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International Physics Olympiad (IPhO) Practice

15 challenging physics problems covering mechanics, electrodynamics, thermodynamics, optics, and modern physics.

Mechanics

1 Precessing Gyroscope on a Turntable Rigid Body Hard
A gyroscope with moment of inertia \(I\) spins at angular velocity \(\omega_s\) about its symmetry axis, which is horizontal. It is placed on a turntable rotating at angular velocity \(\Omega\) about a vertical axis. The gyroscope's center of mass is at distance \(R\) from the turntable axis and height \(h\) above the turntable surface.
  1. Find the steady-state precession rate of the gyroscope axis in the turntable frame.
  2. Determine the condition on \(\omega_s\) for stable precession.
  3. If the gyroscope's spin decreases due to friction (time constant \(\tau\)), describe the qualitative behavior.
Hint
In the rotating frame of the turntable, there are Coriolis and centrifugal pseudo-forces. The precession in the lab frame is \(\omega_p = \frac{mgh}{I\omega_s} + \Omega\). For stability, the spin must be fast enough that gyroscopic effects dominate: \(\omega_s \gg \frac{mgh}{I\Omega}\). As spin decays, the nutation angle increases until the gyroscope topples.
2 Lagrangian of a Bead on a Rotating Hoop Lagrangian Mechanics Hard
A circular hoop of radius \(R\) rotates about a vertical diameter at constant angular velocity \(\omega\). A bead of mass \(m\) slides frictionlessly along the hoop. Let \(\theta\) be the angle from the bottom of the hoop.
  1. Write the Lagrangian and derive the equation of motion.
  2. Find all equilibrium positions and determine their stability as a function of \(\omega\).
  3. Find the frequency of small oscillations about each stable equilibrium.
Hint
The Lagrangian is \(\mathcal{L} = \tfrac{1}{2}mR^2(\dot{\theta}^2 + \omega^2\sin^2\theta) - mgR(1 - \cos\theta)\). The effective potential is \(V_{\text{eff}}(\theta) = -\tfrac{1}{2}mR^2\omega^2\sin^2\theta + mgR(1-\cos\theta)\). Setting \(\frac{dV_{\text{eff}}}{d\theta} = 0\): \(\sin\theta\!\left(mR\omega^2\cos\theta - \frac{mg}{R}\right) = 0\), giving \(\theta = 0\) and \(\cos\theta = \frac{g}{R\omega^2}\). The non-trivial equilibrium exists only when \(\omega > \omega_c = \sqrt{g/R}\).
3 Rolling Cylinder Inside a Cylinder Rigid Body Medium
A solid cylinder of mass \(m\) and radius \(r\) rolls without slipping inside a fixed hollow cylinder of radius \(R\). Assuming small oscillations:
  1. Show that the motion is simple harmonic and find the period.
  2. Compare this with a point mass sliding inside the same cylinder (no friction).
Hint
Using the rolling constraint \(v = r\omega_{\text{roll}}\) and the geometry of contact, the effective gravitational acceleration for oscillations is \(g_{\text{eff}} = \frac{2g}{3}\). The period is \(T = 2\pi\sqrt{\frac{3(R-r)}{2g}}\). For a sliding point mass, \(T = 2\pi\sqrt{\frac{R-r}{g}}\), which is shorter by a factor of \(\sqrt{2/3}\).

Electrodynamics

4 Infinite Resistor Ladder Circuit Analysis Medium
An infinite ladder network is constructed with resistors: each rung has resistance \(R_2\) and each rail segment has resistance \(R_1\).
  1. Find the equivalent resistance between the two input terminals.
  2. Now suppose \(R_1\) is replaced by an inductor \(L\) and \(R_2\) by a capacitor \(C\). Find the input impedance as a function of frequency and determine the cutoff frequency.
Hint
For part (a): Let the equivalent resistance be \(R_{\text{eq}}\). The infinite ladder is self-similar, so \(R_{\text{eq}} = R_1 + R_2 \| R_{\text{eq}} = R_1 + \frac{R_2 R_{\text{eq}}}{R_2 + R_{\text{eq}}}\). Solve the quadratic. For part (b): Replace \(R_1\) with \(j\omega L\) and \(R_2\) with \(\frac{1}{j\omega C}\). The cutoff frequency where behavior changes is \(\omega_c = \frac{2}{\sqrt{LC}}\). This is the low-pass filter model for a transmission line.
5 Falling Magnet in a Conducting Tube EM Induction Hard
A small strong magnet (magnetic dipole moment \(\mathbf{m}\), mass \(M\)) falls vertically through a long copper tube of inner radius \(a\), wall thickness \(t \ll a\), and conductivity \(\sigma\).
  1. Derive an expression for the terminal velocity of the magnet.
  2. Estimate the terminal velocity for a neodymium magnet (\(m = 1\;\text{A}\!\cdot\!\text{m}^2\), \(M = 50\;\text{g}\)) in a copper tube (\(a = 1.5\;\text{cm}\), \(t = 1\;\text{mm}\), \(\sigma = 5.8 \times 10^7\;\text{S/m}\)).
  3. How does the terminal velocity depend on tube radius?
Hint
The changing flux from the falling dipole induces eddy currents in the tube wall. The retarding force is proportional to velocity. At terminal velocity, \(Mg = F_{\text{drag}}\). The drag force can be shown to be \(F = \frac{9\pi\mu_0^2 \sigma t m^2 v}{4a^3}\) (approximate for thin wall). Terminal velocity \(v_t = \frac{4Mga^3}{9\pi\mu_0^2 \sigma t m^2}\). Note \(v_t \propto a^3\): wider tubes give faster terminal speeds.
6 Electromagnetic Wave in a Plasma EM Waves Olympiad Level
An electromagnetic wave of frequency \(f\) enters a plasma with free electron density \(n_e\).
  1. Derive the plasma frequency \(\omega_p = \sqrt{\dfrac{n_e e^2}{m_e \epsilon_0}}\).
  2. Show that the dispersion relation is \(\omega^2 = \omega_p^2 + c^2 k^2\) and find the phase and group velocities.
  3. What happens when \(\omega < \omega_p\)? Explain why the ionosphere reflects AM radio but not FM radio.
Hint
Model each electron as a driven harmonic oscillator without damping: \(m_e \ddot{x} = -eE\). The polarization \(P = -n_e e x\) gives a dielectric function \(\epsilon(\omega) = 1 - \frac{\omega_p^2}{\omega^2}\). Phase velocity \(v_p = \frac{c}{\sqrt{1 - \omega_p^2/\omega^2}} > c\), group velocity \(v_g = c\sqrt{1 - \omega_p^2/\omega^2} < c\). Below \(\omega_p\), the wave vector becomes imaginary — evanescent wave, total reflection. Ionospheric \(\omega_p \sim 5\text{--}10\;\text{MHz}\); AM (~1 MHz) reflects, FM (~100 MHz) passes through.

Thermodynamics & Statistical Physics

7 Carnot Engine with Finite Heat Reservoirs Thermodynamics Medium
Two identical metal blocks, each with heat capacity \(C\), are initially at temperatures \(T_1\) and \(T_2\) (\(T_1 > T_2\)). A heat engine operates between them until they reach thermal equilibrium.
  1. What is the final equilibrium temperature if a Carnot engine is used?
  2. What is the maximum total work extracted?
  3. Compare with the case where the blocks are simply placed in thermal contact.
Hint
For a Carnot engine, total entropy change is zero. \(C \ln\!\left(\frac{T_f}{T_1}\right) + C \ln\!\left(\frac{T_f}{T_2}\right) = 0\), so \(T_f = \sqrt{T_1 T_2}\) (geometric mean). Maximum work \(W = C\!\left(T_1 + T_2 - 2\sqrt{T_1 T_2}\right) = C\!\left(\sqrt{T_1} - \sqrt{T_2}\right)^2\). Without the engine, \(T_f = \frac{T_1 + T_2}{2}\) (arithmetic mean > geometric mean) and \(W = 0\).
8 Adiabatic Atmosphere Statistical Physics Hard
In an adiabatic atmosphere (no heat transfer between adjacent layers), air parcels rise and fall adiabatically.
  1. Derive the adiabatic lapse rate: \(\dfrac{dT}{dz} = -\dfrac{g(\gamma - 1)}{\gamma R/M}\), where \(M\) is the molar mass of air.
  2. Calculate the numerical value for Earth's atmosphere.
  3. Explain why the actual lapse rate is less than the adiabatic value and what happens when a parcel is displaced vertically.
Hint
For an adiabatic process, \(TV^{\gamma-1} = \text{const}\). With the hydrostatic equation \(dP = -\rho g\,dz\) and ideal gas law \(P = \frac{\rho RT}{M}\), eliminate \(P\) and \(\rho\) to get \(\frac{dT}{dz} = -\frac{Mg}{\gamma R/(\gamma-1)}\). For air (\(\gamma = 1.4\), \(M = 29\;\text{g/mol}\)): \(\frac{dT}{dz} \approx -9.8\;\text{K/km}\). Actual \(\approx -6.5\;\text{K/km}\) due to moisture and radiation. If actual lapse rate exceeds adiabatic, convective instability results.

Optics

9 Thin Film Interference Interference Medium
A soap film (\(n = 1.33\)) of thickness \(d\) is illuminated by white light at near-normal incidence.
  1. Derive the condition for constructive and destructive interference in reflected light, accounting for phase changes at both surfaces.
  2. If the film appears green (\(\lambda = 530\;\text{nm}\)) at the thinnest point, find the minimum thickness.
  3. What color does the transmitted light appear at this thickness?
Hint
At normal incidence, light reflects from top surface (low-to-high \(n\): \(\pi\) phase shift) and bottom surface (high-to-low \(n\): no phase shift). Net phase difference: \(\frac{2nd \cdot 2\pi}{\lambda} + \pi\). Constructive in reflection: \(2nd = \left(m + \tfrac{1}{2}\right)\lambda\). For minimum thickness (\(m=0\)): \(d = \frac{\lambda}{4n} = \frac{530}{4 \times 1.33} \approx 99.6\;\text{nm}\). Transmitted light is complementary — appears reddish/magenta.
10 Fiber Optic Modal Analysis Fiber Optics Hard
A step-index optical fiber has core radius \(a = 25\;\mu\text{m}\), core index \(n_1 = 1.48\), and cladding index \(n_2 = 1.46\).
  1. Calculate the numerical aperture and maximum acceptance angle.
  2. Estimate the number of guided modes at \(\lambda = 850\;\text{nm}\) using the V-number.
  3. What core radius would make this fiber single-mode at \(\lambda = 1550\;\text{nm}\)?
Hint
\(\text{NA} = \sqrt{n_1^2 - n_2^2} = \sqrt{1.48^2 - 1.46^2} \approx 0.242\). Acceptance angle \(\theta_{\max} = \arcsin(\text{NA}) \approx 14°\). \(V = \frac{2\pi a \cdot \text{NA}}{\lambda} = \frac{2\pi(25 \times 10^{-6})(0.242)}{850 \times 10^{-9}} \approx 44.8\). Number of modes \(\approx \frac{V^2}{2} \approx 1004\). For single-mode: \(V < 2.405\), so \(a < \frac{2.405\lambda}{2\pi \cdot \text{NA}} = \frac{2.405(1550\;\text{nm})}{2\pi \times 0.242} \approx 2.45\;\mu\text{m}\).

Modern Physics

11 Relativistic Doppler and Aberration Special Relativity Hard
A star emits light at frequency \(f_0\). An observer moves at velocity \(v\) at angle \(\theta\) to the line connecting them.
  1. Derive the relativistic Doppler formula: \(f = f_0\,\gamma(1 - \beta\cos\theta)\) where \(\beta = v/c\).
  2. Show that when \(\theta = \pi/2\) (transverse motion), there is still a Doppler shift (the transverse Doppler effect). Explain its origin.
  3. Derive the relativistic aberration formula relating the emission angle in the star's frame to the observed angle.
Hint
Use four-vectors: the photon four-momentum is \(k^\mu = (\omega/c, \mathbf{k})\). Lorentz-transform to the observer's frame. The transverse Doppler effect \(f = f_0/\gamma\) is purely due to time dilation — a moving clock runs slow, so the frequency is redshifted. For aberration: \(\cos\theta' = \frac{\cos\theta - \beta}{1 - \beta\cos\theta}\). This causes relativistic headlight effect at high speeds.
12 Compton Scattering Quantum Medium
A photon of energy \(E\) scatters off a stationary electron.
  1. Derive the Compton wavelength shift formula \(\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)\).
  2. For a 100 keV photon scattered at \(90°\), find the energies of the scattered photon and recoiling electron.
  3. What is the maximum energy that can be transferred to the electron? Express in terms of \(E\) and \(m_e c^2\).
Hint
Use conservation of four-momentum: \(p_\gamma + p_e = p'_\gamma + p'_e\). Square \((p'_e)^2 = (p_\gamma + p_e - p'_\gamma)^2\) and use \(p_e^2 = m_e^2 c^2\), \(p_\gamma^2 = 0\). For 100 keV at \(90°\): \(\lambda' = \lambda + \lambda_C = \frac{hc}{E} + \frac{h}{m_e c}\). \(E' \approx 83.6\;\text{keV}\), \(E_e \approx 16.4\;\text{keV}\). Maximum transfer at \(\theta = 180°\): \(E_{e,\max} = \frac{2E^2}{2E + m_e c^2}\).
13 Hydrogen Atom in a Magnetic Field Quantum Olympiad Level
A hydrogen atom in the \(n = 2\) state is placed in a uniform magnetic field \(B\).
  1. How many distinct energy levels does the \(n = 2\) state split into? List the quantum numbers for each.
  2. Calculate the energy splitting between adjacent levels for \(B = 1\;\text{T}\).
  3. Which transitions between these levels are allowed by selection rules (\(\Delta l = \pm 1\), \(\Delta m = 0, \pm 1\))? Sketch the spectrum.
Hint
\(n = 2\) has \(l = 0\) (\(m = 0\)) and \(l = 1\) (\(m = -1, 0, +1\)), giving 4 states total. In a weak field (normal Zeeman), the energy shift is \(\Delta E = m_l \mu_B B\) where \(\mu_B = 9.274 \times 10^{-24}\;\text{J/T}\). The \(l = 0\) state doesn't split; \(l = 1\) splits into 3 levels. Adjacent splitting: \(\mu_B B \approx 5.79 \times 10^{-5}\;\text{eV}\) for \(B = 1\;\text{T}\). Allowed transitions from \(2p\) to \(1s\): three lines (normal Zeeman triplet).

Experimental Design

14 Measuring the Speed of Sound Experimental Medium
You are given: a hollow tube (1 m long, open at both ends), a speaker connected to a signal generator, a microphone connected to an oscilloscope, a ruler, and a thermometer.
  1. Design an experiment to measure the speed of sound using resonance in the tube.
  2. What systematic errors might affect your measurement? How would you minimize them?
  3. How would you modify the experiment to measure the speed of sound in a gas other than air?
Hint
Drive the speaker at varying frequencies and detect resonance peaks with the microphone. For an open-open tube, resonances occur at \(f_n = \frac{nv}{2L}\). Plot \(f\) vs \(n\); slope \(= \frac{v}{2L}\). End correction: effective length is \(L + 2 \times 0.6r\). Temperature affects \(v\): \(v = 331.3\sqrt{T/273.15}\;\text{m/s}\). For other gases: seal the tube, fill with gas, repeat. Systematic errors: end corrections, temperature gradients, speaker non-linearity.
15 Determining Moment of Inertia Experimental Hard
You are given an irregularly shaped metal plate, a thin wire, a stopwatch, a ruler, a balance, and a set of known masses.
  1. Design an experiment to measure the moment of inertia of the plate about an axis through its center of mass.
  2. How would you locate the center of mass experimentally?
  3. Estimate the percentage uncertainty in your measurement, identifying the dominant source of error.
Hint
Use a torsion pendulum: suspend the plate from the wire at its center of mass. The period is \(T = 2\pi\sqrt{I/\kappa}\) where \(\kappa\) is the torsion constant. First calibrate \(\kappa\) using a known object (e.g., a disk). Find center of mass by suspending the plate from two different points and finding where plumb lines intersect. Dominant error is likely in \(T\) measurement (timing multiple oscillations helps) or in locating the exact center of mass. Typical uncertainty: 3-5%.