University Physics III: Fluids, Waves, Thermodynamics and Optics

Units

1. 1

   Fluid statics and dynamics

   4 lessons

   1. Pressure and depthDefine pressure as force per area, derive p = p₀ + ρgh from dp/dy = −ρg on a fluid slab, tell gauge from absolute pressure, use Pascal's principle for a hydraulic lift, and read manometers and barometers.
   2. Buoyancy and forces on submerged surfacesDerive Archimedes' principle from the pressure difference across a submerged body, find the floating fraction ρ_object/ρ_fluid and apparent weight, and integrate pressure over depth to get the total force on a dam wall.
   3. Continuity and Bernoulli's equationUse A₁v₁ = A₂v₂ for ideal flow, derive Bernoulli's equation p + ½ρv² + ρgy = constant from the work-energy theorem on a fluid element, and apply it to the Venturi meter, Torricelli's efflux speed and lift.
   4. Viscosity and real flowUse the viscosity coefficient in F = ηAv/d, derive Poiseuille's law with its r⁴ dependence from a force balance on a fluid cylinder, find the terminal speed of a sphere from Stokes drag, and use the Reynolds number to tell laminar from turbulent flow.
2. 2

   Damped and driven oscillations

   2 lessons

   1. Damped oscillationsWrite the equation of motion of a mass on a spring with a drag force, solve it for light damping, tell underdamped, critically damped and overdamped motion apart, and use the decay of energy to define the quality factor Q.
   2. Driven oscillations and resonanceSolve for the steady-state response of a damped oscillator driven by F₀ cos ωt, read the amplitude and phase against driving frequency, and explain how damping and Q set the height, width and power of the resonance peak.
3. 3

   Mechanical waves

   4 lessons

   1. Travelling wavesTell transverse from longitudinal waves, write the wave function y(x, t) = A sin(kx − ωt) and connect k, ω, λ, f and v, read snapshot and history graphs, and separate the velocity of a wave from the velocity of the particles it passes through.
   2. The wave equation and waves on a stringDerive the wave equation ∂²y/∂x² = (1/v²) ∂²y/∂t² from Newton's second law on a short segment of string, get v = √(F/μ), and find the energy and power P = ½μω²A²v that a sinusoidal wave carries.
   3. Superposition, interference and reflectionAdd waves by the principle of superposition, find the resultant amplitude 2A cos(φ/2) from the phase difference, turn path differences into phase differences, and predict the reflection at fixed and free ends and the reflection and transmission at a junction of two strings.
   4. Standing waves and normal modesShow that two equal waves travelling in opposite directions add to a standing wave, locate its nodes and antinodes, and find the normal modes f_n = nv/2L of a string fixed at both ends.
4. 4

   Sound

   4 lessons

   1. Sound waves and the speed of soundDescribe sound as a longitudinal wave of displacement and pressure with the two 90° out of phase, relate the pressure amplitude to the displacement amplitude, and find the speed of sound from v = √(B/ρ) and its dependence on temperature.
   2. Intensity and the decibel scaleDefine intensity as power per area, use the inverse-square law for a point source, relate intensity to the square of the pressure amplitude, and convert intensities to sound levels with β = 10 log(I/I₀) in decibels.
   3. Pipes, harmonics and beatsFind the standing sound waves in a pipe open at both ends and in a pipe closed at one end, state which harmonics each supports and how resonance picks them out, and find the beat frequency of two close frequencies from f_beat = |f₁ − f₂|.
   4. The Doppler effect and shock wavesDerive the frequency heard from a moving source and for a moving observer, combine them as f′ = f(v ± v_o)/(v ∓ v_s) with the sign rules, apply the shift to ultrasound and radar, and find the Mach cone of a source faster than sound from sin θ = v/v_s.
5. 5

   Temperature and heat

   3 lessons

   1. Temperature and thermal expansionDefine temperature through thermal equilibrium and the zeroth law, convert between the Celsius, Fahrenheit and kelvin scales, read absolute zero from a constant-volume gas thermometer, and use ΔL = αLΔT and ΔV = βVΔT, including water's anomaly and thermal stress.
   2. Heat, heat capacity and calorimetryTreat heat as energy transferred because of a temperature difference, use Q = mcΔT, Q = nCΔT and the latent heats Q = mL, read a heating curve, and solve calorimetry problems for the final equilibrium temperature.
   3. Conduction, convection and radiationUse Fourier's law dQ/dt = kAΔT/L for conduction, add thermal resistances of layers in series, describe convection, and use the Stefan-Boltzmann law P = eσAT⁴ with net exchange P = eσA(T⁴ − T_s⁴).
6. 6

   Kinetic theory of gases

   4 lessons

   1. The ideal gasUse the ideal-gas law pV = nRT = NkT with moles, Avogadro's number and Boltzmann's constant, read isotherms on a pV diagram, and say when a real gas departs from ideal behaviour.
   2. The molecular model of pressure and temperatureDerive p = (1/3)(N/V)m⟨v²⟩ from molecular collisions with the walls, show that the average translational kinetic energy is (3/2)kT, find the rms speed √(3kT/m), and read temperature as a measure of molecular motion.
   3. Equipartition and heat capacities of gasesCount the degrees of freedom of a monatomic or diatomic molecule, use the equipartition theorem to write U, C_V, C_p and γ, and explain why hydrogen's heat capacity climbs in steps with temperature as rotations and vibrations switch on.
   4. The Maxwell-Boltzmann speed distributionUse the Maxwell-Boltzmann distribution f(v) to read the fraction of molecules in a speed range, compute the most probable, mean and rms speeds, say how the curve shifts and flattens with temperature and molecular mass, and apply the fast tail to evaporation, atmospheric escape and the mean free path.
7. 7

   The first law of thermodynamics

   3 lessons

   1. Work and pV diagramsWrite the work done by a gas as W = ∫p dV, read it as the area under the path on a pV diagram, compute it for constant-pressure, isothermal and straight-line paths, explain its sign and why it depends on the path, and find the net work of a cycle as the enclosed area.
   2. The first law and internal energyState the first law ΔU = Q − W with its sign convention, explain why internal energy is a state function while heat and work are not, apply it to each leg and to a whole cycle, and show with free expansion that the internal energy of an ideal gas depends on temperature alone.
   3. Isobaric, isochoric, isothermal and adiabatic processesFind Q, W and ΔU for isobaric, isochoric, isothermal and adiabatic processes of an ideal gas, derive C_p − C_V = R and the adiabat pVᵞ = constant with TVᵞ⁻¹ = constant, and explain why an adiabat is steeper than an isotherm on a pV diagram.
8. 8

   Heat engines, entropy and the second law

   5 lessons

   1. Heat engines and efficiencyTreat a heat engine as a cycle with W = Q_H − Q_C, find its efficiency e = W/Q_H = 1 − Q_C/Q_H, draw its energy flow, and trace the Otto cycle of a petrol engine on a pV diagram to show e = 1 − 1/rᵞ⁻¹.
   2. Refrigerators, heat pumps and the second lawRun an engine backwards as a refrigerator or heat pump, find coefficients of performance, state the Kelvin-Planck and Clausius forms of the second law, and show that each statement implies the other.
   3. Reversibility and the Carnot cycleTell reversible from irreversible processes, trace the Carnot cycle of two isotherms and two adiabats on a pV diagram, find its heats and work, and show that e = 1 − T_C/T_H is the most any engine between two reservoirs can reach.
   4. EntropyDefine entropy by dS = dQ_rev/T, find ΔS for reversible heating, phase changes and isothermal expansion, show that the entropy of an isolated system never decreases, and draw the Carnot cycle as a rectangle on a TS diagram.
   5. Microstates and the statistical meaning of entropyCount microstates for coins and gas molecules, state Boltzmann's S = k ln W, show why spreading out is overwhelmingly probable, find the entropy of a free expansion both from heat and from counting, and say what the arrow of time is.
9. 9

   Electromagnetic waves

   3 lessons

   1. Electromagnetic waves from Maxwell's equationsShow that Maxwell's equations in empty space give a wave equation with speed c = 1/√(μ₀ε₀), describe the plane wave with E and B perpendicular, in phase and linked by E = cB, and place the electromagnetic spectrum from radio to gamma rays by wavelength.
   2. Energy, intensity and radiation pressureShow that E and B carry equal energy densities, find the Poynting vector S = E × B/μ₀ and the intensity I = E_max²/(2μ₀c), use the inverse-square law for a source, and find radiation pressure I/c for absorption and 2I/c for reflection, as for a solar sail.
   3. PolarizationDescribe linearly polarized light, use Malus's law I = I₀cos²θ for one filter and for several in a row, explain why an unpolarized beam loses half its intensity at the first filter, and find the Brewster angle tan θ_B = n₂/n₁ for polarization by reflection, with polarization by scattering.
10. 10

    Geometric optics

    5 lessons

    1. Reflection, refraction and dispersionUse wavefronts and rays to derive the law of reflection and Snell's law n₁ sin θ₁ = n₂ sin θ₂ from Huygens' principle, use n = c/v to find speed and wavelength in a medium, and explain how a prism spreads white light into colours.
    2. Total internal reflectionFind the critical angle sin θ_c = n₂/n₁ for light going from a denser to a less dense medium, explain total internal reflection in prisms, diamonds and optical fibres, and find the apparent depth of an object seen through a flat surface.
    3. Plane and spherical mirrorsLocate the image in a plane mirror, show that a spherical mirror focuses parallel rays at f = R/2, draw the principal rays for concave and convex mirrors, and use 1/d_o + 1/d_i = 1/f and m = −d_i/d_o with signs to say where an image is, how big, and whether it is real or virtual.
    4. Thin lensesDerive the paraxial refraction equation for one spherical surface, build the lensmaker's equation 1/f = (n − 1)(1/R₁ − 1/R₂) for a thin lens, draw principal rays for converging and diverging lenses, and use 1/d_o + 1/d_i = 1/f and m = −d_i/d_o with signs to find where real and virtual images form.
    5. Lens systems and optical instrumentsFind the final image of a two-lens system by treating the first image as the object of the second, describe the eye with its near and far points and the lenses that correct it, and find the angular magnifications of the simple magnifier, the compound microscope and the refracting telescope.
11. 11

    Interference and diffraction

    4 lessons

    1. Young's double-slit interferenceExplain why two coherent sources make bright and dark fringes, derive the path-difference conditions d sin θ = mλ (bright) and (m + ½)λ (dark), find the fringe spacing y = mλL/d on a distant screen, and use the intensity pattern I = I₀cos²(πd sin θ/λ).
    2. Thin-film interferenceState when reflection from a boundary flips the phase by half a wavelength, find the path and phase difference between the two waves reflected by a thin film, and use it for bright and dark reflection in soap films and oil slicks, for anti-reflection coatings and for air wedges.
    3. Single-slit diffractionShow with Huygens wavelets that a single slit of width a gives dark fringes at a sin θ = mλ, use the pattern I = I₀(sin β/β)² with β = πa sin θ/λ, say why the central maximum is twice as wide as the others, and combine it with two-slit interference to find missing orders.
    4. Diffraction gratings and resolutionShow that N equally spaced slits keep the principal maxima at d sin θ = mλ and make them sharper by a factor N, use a grating to separate wavelengths and find its resolving power R = λ/Δλ = Nm, and use the Rayleigh criterion θ = 1.22λ/D for the smallest angle a circular aperture can resolve.