{ "schemaVersion": "1.0.0", "batchId": "ipho-theory-pilot-001", "status": "source-fidelity-pilot", "importedAt": null, "provenanceCheckedOn": "2026-10-28", "notes": "Five existing official-source PDFs indexed across IPhO 2011, 2015, and 2025. No claim of cross-competition coverage. Native text is an editorial restatement, not a complete verbatim transcription; original pages remain authoritative for shared context, constants, figures, and tables. No publication date is fabricated for existing files.", "reuse": { "status": "permission-review-required", "license": null, "evidenceUrls": [ "https://www.ipho-new.org/documentations/", "https://www.ipho2025.fr/mentions-legales" ], "notes": "No explicit redistribution or commercial-use permission was verified. Public access is not a reuse license. Do not treat this pilot, including embedded third-party figures, as cleared for wider republication." }, "grading": { "mode": "self-review", "rated": false, "elo": null, "notes": "No automatic correctness, accepted-solution, leaderboard, or Elo claims. Existing isolated calculator adaptations are not full-problem judges." }, "review": { "status": "cofounder-review-pending", "checks": [ "Original assets and subpart/point counts validated", "2011 equal-mass assumption and 2015 constants checked visually" ], "remaining": [ "Independent physics review of all restatements and solution-page mappings", "Full native transcription of shared setup and figures", "Rights clearance before expansion or wider republication" ] }, "problems": [ { "id": "ipho-2011-t1", "competition": "IPhO", "year": 2011, "code": "T1", "totalPoints": 10, "documents": { "questions": { "pdf": "/ipho/2011-t1-questions.pdf", "images": [ "/ipho/2011-t1-page-1.webp", "/ipho/2011-t1-page-2.webp", "/ipho/2011-t1-page-3.webp" ] }, "solutions": { "pdf": "/ipho/2011-t1-solutions.pdf", "images": [ "/ipho/2011-t1-solution-page-1.webp", "/ipho/2011-t1-solution-page-2.webp", "/ipho/2011-t1-solution-page-3.webp", "/ipho/2011-t1-solution-page-4.webp", "/ipho/2011-t1-solution-page-5.webp", "/ipho/2011-t1-solution-page-6.webp", "/ipho/2011-t1-solution-page-7.webp" ] } }, "title": "A Three-body Problem and LISA", "topics": [ "Mechanics", "Waves" ], "tags": [ "Circular motion", "Gravitation", "Lagrange points", "Small oscillations", "Interferometry" ], "summary": "From a co-rotating three-body system to the relative motion of a triangular space-based gravitational-wave detector.", "attribution": "42nd International Physics Olympiad, Bangkok, Thailand, theoretical competition, 12 July 2011. Figure 2 credits D. A. Shaddock, An Overview of the Laser Interferometer Space Antenna, Publications of the Astronomical Society of Australia 26 (2009), 128–132.", "sourceUrl": "https://www.ipho-new.org/documentations/", "pages": 3, "solutionPages": 7, "sections": [ { "id": "1", "title": "A three-body problem and LISA" } ], "context": [ { "type": "text", "value": "Two gravitating masses M and m move in circular orbits of radii R and r about their common centre of mass O. A third body of infinitesimal mass μ may move in a circular orbit in the same plane, remaining stationary relative to the other two masses. The geometry is Figure 1 on original page 1." }, { "type": "math", "value": "$$MR=mr,\\qquad G=\\text{universal gravitational constant}$$" }, { "type": "text", "value": "For 1.4, use the LISA design described in the 2011 paper, not a present-day mission specification. Three identical spacecraft lie at the corners of an equilateral triangle with arms about 5.0 million kilometres long. The constellation follows an Earth-like orbit around the Sun, trailing Earth by 20 degrees. Each spacecraft follows a slightly inclined individual orbit; the triangle appears to roll around its centre once per year. Laser interferometry detects tiny changes in arm lengths. Figures 2 and 3 are on original pages 2 and 3." } ], "parts": [ { "id": "1.1", "section": "1", "title": "Angular velocity of two gravitating bodies", "points": 1.5, "page": 1, "solutionPage": 1, "prompt": "Determine the angular velocity of the line joining M and m in terms of R, r, M, m, and G.", "model": "$$\\omega_0=\\text{angular velocity of the line joining the two masses}$$" }, { "id": "1.2", "section": "1", "title": "A stationary third body", "points": 3.5, "page": 1, "solutionPage": 2, "prompt": "Place an infinitesimal mass μ in a coplanar circular orbit about O, stationary relative to M and m and not collinear with them, as in Figure 1. In terms of R and r, determine each of the following distances.", "items": [ "1.2.1: from μ to M.", "1.2.2: from μ to m.", "1.2.3: from μ to the centre of mass O." ] }, { "id": "1.3", "section": "1", "title": "Radial oscillations for equal masses", "points": 3.2, "page": 2, "solutionPage": 4, "prompt": "Now take M = m. Give μ a small radial perturbation along Oμ. Assuming its angular momentum is conserved, find the angular frequency of its oscillation about the unperturbed position, in terms of the unperturbed orbital angular velocity.", "model": "$$\\omega_0=\\text{unperturbed orbital angular velocity}$$" }, { "id": "1.4", "section": "1", "title": "Relative speed in the LISA constellation", "points": 1.8, "page": 3, "solutionPage": 6, "prompt": "Using the LISA configuration in the shared setup and original Figures 2–3, determine the relative speed of one spacecraft with respect to another in the plane containing the three spacecraft." } ], "statement": { "mode": "original-pages-with-editorial-index", "pdf": "/ipho/2011-t1-questions.pdf", "pages": [ "/ipho/2011-t1-page-1.webp", "/ipho/2011-t1-page-2.webp", "/ipho/2011-t1-page-3.webp" ] }, "solution": { "mode": "official-pdf", "pdf": "/ipho/2011-t1-solutions.pdf", "pages": [ "/ipho/2011-t1-solution-page-1.webp", "/ipho/2011-t1-solution-page-2.webp", "/ipho/2011-t1-solution-page-3.webp", "/ipho/2011-t1-solution-page-4.webp", "/ipho/2011-t1-solution-page-5.webp", "/ipho/2011-t1-solution-page-6.webp", "/ipho/2011-t1-solution-page-7.webp" ] }, "rating": { "elo": null, "status": "uncalibrated" }, "importedAt": null }, { "id": "ipho-2015-t1", "competition": "IPhO", "year": 2015, "code": "T1", "totalPoints": 10, "documents": { "questions": { "pdf": "/ipho/2015-t1-questions.pdf", "images": [ "/ipho/2015-t1-page-1.webp", "/ipho/2015-t1-page-2.webp" ] }, "solutions": { "pdf": "/ipho/2015-t1-solutions.pdf", "images": [ "/ipho/2015-t1-solution-page-1.webp", "/ipho/2015-t1-solution-page-2.webp", "/ipho/2015-t1-solution-page-3.webp", "/ipho/2015-t1-solution-page-4.webp", "/ipho/2015-t1-solution-page-5.webp", "/ipho/2015-t1-solution-page-6.webp", "/ipho/2015-t1-solution-page-7.webp" ] } }, "title": "Particles from the Sun", "topics": [ "Modern", "Thermodynamics", "Optics", "Mechanics" ], "tags": [ "Blackbody radiation", "Solar cells", "Neutrinos", "Cherenkov radiation", "Nuclear fusion" ], "summary": "Use photons and neutrinos to investigate solar temperatures, solar-cell efficiency, gravitational contraction, and nuclear fusion.", "attribution": "46th International Physics Olympiad, Mumbai, India, Theory 1. Principal authors: Amol Dighe (TIFR), Anwesh Mazumdar (HBCSE-TIFR), and Vijay A. Singh; contributions from the Academic Committee, Academic Development Group, and International Board are acknowledged in the original.", "sourceUrl": "https://www.ipho-new.org/documentations/", "pages": 2, "solutionPages": 7, "sections": [ { "id": "A", "title": "Radiation from the Sun" }, { "id": "B", "title": "Neutrinos from the Sun" } ], "context": [ { "type": "text", "value": "Photons from the solar surface and neutrinos from the core provide information about temperatures and the source of solar energy. Use the supplied solar constants throughout. Fundamental constants are denoted by their usual symbols; d⊙ is the Earth–Sun distance." }, { "type": "math", "value": "$$M_\\odot=2.00\\times10^{30}\\,\\mathrm{kg},\\quad R_\\odot=7.00\\times10^8\\,\\mathrm{m}$$" }, { "type": "math", "value": "$$L_\\odot=3.85\\times10^{26}\\,\\mathrm{W},\\quad d_\\odot=1.50\\times10^{11}\\,\\mathrm{m}$$" }, { "type": "text", "value": "The original supplies the following integration identities, with an arbitrary integration constant C." }, { "type": "math", "value": "$$\\int xe^{ax}\\,dx=\\left(\\frac{x}{a}-\\frac{1}{a^2}\\right)e^{ax}+C$$" }, { "type": "math", "value": "$$\\int x^2e^{ax}\\,dx=\\left(\\frac{x^2}{a}-\\frac{2x}{a^2}+\\frac{2}{a^3}\\right)e^{ax}+C$$" }, { "type": "math", "value": "$$\\int x^3e^{ax}\\,dx=\\left(\\frac{x^3}{a}-\\frac{3x^2}{a^2}+\\frac{6x}{a^3}-\\frac{6}{a^4}\\right)e^{ax}+C$$" }, { "type": "text", "value": "For A2–A8, approximate solar radiation by Wien’s distribution. A thin solar cell of area A is perpendicular to the incident rays. The incident energy per unit time per unit frequency interval is u(ν), where ν is frequency and T_s is the solar-surface temperature." }, { "type": "math", "value": "$$u(\\nu)=A\\frac{R_\\odot^2}{d_\\odot^2}\\frac{2\\pi h}{c^2}\\nu^3\\exp\\!\\left(-\\frac{h\\nu}{k_B T_s}\\right)$$" }, { "type": "text", "value": "Each photon with energy at least the semiconductor band gap E_g excites one electron. Each excited electron contributes E_g as useful output; excess photon energy becomes heat. Photons below the band gap produce no useful output." }, { "type": "math", "value": "$$E_g=h\\nu_g,\\qquad x_g=\\frac{h\\nu_g}{k_B T_s}$$" }, { "type": "text", "value": "For A9–A10, the Kelvin–Helmholtz hypothesis starts with a very large cloud of solar mass and negligible density, shrinking continuously. Its light comes from released gravitational potential energy. Compare the resulting lifetime with the meteorite age of the Solar System: the original notes that a purely gravitational energy source is insufficient." }, { "type": "text", "value": "For B1–B2, solar energy comes entirely from hydrogen fusion into helium. Treat electron neutrinos as massless and neglect the energy they carry away. The net reaction and energy released per reaction are:" }, { "type": "math", "value": "$$4\\,{}^1\\mathrm{H}\\longrightarrow{}^4\\mathrm{He}+2e^++2\\nu_e,\\qquad \\Delta E=4.0\\times10^{-12}\\,\\mathrm{J}$$" }, { "type": "text", "value": "Some electron neutrinos nu_e convert into other types nu_x en route to Earth. Detector efficiency for nu_x is one sixth of that for nu_e. Without conversion, N_1 neutrinos per year would be detected; the actual annual detection count, combining both types, is N_2." }, { "type": "text", "value": "For B3, an electron knocked out of a water molecule emits Cherenkov radiation while its speed exceeds c/n, where n is the refractive index of water. Treat the electron as initially at rest before the interaction with the neutrino." }, { "type": "text", "value": "For B4, a beryllium-7 nucleus captures an electron to produce lithium-7 and an electron neutrino. The lithium nucleus has a smaller rest mass. When both initial particles are at rest, the neutrino has the given energy. Thermal motion of beryllium nuclei in the solar core broadens the observed neutrino energy." }, { "type": "math", "value": "$${}^7\\mathrm{Be}+e^-\\longrightarrow{}^7\\mathrm{Li}+\\nu_e$$" }, { "type": "math", "value": "$$m_{\\mathrm{Be}}=11.65\\times10^{-27}\\,\\mathrm{kg},\\quad E_\\nu=1.44\\times10^{-13}\\,\\mathrm{J},\\quad \\Delta E_{\\mathrm{rms}}=5.54\\times10^{-17}\\,\\mathrm{J}$$" } ], "parts": [ { "id": "A1", "section": "A", "title": "Solar-surface temperature", "points": 0.3, "page": 1, "solutionPage": 1, "prompt": "Assume the Sun is a perfect blackbody. Calculate its surface temperature T_s using the supplied solar radius and luminosity." }, { "id": "A2", "section": "A", "title": "Incident solar power", "points": 0.3, "page": 1, "solutionPage": 1, "prompt": "Using the Wien approximation in the shared setup, express the total power P_in incident on the solar cell in terms of A, R⊙, d⊙, T_s, h, c, and k_B." }, { "id": "A3", "section": "A", "title": "Photon arrival spectrum", "points": 0.2, "page": 1, "solutionPage": 2, "prompt": "Find the number nγ(ν) of photons incident on the solar cell per unit time per unit frequency interval in terms of A, R⊙, d⊙, T_s, ν, h, c, and k_B." }, { "id": "A4", "section": "A", "title": "Useful solar-cell output", "points": 1, "page": 1, "solutionPage": 2, "prompt": "Use the band-gap model and the definition of x_g in the shared setup. Express the useful output power P_out in terms of x_g, A, R⊙, d⊙, T_s, h, c, and k_B." }, { "id": "A5", "section": "A", "title": "Solar-cell efficiency", "points": 0.2, "page": 1, "solutionPage": 2, "prompt": "Express the solar-cell efficiency η as a function of x_g." }, { "id": "A6", "section": "A", "title": "Sketch the efficiency curve", "points": 1, "page": 1, "solutionPage": 3, "prompt": "Sketch η against x_g qualitatively, showing the values at x_g = 0 and as x_g tends to infinity. Determine the slope at each of those limits." }, { "id": "A7", "section": "A", "title": "Optimal band gap", "points": 1, "page": 1, "solutionPage": 3, "prompt": "Let x_0 maximize η. Derive the cubic equation determining x_0. Estimate x_0 to within ±0.25 and calculate the maximum efficiency η(x_0)." }, { "id": "A8", "section": "A", "title": "Silicon solar cell", "points": 0.2, "page": 1, "solutionPage": 4, "prompt": "Pure silicon has a band gap of 1.11 eV. Use this value to calculate the efficiency of a silicon solar cell in the given model." }, { "id": "A9", "section": "A", "title": "Gravitational energy of the Sun", "points": 0.3, "page": 2, "solutionPage": 4, "prompt": "Assume a uniform density inside the Sun. Find its present total gravitational potential energy Ω in terms of G, M⊙, and R⊙." }, { "id": "A10", "section": "A", "title": "Kelvin–Helmholtz lifetime", "points": 0.5, "page": 2, "solutionPage": 4, "prompt": "Under the Kelvin–Helmholtz hypothesis in the shared setup, estimate the maximum time t_KH, in years, for which the Sun could have been shining. Assume its luminosity has remained constant." }, { "id": "B1", "section": "B", "title": "Solar-neutrino flux", "points": 0.6, "page": 2, "solutionPage": 5, "prompt": "Using the net fusion reaction and released energy in the shared setup, calculate the neutrino number-flux density at Earth in m⁻² s⁻¹. Assume this reaction accounts for all of the solar radiation." }, { "id": "B2", "section": "B", "title": "Fraction of converted neutrinos", "points": 0.4, "page": 2, "solutionPage": 5, "prompt": "Using the expected and actual detection counts N_1 and N_2, and the relative detector efficiency in the shared setup, determine the fraction of electron neutrinos converted into other types." }, { "id": "B3", "section": "B", "title": "Energy of a Cherenkov-emitting electron", "points": 2, "page": 2, "solutionPage": 6, "prompt": "An electron knocked out by a neutrino loses energy at a constant rate α per unit time while travelling through water. It emits Cherenkov radiation for a duration Δt. Determine the energy imparted to the initially stationary electron by the neutrino, in terms of α, Δt, n, m_e, and c." }, { "id": "B4", "section": "B", "title": "Temperature of the solar core", "points": 2, "page": 2, "solutionPage": 6, "prompt": "Using the beryllium mass, neutrino energy, and rms energy spread in the shared setup, calculate the rms speed of the beryllium nuclei and estimate the solar-core temperature T_c. Hint: the energy spread depends on the rms velocity component along the line of sight." } ], "statement": { "mode": "original-pages-with-editorial-index", "pdf": "/ipho/2015-t1-questions.pdf", "pages": [ "/ipho/2015-t1-page-1.webp", "/ipho/2015-t1-page-2.webp" ] }, "solution": { "mode": "official-pdf", "pdf": "/ipho/2015-t1-solutions.pdf", "pages": [ "/ipho/2015-t1-solution-page-1.webp", "/ipho/2015-t1-solution-page-2.webp", "/ipho/2015-t1-solution-page-3.webp", "/ipho/2015-t1-solution-page-4.webp", "/ipho/2015-t1-solution-page-5.webp", "/ipho/2015-t1-solution-page-6.webp", "/ipho/2015-t1-solution-page-7.webp" ] }, "rating": { "elo": null, "status": "uncalibrated" }, "importedAt": null }, { "id": "ipho-2025-t1", "competition": "IPhO", "year": 2025, "code": "T1", "totalPoints": 10, "documents": { "questions": { "pdf": "/ipho/2025-t1-questions.pdf", "images": [ "/ipho/2025-t1-page-1.webp", "/ipho/2025-t1-page-2.webp", "/ipho/2025-t1-page-3.webp", "/ipho/2025-t1-page-4.webp", "/ipho/2025-t1-page-5.webp", "/ipho/2025-t1-page-6.webp" ] }, "solutions": { "pdf": "/ipho/2025-t1-solutions.pdf", "images": [ "/ipho/2025-t1-solution-page-1.webp", "/ipho/2025-t1-solution-page-2.webp", "/ipho/2025-t1-solution-page-3.webp", "/ipho/2025-t1-solution-page-4.webp", "/ipho/2025-t1-solution-page-5.webp", "/ipho/2025-t1-solution-page-6.webp", "/ipho/2025-t1-solution-page-7.webp", "/ipho/2025-t1-solution-page-8.webp", "/ipho/2025-t1-solution-page-9.webp", "/ipho/2025-t1-solution-page-10.webp", "/ipho/2025-t1-solution-page-11.webp", "/ipho/2025-t1-solution-page-12.webp", "/ipho/2025-t1-solution-page-13.webp", "/ipho/2025-t1-solution-page-14.webp", "/ipho/2025-t1-solution-page-15.webp" ] } }, "title": "Hydrogen and galaxies", "topics": [ "Modern", "Mechanics" ], "tags": [ "Bohr model", "Hydrogen spectrum", "Rotation curves", "Dark matter", "MOND" ], "summary": "Follow hydrogen from the Bohr model and the 21 cm line to galactic rotation curves, mass distributions, and modified Newtonian dynamics. The original figures and physical constants are available in the paper reader below.", "attribution": "55th International Physics Olympiad, France, 2025, T1. Original statements, figures, answer sheets, and solutions are credited to the IPhO 2025 organisers; see the original paper for contributor and image credits.", "sourceUrl": "https://www.ipho2025.fr/sujets-officiels-ipho-france-2025", "pages": 6, "solutionPages": 15, "sections": [ { "id": "A", "title": "Introduction: hydrogen" }, { "id": "B", "title": "Rotation curves of galaxies" }, { "id": "C", "title": "Mass distribution in our galaxy" }, { "id": "D", "title": "Tully–Fisher relation and MOND" } ], "context": [ { "type": "text", "value": "Follow hydrogen from the Bohr model and the 21 cm line to galactic rotation curves, mass distributions, and modified Newtonian dynamics. The original figures and physical constants are available in the paper reader below." } ], "parts": [ { "id": "A.1", "section": "A", "title": "Electron orbital speed", "points": 0.2, "page": 1, "solutionPage": 1, "prompt": "A non-relativistic electron moves on a circular orbit around a fixed proton. Determine its speed as a function of the orbit radius." }, { "id": "A.2", "section": "A", "title": "Bohr radius and ground-state speed", "points": 0.5, "page": 1, "solutionPage": 1, "prompt": "Use quantization of the electron’s angular momentum to show that each allowed radius is the ground-state radius multiplied by the square of the principal quantum number. Express the Bohr radius in terms of the fine-structure constant, electron mass, speed of light, and reduced Planck constant. Calculate it to three significant digits, and express the ground-state speed in terms of the fine-structure constant and speed of light.", "model": "$$L=n\\hbar,\\qquad \\alpha=\\frac{e^2}{4\\pi\\varepsilon_0\\hbar c}$$" }, { "id": "A.3", "section": "A", "title": "Hydrogen energy levels", "points": 0.5, "page": 1, "solutionPage": 2, "prompt": "Determine the electron’s mechanical energy on each allowed orbit in terms of elementary charge, vacuum permittivity, Bohr radius, and principal quantum number. Express the ground-state energy using the fine-structure constant, electron mass, and speed of light, then evaluate it in electronvolts." }, { "id": "A.4", "section": "A", "title": "Magnetic field in the electron frame", "points": 0.5, "page": 1, "solutionPage": 2, "prompt": "In the electron’s reference frame, the proton moves around it at the Bohr radius. Determine the magnitude of the magnetic field at the electron using vacuum permeability, elementary charge, the fine-structure constant, speed of light, and Bohr radius." }, { "id": "A.5", "section": "A", "title": "Fine structure and the 21 cm line", "points": 0.5, "page": 2, "solutionPage": 3, "prompt": "Express the fine-structure energy splitting in terms of the fine-structure constant and ground-state energy. Use the magnetic-moment approximation and hyperfine relation below to obtain the wavelength emitted in a hyperfine transition, and evaluate it to two significant digits.", "model": "$$\\mathcal M_s\\simeq\\frac{e\\hbar}{m_e},\\qquad \\Delta E_{\\mathrm{HF}}\\simeq3.72\\frac{m_e}{m_p}\\Delta E_{\\mathrm F}$$" }, { "id": "B.1", "section": "B", "title": "Circular speed in a spherical galaxy", "points": 0.2, "page": 2, "solutionPage": 4, "prompt": "A spherical galaxy has a gravitational potential per unit mass that depends only on distance from its centre. Express the circular orbital speed at a given radius in terms of that radius and the radial derivative of the potential." }, { "id": "B.2", "section": "B", "title": "Mass of the galactic bulge", "points": 0.5, "page": 2, "solutionPage": 4, "prompt": "Use the red model rotation curve of NGC 6946 in Figure 1(B) to infer the bulge mass, expressed in solar masses. The bulge is approximately homogeneous inside one kiloparsec, and the model outside is Keplerian. Use the axis values and unit conversions from the original page." }, { "id": "B.3", "section": "B", "title": "Dark-halo rotation curve", "points": 1.8, "page": 3, "solutionPage": 5, "prompt": "For the mass-density model below, derive the stated circular-speed form and identify both constants. Simplify it far inside and far outside the characteristic radius. At large radius, show how enclosed mass depends only on the density constant and radius. Use Figure 1 to estimate the mass of NGC 6946 visible within the photograph.", "model": "$$\\rho_m(r)=\\frac{C_m}{r_m^2+r^2},\\qquad v_{c,m}(r)=\\sqrt{k_1-\\frac{k_2\\arctan(r/r_m)}r}$$" }, { "id": "C.1", "section": "C", "title": "Vertical stability of the galactic plane", "points": 0.5, "page": 3, "solutionPage": 7, "prompt": "For the potential below, derive the equation of vertical motion at a fixed cylindrical radius smaller than the reference radius. Show that the galactic plane is a stable equilibrium and determine the angular frequency of small vertical oscillations.", "model": "$$\\varphi_G(r,z)=\\varphi_0\\ln\\!\\left(\\frac r{r_0}\\right)\\exp\\!\\left[-\\left(\\frac z{z_0}\\right)^2\\right],\\qquad\\varphi_0>0$$" }, { "id": "C.2", "section": "C", "title": "Recovering a flat rotation curve", "points": 0.6, "page": 3, "solutionPage": 7, "prompt": "With motion restricted to the galactic plane, identify whether the small-radius or large-radius limit of the density model in Part B recovers the logarithmic potential in C.1. Specify the corresponding potential constant and express the radius-independent circular speed in terms of it." }, { "id": "C.3", "section": "C", "title": "Galactic Doppler geometry", "points": 0.7, "page": 4, "solutionPage": 8, "prompt": "The Sun and a hydrogen-emitting galactic object move on circular orbits with the same speed. Using Figure 2 and the longitude of the line of sight, determine the object’s radial velocity relative to the Sun. Rearrange the result to express its distance from the galactic centre in terms of the Sun’s orbital radius, Sun’s speed, longitude, and measured relative radial velocity." }, { "id": "C.4", "section": "C", "title": "Locate three hydrogen sources", "points": 0.6, "page": 5, "solutionPage": 9, "prompt": "Using the three spectral peaks in Figure 3 on page 4, determine each source’s relative radial velocity to three significant digits and distance from the galactic centre to two significant digits. Express distances as multiples of the Sun’s orbital radius. The observing longitude is 30 degrees, the reference frequency is 1.42 GHz, and the Sun’s orbital speed is 220 kilometres per second." }, { "id": "C.5", "section": "C", "title": "Map the sources in the Milky Way", "points": 0.6, "page": 5, "solutionPage": 10, "prompt": "On the top-view galactic diagram in the official answer sheet, indicate the possible positions of the three sources identified in Figure 3. Explain what repeated measurements at different galactic longitudes would reveal." }, { "id": "D.1", "section": "D", "title": "The Tully–Fisher exponent", "points": 0.4, "page": 5, "solutionPage": 12, "prompt": "Assuming a galaxy’s radius is independent of its mass, show that the density model in Part B produces a power-law relation between total mass and asymptotic circular speed. Determine its exponent and prefactor. Read the observed Tully–Fisher exponent from Figure 4 and compare the two results.", "model": "$$M_{\\mathrm{tot}}=\\eta\\,v_{c,\\infty}^{\\gamma}$$" }, { "id": "D.2", "section": "D", "title": "Acceleration at a galaxy’s outskirts", "points": 0.2, "page": 6, "solutionPage": 13, "prompt": "Use the rotation data for NGC 6946 in Figure 1 on page 2 to estimate, within Newtonian mechanics, the magnitude of the acceleration of an orbiting mass in the galaxy’s outer regions." }, { "id": "D.3", "section": "D", "title": "Tully–Fisher from MOND", "points": 0.8, "page": 6, "solutionPage": 13, "prompt": "In the deep-MOND regime, determine the mass–speed power-law exponent for circular motion around a fixed central mass. Use the NGC 6946 data and/or the Tully–Fisher relation to estimate the MOND acceleration constant and verify that the assumed low-acceleration regime is consistent.", "model": "$$\\vec F=m\\mu(a/a_0)\\vec a,\\qquad \\mu(x)=\\frac{x}{1+x},\\qquad a\\ll a_0$$" }, { "id": "D.4", "section": "D", "title": "A complete MOND rotation curve", "points": 0.9, "page": 6, "solutionPage": 14, "prompt": "Determine the circular-speed profile at every radius for a homogeneous spherical mass distribution of total mass M and radius Rb using the MOND interpolation law. Distinguish the relevant interior/exterior and acceleration regimes.", "model": "$$\\mu(x)=\\frac{x}{1+x}$$" } ], "statement": { "mode": "original-pages-with-editorial-index", "pdf": "/ipho/2025-t1-questions.pdf", "pages": [ "/ipho/2025-t1-page-1.webp", "/ipho/2025-t1-page-2.webp", "/ipho/2025-t1-page-3.webp", "/ipho/2025-t1-page-4.webp", "/ipho/2025-t1-page-5.webp", "/ipho/2025-t1-page-6.webp" ] }, "solution": { "mode": "official-pdf", "pdf": "/ipho/2025-t1-solutions.pdf", "pages": [ "/ipho/2025-t1-solution-page-1.webp", "/ipho/2025-t1-solution-page-2.webp", "/ipho/2025-t1-solution-page-3.webp", "/ipho/2025-t1-solution-page-4.webp", "/ipho/2025-t1-solution-page-5.webp", "/ipho/2025-t1-solution-page-6.webp", "/ipho/2025-t1-solution-page-7.webp", "/ipho/2025-t1-solution-page-8.webp", "/ipho/2025-t1-solution-page-9.webp", "/ipho/2025-t1-solution-page-10.webp", "/ipho/2025-t1-solution-page-11.webp", "/ipho/2025-t1-solution-page-12.webp", "/ipho/2025-t1-solution-page-13.webp", "/ipho/2025-t1-solution-page-14.webp", "/ipho/2025-t1-solution-page-15.webp" ] }, "rating": { "elo": null, "status": "uncalibrated" }, "importedAt": null }, { "id": "ipho-2025-t2", "competition": "IPhO", "year": 2025, "code": "T2", "totalPoints": 10, "documents": { "questions": { "pdf": "/ipho/2025-t2-questions.pdf", "images": [ "/ipho/2025-t2-page-1.webp", "/ipho/2025-t2-page-2.webp", "/ipho/2025-t2-page-3.webp", "/ipho/2025-t2-page-4.webp", "/ipho/2025-t2-page-5.webp", "/ipho/2025-t2-page-6.webp" ] }, "solutions": { "pdf": "/ipho/2025-t2-solutions.pdf", "images": [ "/ipho/2025-t2-solution-page-1.webp", "/ipho/2025-t2-solution-page-2.webp", "/ipho/2025-t2-solution-page-3.webp", "/ipho/2025-t2-solution-page-4.webp", "/ipho/2025-t2-solution-page-5.webp", "/ipho/2025-t2-solution-page-6.webp", "/ipho/2025-t2-solution-page-7.webp", "/ipho/2025-t2-solution-page-8.webp", "/ipho/2025-t2-solution-page-9.webp", "/ipho/2025-t2-solution-page-10.webp", "/ipho/2025-t2-solution-page-11.webp", "/ipho/2025-t2-solution-page-12.webp", "/ipho/2025-t2-solution-page-13.webp", "/ipho/2025-t2-solution-page-14.webp", "/ipho/2025-t2-solution-page-15.webp", "/ipho/2025-t2-solution-page-16.webp", "/ipho/2025-t2-solution-page-17.webp", "/ipho/2025-t2-solution-page-18.webp", "/ipho/2025-t2-solution-page-19.webp", "/ipho/2025-t2-solution-page-20.webp" ] } }, "title": "Cox’s Timepiece", "topics": [ "Mechanics", "Thermodynamics" ], "tags": [ "Hydrostatics", "Vapour pressure", "Barometric clock", "Friction", "Energy harvesting" ], "summary": "Study a clock powered by atmospheric-pressure fluctuations. Liquids are incompressible, surface tension is neglected, temperature is constant, and atmospheric pressure does not vary with altitude. Use the original diagrams and tables for the tube, bulb, cistern, and pulley geometry.", "attribution": "55th International Physics Olympiad, France, 2025, T2. Original statements, figures, answer sheets, and solutions are credited to the IPhO 2025 organisers; see the original paper for contributor and image credits.", "sourceUrl": "https://www.ipho2025.fr/sujets-officiels-ipho-france-2025", "pages": 6, "solutionPages": 20, "sections": [ { "id": "A", "title": "Pulling on a submerged tube" }, { "id": "B", "title": "Two-part barometric tube" }, { "id": "C", "title": "Cox’s timepiece" } ], "context": [ { "type": "text", "value": "Study a clock powered by atmospheric-pressure fluctuations. Liquids are incompressible, surface tension is neglected, temperature is constant, and atmospheric pressure does not vary with altitude. Use the original diagrams and tables for the tube, bulb, cistern, and pulley geometry." } ], "parts": [ { "id": "A.1", "section": "A", "title": "Pressure and holding force", "points": 0.2, "page": 2, "solutionPage": 2, "prompt": "A thin-walled tube, closed at the top and open at the bottom, is lifted slowly from a liquid bath without initially containing any gas. For configuration b in Figure 2, express the pressure at the top of the liquid and the upward holding-force vector. Use atmospheric pressure, density, empty tube mass, tube cross-sectional area, top height, gravitational acceleration, and the upward unit vector." }, { "id": "A.2", "section": "A", "title": "Cavitation and force plateaus", "points": 0.8, "page": 2, "solutionPage": 3, "prompt": "For each of the three water experiments in Table 1, identify whether force–height behaviour A or B occurs. Complete the official answer-sheet table with the maximum holding force and, when relevant, the threshold height where the force plateaus. Use the stated temperatures, densities, saturated vapour pressures, tube dimensions, and atmospheric pressure." }, { "id": "A.3", "section": "A", "title": "Neglecting mercury vapour pressure", "points": 0.3, "page": 3, "solutionPage": 4, "prompt": "For mercury at 20 degrees Celsius, express the relative error in the maximum holding force caused by neglecting saturated vapour pressure compared with atmospheric pressure. Evaluate the error numerically using the mercury data on page 2." }, { "id": "B.1", "section": "B", "title": "Which mercury adds to the weight?", "points": 0.3, "page": 3, "solutionPage": 5, "prompt": "Consider the air-free barometric tube with a wider bulb shown in Figure 3. On the official answer-sheet diagram, shade the volume of mercury responsible for the added-mass term in the holding force. The lower tube and upper bulb have different cross-sectional areas.", "model": "$$\\vec F=(m_{\\mathrm{tb}}+m_{\\mathrm{add}})g\\vec u_z$$" }, { "id": "B.2", "section": "B", "title": "Added mass as the tube rises", "points": 1.4, "page": 3, "solutionPage": 6, "prompt": "At fixed atmospheric pressure, lift the two-part tube from full submersion until its base reaches the bath surface. Sketch added mass against the height of the tube–bulb junction over the full specified interval. Give analytical expressions for the slopes of each segment and the heights of all corners, in terms of the dimensions, cross sections, pressure, density, and gravity.", "model": "$$h_t\\in[-H_b,H_t],\\qquad H_t=80\\,\\mathrm{cm},\\quad H_b=20\\,\\mathrm{cm}$$" }, { "id": "B.3", "section": "B", "title": "Pressure-driven mass variation", "points": 0.3, "page": 4, "solutionPage": 7, "prompt": "Hold the tube–bulb junction at a fixed height so that the mercury free surface stays inside the bulb. Atmospheric pressure follows the triangular perturbation in Figure 4 with amplitude 500 pascals. Derive and evaluate the amplitude of added-mass variation for tube and bulb cross sections of 5 and 200 square centimetres, respectively." }, { "id": "C.1", "section": "C", "title": "The threshold for motion", "points": 1, "page": 5, "solutionPage": 9, "prompt": "Use the cistern, barometric tube, pulley arrangement, dimensions, and initial balance described on pages 4–5. Determine the threshold of the dimensionless friction parameter above which the sliding mass remains permanently at rest. Use its definition in Equation 3 and the periodic pressure forcing." }, { "id": "C.2", "section": "C", "title": "Restoring tension at the stop", "points": 1, "page": 6, "solutionPage": 12, "prompt": "Temporarily block the sliding mass at the positive displacement stop while the atmospheric-pressure perturbation is zero. Derive the vector sum of the two cable tensions on the mass in terms of mercury density, gravity, stop displacement, and the relevant cross-sectional areas." }, { "id": "C.3", "section": "C", "title": "Motion regimes and hysteresis", "points": 2, "page": 6, "solutionPage": 14, "prompt": "When the friction parameter is below the no-motion threshold, distinguish the two possible operating regimes. Complete the official answer-sheet table with conditions expressed as inequalities in the dimensionless friction and displacement parameters defined in Equations 3–4. Sketch the normalized mass displacement over the first three pressure periods, starting from the stated balanced initial condition. Coordinates of special points are not required." }, { "id": "C.4", "section": "C", "title": "Maximize the clock’s energy", "points": 1, "page": 6, "solutionPage": 17, "prompt": "In steady energy-harvesting operation, assume the bulb and cistern areas are approximately equal and the lower-tube area is negligible. Optimize friction force and stop displacement to maximize energy dissipated per pressure cycle. Give the optimal force, displacement, and maximum recovered energy in terms of density, gravity, cistern area, and pressure amplitude. Evaluate the energy for the supplied apparatus and 500-pascal amplitude." }, { "id": "C.5", "section": "C", "title": "Pressure work and efficiency", "points": 1.7, "page": 6, "solutionPage": 17, "prompt": "For the optimal operating configuration, calculate the work supplied to the system by atmospheric-pressure forces during a complete cycle. Determine the ratio of recovered energy to that pressure work. A pressure–volume diagram may be useful." } ], "statement": { "mode": "original-pages-with-editorial-index", "pdf": "/ipho/2025-t2-questions.pdf", "pages": [ "/ipho/2025-t2-page-1.webp", "/ipho/2025-t2-page-2.webp", "/ipho/2025-t2-page-3.webp", "/ipho/2025-t2-page-4.webp", "/ipho/2025-t2-page-5.webp", "/ipho/2025-t2-page-6.webp" ] }, "solution": { "mode": "official-pdf", "pdf": "/ipho/2025-t2-solutions.pdf", "pages": [ "/ipho/2025-t2-solution-page-1.webp", "/ipho/2025-t2-solution-page-2.webp", "/ipho/2025-t2-solution-page-3.webp", "/ipho/2025-t2-solution-page-4.webp", "/ipho/2025-t2-solution-page-5.webp", "/ipho/2025-t2-solution-page-6.webp", "/ipho/2025-t2-solution-page-7.webp", "/ipho/2025-t2-solution-page-8.webp", "/ipho/2025-t2-solution-page-9.webp", "/ipho/2025-t2-solution-page-10.webp", "/ipho/2025-t2-solution-page-11.webp", "/ipho/2025-t2-solution-page-12.webp", "/ipho/2025-t2-solution-page-13.webp", "/ipho/2025-t2-solution-page-14.webp", "/ipho/2025-t2-solution-page-15.webp", "/ipho/2025-t2-solution-page-16.webp", "/ipho/2025-t2-solution-page-17.webp", "/ipho/2025-t2-solution-page-18.webp", "/ipho/2025-t2-solution-page-19.webp", "/ipho/2025-t2-solution-page-20.webp" ] }, "rating": { "elo": null, "status": "uncalibrated" }, "importedAt": null }, { "id": "ipho-2025-t3", "competition": "IPhO", "year": 2025, "code": "T3", "totalPoints": 10, "documents": { "questions": { "pdf": "/ipho/2025-t3-questions.pdf", "images": [ "/ipho/2025-t3-page-1.webp", "/ipho/2025-t3-page-2.webp", "/ipho/2025-t3-page-3.webp", "/ipho/2025-t3-page-4.webp", "/ipho/2025-t3-page-5.webp" ] }, "solutions": { "pdf": "/ipho/2025-t3-solutions.pdf", "images": [ "/ipho/2025-t3-solution-page-1.webp", "/ipho/2025-t3-solution-page-2.webp", "/ipho/2025-t3-solution-page-3.webp", "/ipho/2025-t3-solution-page-4.webp", "/ipho/2025-t3-solution-page-5.webp", "/ipho/2025-t3-solution-page-6.webp", "/ipho/2025-t3-solution-page-7.webp", "/ipho/2025-t3-solution-page-8.webp", "/ipho/2025-t3-solution-page-9.webp", "/ipho/2025-t3-solution-page-10.webp", "/ipho/2025-t3-solution-page-11.webp", "/ipho/2025-t3-solution-page-12.webp" ] } }, "title": "Champagne!", "topics": [ "Thermodynamics", "Waves" ], "tags": [ "Surface tension", "Henry’s law", "Bubble dynamics", "Acoustics", "Ideal gases" ], "summary": "Study bubble nucleation and growth, the sound of bursting bubbles, and the thermodynamics of opening a bottle. Bubbles contain only carbon dioxide; gases are ideal. Henry’s constants, material properties, photographs, and graphs are in the original paper.", "attribution": "55th International Physics Olympiad, France, 2025, T3. Original statements, figures, answer sheets, and solutions are credited to the IPhO 2025 organisers; see the original paper for contributor and image credits.", "sourceUrl": "https://www.ipho2025.fr/sujets-officiels-ipho-france-2025", "pages": 5, "solutionPages": 12, "sections": [ { "id": "A", "title": "Nucleation, growth and rise of bubbles" }, { "id": "B", "title": "Acoustic emission of a bursting bubble" }, { "id": "C", "title": "Popping champagne" } ], "context": [ { "type": "text", "value": "Study bubble nucleation and growth, the sound of bursting bubbles, and the thermodynamics of opening a bottle. Bubbles contain only carbon dioxide; gases are ideal. Henry’s constants, material properties, photographs, and graphs are in the original paper." } ], "parts": [ { "id": "A.1", "section": "A", "title": "Laplace pressure of a bubble", "points": 0.2, "page": 1, "solutionPage": 1, "prompt": "A spherical carbon-dioxide bubble forms in champagne at atmospheric liquid pressure. Express the bubble’s internal pressure in terms of ambient pressure, bubble radius, and surface tension. There is one liquid–gas interface." }, { "id": "A.2", "section": "A", "title": "The critical nucleation radius", "points": 0.5, "page": 1, "solutionPage": 1, "prompt": "Using Henry’s law and the Laplace pressure jump, express the critical bubble radius above which a bubble grows, in terms of atmospheric pressure, surface tension, dissolved carbon-dioxide concentration, and its equilibrium concentration at atmospheric pressure. Evaluate the radius when the dissolved concentration is four times the equilibrium value." }, { "id": "A.3", "section": "A", "title": "Which bubble-growth law fits?", "points": 1.2, "page": 2, "solutionPage": 2, "prompt": "For a visible bubble, neglect the pressure excess due to surface tension and write its carbon-dioxide content using the ideal-gas law. Derive the radius as a function of time for both interfacial-flux models below, starting from the given initial radius. Use the measured radius–time graph in Figure 2 to select the model and determine the appropriate transport coefficient numerically.", "model": "$$j_1=\\frac D a(c_\\ell-c_b),\\qquad j_2=K(c_\\ell-c_b),\\qquad c_\\ell\\simeq4c_0$$" }, { "id": "A.4", "section": "A", "title": "Rising bubbles and viscosity", "points": 0.8, "page": 2, "solutionPage": 3, "prompt": "List the main forces on a vertically rising bubble and obtain its terminal speed as a function of radius using Stokes drag. Estimate the liquid’s dynamic viscosity using Figure 3, liquid density, gravitational acceleration, and the bubble-emission frequency of 20 hertz.", "model": "$$F_{\\mathrm{drag}}=6\\pi\\eta a v$$" }, { "id": "A.5", "section": "A", "title": "Bubble radius at the surface", "points": 0.5, "page": 2, "solutionPage": 4, "prompt": "Assume a constant bubble-radius growth rate throughout the rise and a final radius much larger than the initial radius. Express the radius at the free surface in terms of the distance risen, growth rate, and any needed constants. Evaluate it for a 10-centimetre liquid depth using the growth rate inferred from Figure 2." }, { "id": "A.6", "section": "A", "title": "How quickly does champagne go flat?", "points": 1.1, "page": 3, "solutionPage": 5, "prompt": "Bubbles are produced at a constant frequency at a specified number of nucleation sites in a glass of known liquid height and volume. Neglect initial bubble radius and direct carbon-dioxide diffusion through the free surface. Derive the differential equation for dissolved concentration versus time and obtain its characteristic decay time." }, { "id": "B.1", "section": "B", "title": "Retraction speed of a pierced film", "points": 1.1, "page": 3, "solutionPage": 6, "prompt": "A circular hole expands at constant speed through a liquid film of uniform thickness. The surrounding film is initially at rest and liquid accumulates in the rim. Only half the decrease in surface energy becomes kinetic energy; ignore the rim’s surface-area change relative to the film’s. Express the retraction speed in terms of liquid density, surface tension, and film thickness, using Figure 4." }, { "id": "B.2", "section": "B", "title": "The Helmholtz resonator", "points": 1.1, "page": 4, "solutionPage": 7, "prompt": "A gas-filled cavity at atmospheric pressure has an opening of cross-sectional area S containing an effective piston mass. Small piston displacements compress or expand the cavity gas adiabatically; gravity on the piston is negligible. Express the oscillation frequency using equilibrium cavity volume, opening area, piston mass, pressure, and adiabatic coefficient. Refer to Figure 5." }, { "id": "B.3", "section": "B", "title": "Infer bubble size from its sound", "points": 1.1, "page": 4, "solutionPage": 8, "prompt": "Apply the Helmholtz model to a bursting spherical bubble. Use the aperture-dependent effective gas mass, maximum aperture radius, and gas density given on page 4. The maximum emitted frequency is 40 kilohertz and the bursting time is 0.03 milliseconds. Determine the bubble radius and the thickness of the liquid film separating it from the atmosphere.", "model": "$$m_p=\\frac83\\rho_g r^3,\\qquad r_c=\\frac2{\\sqrt3}a^2\\sqrt{\\frac{\\rho_\\ell g_0}{\\sigma}}$$" }, { "id": "C.1", "section": "C", "title": "Pressure inside a sealed bottle", "points": 0.4, "page": 5, "solutionPage": 9, "prompt": "A sealed bottle contains a total of 0.2 moles of carbon dioxide, partitioned between 750 millilitres of liquid and 25 millilitres of gas. Calculate the equilibrium gas pressure at 6 and 20 degrees Celsius using Henry’s law, the ideal-gas law, and the temperature-dependent Henry constants on page 1." }, { "id": "C.2", "section": "C", "title": "Blue fog or white fog?", "points": 0.7, "page": 5, "solutionPage": 10, "prompt": "For rapid reversible adiabatic expansion on opening, calculate the final gas temperature starting at 6 and 20 degrees Celsius, first assuming no phase transition. For these questions use initial pressures of 4.69 and 7.45 bar, respectively. Use the supplied carbon-dioxide sublimation curve to decide which of the four statements on page 5 are true: grey-white water fog or blue solid-carbon-dioxide fog at each initial temperature. Several statements may be true.", "model": "$$\\log_{10}\\!\\left(\\frac{P_{\\mathrm{CO}_2}^{\\mathrm{sat}}}{P_0}\\right)=6.81-\\frac{1.30\\times10^3\\,\\mathrm K}{T-3.49\\,\\mathrm K},\\qquad\\gamma=1.3$$" }, { "id": "C.3", "section": "C", "title": "How high does the cork fly?", "points": 1.3, "page": 5, "solutionPage": 11, "prompt": "Determine the maximum height reached by the cork when a bottle initially at 6 degrees Celsius is opened. Friction is proportional to the cork’s contact area, and the initial net pressure force only just overcomes friction. The cork mass is 10 grams, its diameter is 1.8 centimetres, and its initially inserted cylindrical length is 2.5 centimetres. Use the specified initial pressure and reversible adiabatic expansion. Neglect net pressure force after the cork clears the neck." } ], "statement": { "mode": "original-pages-with-editorial-index", "pdf": "/ipho/2025-t3-questions.pdf", "pages": [ "/ipho/2025-t3-page-1.webp", "/ipho/2025-t3-page-2.webp", "/ipho/2025-t3-page-3.webp", "/ipho/2025-t3-page-4.webp", "/ipho/2025-t3-page-5.webp" ] }, "solution": { "mode": "official-pdf", "pdf": "/ipho/2025-t3-solutions.pdf", "pages": [ "/ipho/2025-t3-solution-page-1.webp", "/ipho/2025-t3-solution-page-2.webp", "/ipho/2025-t3-solution-page-3.webp", "/ipho/2025-t3-solution-page-4.webp", "/ipho/2025-t3-solution-page-5.webp", "/ipho/2025-t3-solution-page-6.webp", "/ipho/2025-t3-solution-page-7.webp", "/ipho/2025-t3-solution-page-8.webp", "/ipho/2025-t3-solution-page-9.webp", "/ipho/2025-t3-solution-page-10.webp", "/ipho/2025-t3-solution-page-11.webp", "/ipho/2025-t3-solution-page-12.webp" ] }, "rating": { "elo": null, "status": "uncalibrated" }, "importedAt": null } ], "integrity": { "algorithm": "sha256", "assets": 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