Below is a summary of the topics coverered in the lecture sections.Date Topics ___________________________________________________________ 9.3 Course outline. Fluids. Pressure and volume forces. Estimates of the magnitudes of some forces at boundaries of fluids. 9.5 Gases v. Liquids. Compressible and incompressible fluids. Variation of pressure in a column of incompressible fluid. Variation of pressure in column of ideal gas. Pascal's principle. 9.8 Forces on curved surfaces. Compare forces on two halves of a hemispherical partition. Archimedes principle. No Archimedes principle for constant pressure. Calculation of Archimedes force for an object in an incompressible fluid. Generalization to compressible fluids. Force density as gradient of the pressure.
9.10 Surface tension. Definition of surface tension and contact line. Surface tension on supported rectangular soap film. Surface tension supporting a paper clip. Why surface tension can support small objects but not big ones. Some graphics illustrating effect of surface tension in microgravity environment:
A ball of oj Nasa graphic1 Nasa graphic2 9.12 Hydrodynamics. Newton's Second Law for fluid flow. Steady flow and the convective derivative. Example for steady fluid flow in a circular channel. Centripetal acceleration from the convective derivative (!). Introduction to Bernoulli's Law. 9.15 More on Bernoulli's Law for steady, irrotational, incompressible, inviscid flow. ( See a graphic of a fluid flow that is not irrotational. ) Equation of continuity. Application of equation of continuity in traffic flow. Velocity and force distribution in a syringe. Flow of fluid from a leaky column. 9.17 Bernoulli's principle and Torricelli's law. Demonstration of fluid flow from a leaky column. Pressure in a water main (estimated). Fluid flow in uniform pipe according to Bernoulli. 9.19 Shearing fluids: static shear (no resistance to static shear) vs. dynamic shear (resists dynamic shear). A constant shear rate requires a constant applied force. Microscopic model for viscosity. Macroscopic description of the viscous force on a plate. Coefficients of viscosity. Calculation of viscous force from a fluid with a uniform shear rate. Exponential decay of velocity produced by a viscous force. No slip boundary condition for fluid near a wall. Velocity field for steady flow in a uniform pipe with circular cross section (solution by balancing forces). 9.22 More on Poiseuille flow. Numerical estimate of flow rate in a long thin pipe. Effect of changing the radius of the pipe. Analogy with current flowing in a wire. Motion of trapped air bubble in a viscous fluid. Calculation of the terminal velocity. Small bubbles drift slowly. 9.24 Calculation of pressure in fluid from kinetic theory. Connection with equation of state for ideal gas. Average kinetic energy. Thermodynamic systems in thermal contact. Fluctuations of energy. Temperature as a measure of energy fluctuations. Interpretation of temperature of an isolated macroscopic object. 9.26 Examples using ideal gas equation of state. Pressures from two different gases in thermal contact. Pressures from mixture of two gases. Objects at different temperatures in thermal contact. Heat. Heat capacity of monatomic ideal gas. Observed molar heat capacities at constant volume. 9.29 More on molar heat capacities. Translational kinetic energy of Argon atoms in the gas phase. Absence of rotational energy in the thermal energy of Argon gas. Energies of diatomic molecules. Effects of rigid rotation and bond stretching. Application to molar heat capacity of diatomic gases. 10.1 Latent heat at first order phase transition. Exercise with heat capacity of ice, and latent heat at its melting transition (calculate warming time and melting time for fixed input power). Thermal conductivity. Calculation of heat loss through a window pane. Return to kinetic theory: distribution of stopping heights in a column of gas. Distribution of velocities. 10.3 Maxwell Boltzmann distribution. Using the equilibrium height distribution, extract the equilibrium velocity distribution. Properties of the one dimensional velocity distribution. Normalization, most probable velocity, average velocity (it's zero of course) and average of the square of the velocity (good exercise in Gaussian integration). 10.6 Review calculation of mean square velocity using Maxwell distribution. Puzzler: what is the effect of temperature on f(v_z). Click here for a graphic, showing the effect of increasing temperature on the magnitude and width of the distribution (the number is brackets is the absolute temperature, the horizontal scale is velocity in m/sec). Maxwell velocity distribution in 3D. Finding the *speed* distribution in 3D. 10.8 Review some results on the M-B velocity distribution (1D) and M-B speed distribution (3D). Translating the speed distribution into an energy distribution. Properties of the energy distribution. Short discussion of the 2001 Nobel Prize. Take the introduction to thermodynamics quiz. 10.10 Compression of a gas at constant pressure. Comparison to isothermal compression. Calculate change of internal energy, work done, heat into/out of gas. Adiabatic reversible compression. Definition of adiabatic reversible (fast but not too fast). Discussion of adiabatic compression in the p-V plane. 10.15 Compare an isothermal and adiabatic compression. pV^gamma is constant along adiabat. Energy change, work done and heat transferred (zero) in a reversible adiabatic compression. Return to three step process: 1. isothermal compression to half initial volume, 2. reversible adiabatic expansion to original volume, 3. heating at constant volume: calculate change of internal energy, work done and heat transferred in each step, and in total process. 10.17 Midterm Exam 10.20 Running a thermodynamic cycle forward but not backwards. Introduction to entropy. Applying entropy to the isothermal- adiabatic-wait cycle. Change of entropy of gas. Change of entropy of reservoir. 10.22 Exercise: entropy change when two pans are heated. Comparison of reversible adiabatic expansion and adiabatic free expansion. Entropy change in the relaxation of a nonequilbrium state. Some thermodynamicecycles that do work p-v-t model otto cycle carnot cycle with same temperature range. 10.24 A short course in thermodynamics (summary) Analysis of some closed thermodynamic cycles that do work. Efficiency of a cycle. Analysis of p-V-T cycle, and comparison with Otto cycle. Maximum efficiency of a cycle. The Carnot cycle. Running the Carnot cycle backwards. Refrigerators and coefficient of performance of a refrigerator. 10.27 A wave is energy in motion. Demonstration of energy transfer between two coupled oscillators. Quick review of basic facts about the harmonic oscillator. Natural frequency, amplitude and phase. Exercise: maximum speed and displacement at 1/3 maximum speed for free oscillator. The oscillator in the complex plane. Roots of the characteristic polynomial. Interpretation of complex roots. Evolution of complex roots as a function of damping. Underdamped and overdamped solutions. Animated graphic showing complex roots of the characteristic polynomial 10.29 More on the damped oscillator. Example of motion for the underdamped oscillator overdamped oscillator and critically damped oscillator. No inertial mass for the heavily overdamped oscillator. Solutions for the critically damped oscillator. Calculation of the relaxation time for the critically damped oscillator. 1.31 Introduction to the forced oscillator. Graphics illustrating response when the oscillator is driven below its natural frequency. See also the graphic giving the spring length as a function of time. Transients when the oscillator is driven above it's natural frequency decay are absent in steady state motion. Solution for the steady state reponse of the driven oscillator in terms of complex exponentials. 11.3 Oscillator driven on resonance. See the tips for studying forced oscillations. Short time and long time response of the oscillator driven on resonance. Frequency response of the driven oscillator at Q=20 and as a function of Q. Finding the Q of a tuning fork, and demonstrating by an experiment that it is large. 11.5 Coupled oscillators. Solution by symmetry. Animation of the modes of two coupled oscillators that cannot be determined by symmetry alone (red=slow mode for unequal masses, blue=fast mode). (See the graphic of a person who has just discovered that many problems with coupled oscillators cannot be solved by symmetry.) Constructing and diagonalizing the dynamical matrix. Determination of normal mode frequencies and the associated eigenvectors. 11.7 Brief review of two ways to solve motion of linearly coupled oscillators: solution by symmetry and a solution as an eigenvalue problem. Fitting to the initial conditions. Separating the "fast" and "slow" component of resulting motion ("fast" at average frequency, "slow" at half the difference frequency). See plots of the motion for coupled oscillators for k/K = 0.1. See a plot showing resonant energy transfer between coupled oscillators. 11.10 Coupled oscillators with different masses. Calculation of equation of motion and its solution for the frequencies of the normal modes. Plots of the displacements of the heavy (bold) and light (thin) displacements (m/M = 0.1) for the low frequency and for the high frequency modes. Animation of the motion in the two normal modes. Introduction to the wave equation. 11.12 More on the wave equation. Dynamics of the stretched string: derivation of the wave equation for small amplitude displacements. Three methods for solving the wave equation in one dimension. Illustration of the method of propagation of solutions. Solutions that propagate with no shape change at constant speed. Given a snapshot of a gaussian pulse, where is the pulse at a later time. 11.14 Solving the one dimensional wave equation by propagating its solutions. See an example of a right moving pulse, a movie showing right and left moving Gaussian pulses (these have the same displacement pattern at t=0), or a tutorial showing how to distinguish the two based on the initial velocity profile. Separating the right and left moving excitations using the propagation method. Solution for a gaussian disturbance. Movie of the evolution of a gaussian pulse. Introduction to separation of variables. Separation constant and discussion of the sign of the separation constant. 11.17 More on the solution of the wave equation by separation of variables. Constructing the product solutions. See a graphic of a standing wave solution of the wave equation. The standing wave is a superposition of right and left moving solutions of the wave equation. Considerations for a string clamped at both endpoints. See a graphic of the motion of a symmetric triangular pulse on a clamped string. 11.19 Quantized normal modes for a clamped string still image and animated images. Solution for propagation of a triangular pulse by Fourier series. Term by term reconstruction of the Fourier representation of a triangular pulse. Waveform produced by an impulsive force on a stretched string. 11.21 Midterm Exam 11.24 Energy transport and energy densities in wave motion. Some clips illustrating energy in a standing wave in a running wave and during fission of gaussian pulse. Calculation of kinetic energy densities and potential energy densities for the stretched string. Worked exampled for the standing wave and for the harmonic running wave. 11.26 Wave power for a propagating harmonic wave. Numerical estimate of wave power on a stretched string for typical parameters. Energy considerations for mechanically forced wave motion. Balancing forces at a point source for waves on a stretched string. See the animated graphic showing forward and backward moving waves produced by a point source. Work and power provided by the source. Energy density in a Gaussian pulse. Integrating the energy density to find the total energy carried by a gaussian pulse. Dependence of the pulse energy on tension, amplitude and pulse width. Click here for a copy of lecture notes from November 26. 12.1 Waves at boundaries. Junctions between stretched strings. Tension and wave speeds. Animation showing scattering of an incident Gaussian pulse from a denser and a less dense medium. Matching rules for displacements and string slopes at boundaries. Calculation of transmission and reflection coefficients for cosine waveform. 12.3 More on reflection and transmission of Gaussian pulses and harmonic waves. Click here for a graphic showing the reflection of an incident cosine wave from a denser string. Using the reflection coefficient and transmission coefficients to find the reflected and transmitted wave amplitudes. Analysis of nonintuitive physics in the situation where v_2 > v_1. See the (amazing) graphic illustrating the transmission of a large (but wide) Gaussian pulse in this limit. (Don't be alarmed by this graphic, energy is actually conserved since the large wide transmitted pulse carries almost no energy. (Q: Why?)) Formulation of the scattering problem using wave power. Conservation of energy == conservation of wave power. 12.5 Wave packets and wave groups. Click here for Escher's rendering of some wave groups. A toy problem adding the amplitudes of two propagating harmonic waves. Factorization into parts containing motion of crests (these propagate at the phase velocity) and an envelope (these propagate at the group velocity). Phase and group velocity are equal in a nondispersive medium. Examples of dispersive media. Solutions of the two wave problem where vp=vg, or where vg lt vp, or where vg gt vp. 12.8 See the tips for preparing for the final exam. More on wave groups. Synthesis of waves with discrete frequencies to construct a wave train. (It is a good exercise to play "what changes, what doesn't and why" with the wavetrain animation.) Constructing a localized wavepacket by integrating (summing an infinite number) of waves with frequencies in a finite interval. Click here for examples of wavepackets of various widths.
