e.potential energy per unit mass).Both p and o depend only on r OP.The motion of a mass m located at P,due to the field is restricted to a plane containing O. B.1 In the case of a circular orbit,determine the velocity vof an object on a circular 0.2pt orbit passing through P in terms ofr and dr Fig.1(A)is a picture of the spiral galaxy NGC 6946 in the visible band (from the 0.8m Schulman Telescope at the Mount Lemmon Sky Center in Arizona).The little ellipses in Fig.1(B)show experimental measure- ments of v for this galaxy.The central region (r<1kpc)is named the bulge.In this region,the mass distribution is roughly homogeneous.The red curve is a prediction for v.if the system were homoge- neous in the bulge and keplerian(p(r)=-B/r with B>0)outside it,i.e.considering that the total mass of the galaxy is concentrated in the bulge. 200 150 (A) 100 (B) 50 0 0 10 18 kpc r[kpc] Fig.1:NGC 6946 galaxy:Picture (A)and rotation curve(B) B.2 Deduce the mass M,of the bulge of NGC 6946 from the red rotation curve in 0.5pt Fig.1(B),in solar mass units. Comparing the keplerian model and the experimental data makes astronomers confident that part of the mass is invisible in the picture.They thus suppose that the galaxy's actual mass density is given by Theory 0 Q1-3 nternational English(Official) Cm pmn=后+7 (1) where C>0 and rm>0 are constants. B.3 Show that the velocity profile vm(r),corresponding to the mass density in Eq. 1.8pt 1,can be written(r)=Expressk and in terms of C and G. x2 Hints: d=r-a arctan(rla),and:arctan()for) Simplify v(r)when r<rm and when r Show that ifrm the mass M(r)embedded in a sphere of radius r with the mass density given by Eq.1 simplifies and depends only on C and r. Estimate the mass of the galaxy NGC 6946 actually present in the picture in Fig. 1(A). Part C-Mass distribution in our galaxy For a spiral galaxy,the model for Eq.1 is modified and one usually considers the gravitational potential is given by pc(r)n exp- where z is the distance to the galactic plane(defined by z=0 )andr<o is now the axial radius and o>0a constant to be determined.r and z are constant values. C.1 Find the equation of motion on z for the vertical motion of a point mass m 0.5pt in such a potential,assuming r is constant.Show that,if r<,the galactic plane is a stable equilibrium state by giving the angular frequency of small oscillations around it. From here on,we set z=0. C.2 Identify the regime,eitherrorrm,in which the model of Eq.1 recovers 0.6pt a potential of the form c(r,0)with a suitable definition ofo. Under this condition v(r)no longer depends on r.Express it in terms of po Therefore,outside the bulge the velocity modulus v.does not depend on the distance to the galactic center.We will use this fact,as astronomers do,to measure the galaxy's mass distribution from the inside. All galactic objects considered here for astronomical observations,such as stars or nebulae,are primarily composed of hydrogen.Outside the bulge,we assume that they rotate on circular orbits around the galactic center C.S is the sun's position and E that of a given galactic object emitting in the hydrogen spectrum.In the galactic plane,we consider a line of sight SE corresponding to the orientation of an observation,on the unit vector (see Fig.2). Theory 0 Q1-4 English (Official) ⊙ R Fig.2:Geometry of the measurement Let be the galactic longitude,measuring the angle between SC and the SE.The sun's velocity on its circular orbit of radius R =8.00kpc is denoted 7.A galactic object in E orbits on another circle of radius R at velocity v.Using a Doppler effect on the previously studied 21cm line,one can obtain the relative radial velocity s of the emitter E with respect to the sun S:it is the projection of-on the line of sight. C.3 Determine vrEs in terms of R,R and v.Then,express R in terms of Ro.v.0.7pt e and vrEis- Using a radio telescope,we make observations in the plane of our galaxy toward a longitude =30.The frequency band used contains the 21cm line,whose frequency is=1.42GHz.The results are reported in Fig.3. 800 400 0.1 0.10.20.3 0.4 0.5 f-fo MHz Fig.3:Electromagnetic signal as a function of the frequency shift,measured in the radio frequency band at (=30 using EU-HOU RadioAstronomy Theory 0 Q1-5 nternational English(Official) C.4 In our galaxy,v=220km.s-1.Determine the values of the relative radial ve- 0.6pt locity (with 3 significant digits)and the distance from the galactic center(with 2 significant digits)of the 3 sources observed in Fig.3.Distances should be expressed as multiples of R C.5 On the top view of our galaxy(in the answer box),indicate the positions of the 0.6pt sources observed in Fig.3. What could be deduced from repeated measurements changing ( Part D-Tully-Fisher relation and MOND theory The flat external velocity curve of NGC 6946 in Fig.1 is a common property of spiral galaxies,as can be seen in Fig.4(left).Plotting the external constant velocity value v.as a function of the measured total mass M.o of each galaxy gives an interesting correlation called the Tully-Fischer relation,see Fig.4 (right). 12 NGC 3621 NGC 5055 NGC 2903 NGC 3521 NGC 7321 NGC 2841 20 22 logio(veo/1 km/s Fig.4.Left:Rotation curves for typical spiral galaxies-Right:logo(Mo)as a function of logo()on linear scales.Colored dots correspond to different galaxies and different sur- veys.The green line is the Tully-Fischer relation which is in very good agreement with the best fit line of the data (in black). D.1 Assuming that the radius R of a galaxy doesn't depend on its mass,show that 0.4pt the model of Eq.1(part B)gives a relation of the form Mo:=nwhere y and should be specified. Compare this expression to the Tully-Fischer relation by computing yrF. In the extremely low acceleration regime,of the order of a=10-10m.s-2,the MOdified Newtonian Dy- namics(MOND)theory suggests that one can modify Newton's second law using F=mu a where a=is the modulus of the acceleration and the u function is defined by u(x)=- +x Theory 0 Q1-6 English(Official) D.2 Using data for NGC 6946 in Fig.1,estimate,within Newton's theory,the mod-0.2pt ulus of the accelerationm of a mass in the outer regions of NGC 6946. D.3 Let m be a mass on a circular orbit of radiusr with velocity v in the gravity 0.8pt field of a fixed mass M. Within the MOND theory,with a<do,determine the Tully-Fischer exponent. Using data for NGC 6946 and/or Tully-Fischer law,calculate ap to show that MOND operates in the correct regime. D.4 Considering relevant cases,determine v(r)for all values of r in the MOND the- 0.9pt ory in the case of a gravitational field due to a homogeneously distributed mass M with radius R. Theory 0 Q2-1 nternational English(Official) Cox's Timepiece(10 points) In 1765,British clockmaker James Cox invented a clock whose only source of energy is the fluctuations in atmospheric pressure.Cox's clock used two vessels containing mercury.Changes in atmospheric pressure caused mercury to move between the vessels,and the two vessels to move relative to each other.This movement acted as an energy source for the actual clock. We propose an analysis of this device.Throughout, we assume that the Earth's gravitational field g=-g:is uni- form with g=9.8ms-2and正，a unit vector; all liquids are incompressible and their density is denoted p; no surface tension effects will be considered; the variations of atmospheric pressure with al- titude are neglected; the surrounding temperature T is uniform and all transformations are isothermal. Fig.1.Artistic view of Cox's clock Part A-Pulling on a submerged tube We first consider a bath of water that occupies the semi-infinite space zs 0.The air above it is at a pressure P =P.A cylindrical vertical tube of length H=1m,cross-sectional area S=10cm2 and mass m=0.5kg is dipped into the bath.The bottom end of the tube is open,and the top end of the tube is closed.We denote h the altitude of the top of the tube and z,that of the water inside the tube.The thickness of the tube walls is neglected. 2 b h=Z h=z=0 H Fig.2.Sketch of the tube in different configurations Theory 屈0 Q2-2 English(Official) We start from the situation where the tube in Fig.2 contains no gas and its top is at the bath level:in other words,h=0 and z=0(case a).The tube is then slowly lifted until its bottom end reaches the bath level.The pulling force exerted on the tube is denoted F=FZ: A.1 For the configuration shown in Fig.2(case b),express the pressure P in the 0.2pt water at the top of the tube.Also express the force F necessary to maintain the tube at this position.Expressions must be written in terms of P,p,m,S,h,g and7. Three experiments are performed.In each,the tube is lifted from the initial state shown in Fig.2(a) under the conditions specified in Table 1. Experiment Liquid T(C) p (kg.m3) Pa:(Pa) Water 20 1.00×10 2.34×103 Water 80 0.97×10 47.4×103 Water 99 0.96×103 99.8×103 Table 1.Experimental conditions and numerical values of physical quantities for each experiment (P:designates the saturated vapour pressure of the pure fluid) In each case,we study the evolution of the force F that must be applied in order to maintain the tube in equilibrium at an altitude h,the external pressure being fixed at P=P=1.000 x 10 Pa.Two different behaviours are possible Behaviour A Behaviour B Fo h H A.2 For each experiment,complete the table in the answer sheet to indicate the ex- 0.8pt pected behaviour and the numerical values for Fax and for h*(when pertinent). where Fm and h*are defined in the figures illustrating the two behaviours. When we replace the water with liquid mercury(whose properties are given below),behaviour B is ob- served. Liquid T (C) p (kg.m) Psat (Pa) Mercury 20 13.5×103 0.163 Theory 0 Q2-3 nternational English(Official) A.3 Express the relative error,denoted e,committed when we evaluate the maximal 0.3pt force Fmax neglecting Po:compared to P.Give the numerical value of e. Part B-Two-part barometric tube From now on,we work with mercury(density p=13.5 x 103kg.m-3)at the ambient tempera- ture Ta =20C and we take Pt =0. Let us consider a tube with a reservoir on top, modeled as two superposed cylinders of differ- H=20cm ent dimensions,as shown in Fig.3. h. the bottom part (still called the tube) has cross-sectional area S,and height H:=80cm; 0 the top part (called the bulb)has H,=80cm cross-sectional area S>S,and height H、=20cm. This two-part tube is dipped into a semi-infinite liquid bath. Fig.3.Sketch of the two-part barometric tube As in Part A,the system is prepared such that the tube contains no air.We identify the vertical position of the tube by the altitude h of the junction between the tube and the bulb.The height of the column of mercury is again denoted z.The force F that must be exerted to maintain the tube in equilibrium in the configuration shown in Fig.3 can now be written as F=（mb+madd)gu (1) where mb is the total mass of the two-part tube(when empty of mercury). B.1 On the answer sheet,color the area corresponding to the volume of liquid mer-0.3pt cury that is responsible for the term madd appearing in equation(1). The mass mdd depends both on the height Iz and the atmospheric pressure Pa.For the next question, assume that the atmospheric pressure is fixed at P=P=1.000 x 105Pa.Starting from the situation where the system is completely submerged,the tube is slowly lifted until its base is flush with the liquid bath. B.2 Sketch the evolution of the mass madd as a function of h,for[HH.On 1.4pt the graph,provide the expression for the slopes of the different segments,as well as the h,analytical value of any angular points,in terms of P,p,g,Sp S H and H.. Theory 屈0 Q2-4 English(Official) As the system is lifted while P=P=105Pa,we stop when the free surface of the liquid is in the middle of the bulb.The value of is fixed and then we observe variations in the mass mdd due to variations in the atmospheric pressure described by P (t)=Po+P(t) (2) where P designates the average value and P is a perturbative term.We model p by a periodic triangular function of amplitude A=5x 102 Pa and period r of 1 week. B(t) Fig.4.Simplified model of the perturbative term P(r) B.3 Given that S:=5cm2 and Sp=200cm2,express the amplitude Amadd of the varia- 0.3pt tions of the mass mdd over time,then give its numerical value.Assume that the liquid surface always stays in the bulb. Part C-Cox's timepiece The real mechanism developed by Cox is complex(Fig.5).We study a simplified version,depicted in Fig. 6,and described below a cylindrical bottom cistern containing a mercury bath a two-part barometric tube identical to that studied in part B,which is still completely emptied of any air,is dipped into the bath; the cistern and the two-part tube are each suspended by a cable.Both cables (assumed to be inextensible and of negligible mass)pass through a system of ideal pullies and finish attached to either side of the same mass M,which can slide on a horizontal surface the total volume of liquid mercury contained in the system is V=5L. The height,cross-section and masses of each part are given in Table 2.The position of mass M is ref- erenced by the coordinate x of its center of mass.We consider solid friction between the horizontal support and the mass M,without distinction between static and dynamic coefficients;the magnitude of this force when sliding occurs is denoted Fs. Two stops limit the displacement of the mass M such that-XsxsX (with X>0).Assume that the value of X guarantees that the bottom of the two-part tube never touches the bottom of the cistern nor comes out of the liquid bath; the altitude z of the mercury column is always in the upper bulb.