Physics Olympiad: Basic to Advanced Exercises - Theory Notes

  • Physics Olympiad: Basic to Advanced Exercises is a resource for high-school students interested in modern physics.
  • The Committee of Japan Physics Olympiad (JPhO) organizes Physics Challenge, a domestic competition in Japan, and selects students for the International Physics Olympiad (IPhO).
  • The book includes questions from the theoretical and experimental examinations of previous Physics Challenges.
  • Problems range from junior-high-school level to the forefront of advanced physics and technology.
  • The book emphasizes logical thinking, stamina, and interest in science.
  • The book provides basic knowledge and skills in physics.
  • The book aims to bridge the gap between basic concepts and cutting-edge science and technology and is useful in high school physics classes and extracurricular activities.

Chapter 1. General Physics

  • Fundamental units are the units of fundamental physical quantities (length, mass, time) from which other units are derived.
  • SI units: meter (m) for length, kilogram (kg) for mass, second (s) for time.
  • cgs system of units: gram (g) for mass, centimeter (cm) for length, second (s) for time.
  • Dimensional analysis is used to examine the relationship between physical quantities by investigating dimensions.

Elementary Problems

  • Problem 1.1: Conversion between SI and cgs units.
    • Volume: 1 m³ = 10^6 cm³
    • Speed: 1 m/s = 10² cm/s
    • Acceleration: 1 m/s² = 10² cm/s²
    • Force: 1 N = 10^5 dyn
    • Energy: 1 J = 10^7 erg
    • Pressure: 1 Pa = 10 dyn/cm²
  • Problem 1.2: Pressure exerted by high heels vs. elephant feet.
    • Pressure = Force / Area
    • Pressure of high heels is approximately 10 times larger than the pressure exerted by an elephant.
  • Problem 1.3: Fraction of iceberg above the sea.
    • Buoyant force equals the weight of displaced seawater.
    • Ratio of volume above sea to total volume = 1 - (density of ice / density of seawater) = 10.4%.
  • Problem 1.4: Altitude angle of the Sun.
    • Difference in altitude angles is equal to the difference in latitudes.
    • θ = (distance between locations / meridian length) × 90° = 3.0°.

Advanced Problems

  • Problem 1.5: Dimensional analysis and scale transformation.

    • Dimensional analysis can be used except for some numerical factor
    • scale of length as r → r1 = α r and the scale of time as t → t1 = β t where α and β are some numbers.
    • i = 1, j = −1, k = 1, l = −2.

  • Problem 1.6: Why don’t clouds fall?

    • Clouds are made up of minute drops of water and water vapor; density of a cloud is approximately equal to that of air so it doesn't fall.
    • Relative motion of water droplets and vapor, viscous force (Stokes’ law), and droplet size determine rain formation.

Chapter 2. Mechanics

Elementary Course

  • Motion with Constant Acceleration
    • Velocity: v = v_0 + at
    • Displacement: x = x0 + v0t + \frac{1}{2}at^2
    • v^2 - v0^2 = 2a(x - x0)
  • Projectile Motion
    • x = v_0 \cos(\theta) \cdot t
    • y = v_0 \sin(\theta) \cdot t - \frac{1}{2}gt^2
  • Equation of Motion
    • ma = f
  • Law of Conservation of Energy
    • Work done by a force: W = \vec{f} \cdot \vec{r} = f r \cos(\theta)
    • Kinetic Energy: K = \frac{1}{2}mv^2
    • Change in kinetic energy equals work done: \frac{1}{2}mv2^2 - \frac{1}{2}mv1^2 = f(x2 - x1)
    • Conservative forces: work done is path-independent.
    • Potential energy: U(P) = W(P \rightarrow O)
    • Gravitational potential energy: U(h0) = mgh0
    • Elastic potential energy: U(x0) = \frac{1}{2}kx0^2
    • Conservation of mechanical energy: \frac{1}{2}mv2^2 + U(x2) = \frac{1}{2}mv1^2 + U(x1)
  • Energy Transfer between Interacting Bodies
  • Work Done by Non-conservative Forces
    • \Delta \text{Mechanical Energy} = W_{\text{non-conservative}}
  • Newton's Law of Universal Gravitation
    • F = -G\frac{Mm}{r^2}
    • Gravitational potential energy: U(r0) = -G\frac{Mm}{r0}
  • Kepler's Law
    • 1st: Planets move in elliptical orbits with the Sun at one focus.
    • 2nd: A line from the Sun to a planet sweeps out equal areas in equal times.
    • 3rd: T^2 \propto a^3

Elementary Problems

  • Problem 2.1: A ball falling from a bicycle follows a parabolic trajectory (a).
  • Problem 2.2: Identical speeds at ground (f).
    • v = \sqrt{v_0^2 + 2gh}
  • Problem 2.3: Trajectory of a ball depends on the angle thrown (a).
  • Problem 2.4: Motion of a train.
    • Uniform acceleration equations.
  • Problem 2.5: Skydiving.
    • Terminal velocity (d).
    • Relative motion impacts velocity (c).
    • Upward acceleration (a).
  • Problem 2.6: Descendent paths.
    • Object in path B arrives earlier (b).
  • Problem 2.7: Inclined plane A: shorter time (d).
  • Problem 2.8: A space probe converges with the orbit of Pluto.
    • Period of Pluto ≈ 253 years (d).

Advanced Course

  • Conservation of Momentum
    • Linear Impulse: I = \int{t1}^{t_2} f dt
    • Change in Momentum: mv2 - mv1 = I
  • Law of Conservation of Momentum
    • If no external forces act, total momentum is conserved.
  • Moment of Force and Angular Momentum
    • Moment of force: \vec{m} = \vec{r} \times \vec{f}
    • Angular momentum: \vec{l} = \vec{r} \times \vec{p}
    • \frac{d\vec{l}}{dt} = \vec{m}
  • The Keplerian Motion
    • Two-Dimensional Polar Coordinates
    • Cartesian coordinate: x = r \cos {\varphi}, y = r \sin {\varphi}
    • Velocity: vr = \dot{r}, v{\varphi} = r\dot{\varphi}
    • Acceleration: ar = \ddot{r} - r\dot{\varphi}^2, a{\varphi} = 2\dot{r}\dot{\varphi} + r\ddot{\varphi}
  • Universal Gravitation Acting on Planets: The motion of equation is mr¨ = −GmM r2 \frac{r}{r}
  • The angular momentum of a particle is conserved during the motion of the particle on which only central forces act.
    From Kepler’s second law, the relation between r and vθ becomes \frac{1}{2}rv_\theta = k = constant

Advanced Problems

  • Problem 2.9: The Atwood machine with friction
  • Problem 2.10: The rotation of rods
  • Problem 2.11: The expanding universe

Chapter 3. Oscillations and Waves

Elementary Course

  • Simple Harmonic Oscillation
    • m \frac{d^2x}{dt^2} = -kx
    • Solution: x(t) = A \cos(\omega_0 t + \alpha)
    • \omega_0 = sqrt(k/m).
    • For a simple pendulum: \omega_0 = \sqrt{\frac{g}{L}}
  • Waves
    • Equation of a wave: y(x, t) = A \cos\left(2\pi \left( \frac{t}{T} - \frac{x}{\lambda} \right)\right)
    • V = f \lambda

Elementary Problems

  • Problem 3.1: Wave propagation: use graph to determine values.
  • Problem 3.2: Using microphones: the relationship between wavelength, phase, and distance.

Advanced Course

  • Superposition of Waves
    • Constructive Interferences occur at δ=2nπ with Intensity I=(√I1 + √I2 )2
  • Standing Waves
    Two waves are travelling in opposite directions with the same period, (T), amplitude, (A), and wavelength, (λ).
    Velocity of the wave is V = αβ−1V and A → A1 = αβ−2A, respectively

Chapter 4. Electromagnetism

Elementary Course

  • Direct-Current Circuits
    • Ohm’s law: V = RI
    • Resistors in series: R = R1 + R2 + … + R_n
    • Resistors in parallel: \frac{1}{R} = \frac{1}{R1} + \frac{1}{R2} + … + \frac{1}{R_n}
    • Kirchhoff’s Rules: junction rule and loop rule.
  • Magnetic Field and Electromagnetic Induction
    • Fleming’s left-hand rule indicates the direction of force on a current-carrying wire in a magnetic field, the direction of his Thumb.

Chapter 5. Thermodynamics

Elementary Course

  • Heat and Temperature
    • Boyle’s law, Volume inverse proportional to temperature
    • Absolute temperature T related to pressure and volume state is given by: pV = nRT
    • Quantity of Heat and Heat Capacity: Amount of energy it takes to cause reaction
    • Specific heat: Quantity of heat needed to raise the temperature of matter of 1kg by 1K

Elementary Problems

Thermodynamic Laws such as zeroth, first (conservation of energy), second and thrid law deal with heat transfer and energy conservation with temperature being the state variable

Advanced Course

Provides kinetic theory of gasses and related thermal dynamics