Date Topics ___________________________________________________________ 9.8 Course outline. Fluids. Pressure and volume forces. Estimates of the magnitudes of some forces at boundaries of fluids. 9.10 Gases v. Liquids. Compressible and incompressible fluids. Variation of pressure in a column of incompressible fluid. Pascal's principle. List of useful formulae from lecture 9.13 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.
9.15 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
A water strider at a surface
A ball of oj in microgravity
Nasa graphic1
Nasa graphic2
Equation of continuity. Application to traffic flow.
9.17 Equation of motion for a fluid. The convective
derivative. Two illustrative calculations of the convective derivative.
The convective derivative as a gradient of the kinetic energy
density. Application to fluid flow in a syringe.
List of useful formulae from lecture
9.20 Bernoulli's Law. Work energy theorem for steady, incompressible and irrotational
flow. Some examples of Bernoulli's law. Flow speed through a leak at the
bottom of a column of liquid. Speed of water flowing out of a leak in a water main.
Time to evacuate a column of liquid.
9.22 Demo: time to evacuate a column of liquid. Bernoulli's law for constant flow
in a uniform pipe. Viscous forces. Microscopic origin of the viscous force.
Macroscopic description of the viscous force.
9.24 More on viscous flow. Viscosity. Dimensional analysis of viscosity. Estimate
viscous force on a plate. Flow in a cylindrical pipe. Velocity as a function
of radius. Poiseuille's law. Pressure difference to maintain volume flow
rate.
Click here for a copy of lecture notes from September 24.
List of useful formulae from lecture
9.27 More on viscous fluids. Stokes' Law for viscous drag on a moving sphere. Estimates
of drift time for air bubbles in glycerine. Introduction to kinetic theory. Calculation
of the pressure from a gas from kinetic theory. Pressure and the average kinetic energy.
9.29 Kinetic theory for pressure of a gas. Mechanical and thermal equilibrium of two
gases in contact. Temperature and the zeroth law of thermodynamics. Primer
to calculate the pressures of two inequivalent gases from kinetic theory
and from the equation of state.
Tutorial illustrating two
kinds of equilibrium between thermodynamic systems. (Opens in powerpoint)
Primer for connecting the kinetic
theory and the equation of state for two ideal gases. (Opens in powerpoint)
There is a nice book written by my colleague Gino Segre which discusses the "history
of temperature" and scientific attempts to quantify it. It's written for
a nontechnical audience and gives a highly readable discussion of the development of this
concept. Here's the cover:
10.1 More on thermal equilbrium. Temperature of a mixture of two gases. Heat flow in a nonequilibrium state. Definition of heat capacity, specific heat, and molar heat capacity. Prediction of heat capacity from the ideal gas law. Experimental data for heat capacities of gases. Illustrative Table of measured molar heat capacities in various gases. List of useful formulae from lecture 10.4 More on specific heats of gases. Equipartition theorem. Comparison of thermal energy with rotational and vibrational energies of H_2. Specific heats of diatomic molecules. Thermal conductivity. Calculation of heat loss through a single glass window. 10.6 Distributions in thermal equilbrium. Variation of gas density as a function of height. Demonstration that states with high potential energy are exponentially less probable than states with low potential energy. The Boltzmann factor. Discussion of the equilibrium velocity distribution of a three dimensional ideal gas. Application to the one dimensional Maxwell velocity distribution. Normalization of the distribution function. How to do Gaussian integrals. Click here for a plot of the one-dimensional Maxwellian velocity distribution for N_2 at 300 K. Click here for an animation showing the evolution of the Maxwell velocity distribution (1D) as a function of temperature. 10.8 More on the Maxwell velocity distribution. Maxwell velocity distribution in one dimension. Most probable velocity and mean square velocity. Moments of gaussian distributions by differentiating under the integral sign. Generalization to the three dimensional velocity distribution. Normalization of the three dimensional velocity distribution. Distribution of molecular speeds in 3D. Click here for an animation showing the evolution of the 3D speed distribution as a function of temperature. List of useful formulae from lecture 10.11 More on the Maxwell distribution. Effect of temperature on the speed distribution. Calculation of most probable and average velocities. (rms velocity left as a homework exercise). Boltzmann distribution. Equilibrium distribution as a Boltzmann distribution. 10.15 Introduction to thermodynamics. Take the introduction to thermodynamics quiz (opens in powerpoint). Work and heat flow in thermodynamic systems. The first law of thermodynamics. Heat capacity at constant pressure and at constant volume. Heat and work in an isothermal compression. List of useful formulae from lecture 10.18 Reversibly adiabatic compression/expansion of a gas. Change of internal energy, work done, heat flow during three different changes of state. Reversibility of a thermodynamic cycle. 10.20 Reversibility of a thermodynamic cycle. Introduction to entropy. Here is a powerpoint slide tutorial on some basic facts about entropy and its classical and statistical interpretations. Worked example: entropy changes of gas and reservoir in three stage cycle. How the entropy of a temperature reservoir can change with a negligible change of state, 10.22 Entropy changes in three step cycle. Entropy changes while heating two pans. Comparison with entropy change of the ideal gas. Compare entropy change in a reversibly adiabatic expansion and a free exansion (adiabatic but irreversible). List of useful formulae from lecture 10.27 Click here for a summary of key points in thermodynamics. Heat and work in thermodynamic cycles. Here are animations for The p-V-T cycle The Otto cycle The Carnot cycle Calculation of the Carnot efficiency, and the optimum efficiency of a thermodynamic cycle. 10.29 Demonstration of two coupled mechanical oscillators. Review of basic facts for the motion of a harmonic oscillator. Click here for a powerpoint tutorial. Equation of motion for the harmonic oscillator with viscous damping. Animated graphic giving the two roots of the characteristic equation for the damped oscillator. Application to the undamped oscillator. 11.1 Three regimes for the damped harmonic oscillator. Damping constant and frequency of the underdamped oscillator. Click here for a plot of the motion of the underdamped oscillator. Fast and slow decaying solutions of the overdamped oscillator. Click here for a plot of the motions of the overdamped oscillator. Solutions of the critically damped oscillator. Click here for a plot of solutions of the critically damped oscillator. Finding two linearly independent solutions of the critically damped oscillator. List of useful formulae from lecture 11.3 Motion of the critically damped oscillator. Solution for relaxation time. Introduction to forced oscillations. Solutions to the equation of motion for an oscillating driving force when the driving frequency is less than the natural frequency. See also a plot of the spring extension as a function of time. Solutions of the eom when the driving frequency is is greater than the natural frequency. Steady state response after a long wait. 11.5 Forced oscillations. Complex solutions of the eom. Solution for amplitude and phase shift of the forced oscillation. Form of solutions for small frequency, for high frequency. Response at the resonant frequency. Click here for a plot of the frequency response of a driven oscillator. Q factor of the driven oscillator. Demonstration of large Q for tuning fork. and the effect of detuning tuning forks. List of useful formulae from lecture 11.8 Resonance for the driven oscillator. Click here for a plot of the frequency response of the driven harmonic oscillator for various Q's. Introduction to coupled mechanical oscillators. Equations of motion. Decoupling equations of motion by symmetry. Solutions of the decoupled problems. Reconstruct solutions using normal modes. Here is a plot of the solutions for two coupled oscillators. 11.10 Superposition of normal modes for the symmetric two-mass oscillator. Situations where the modes cannot be determined by symmetry alone. Click here for an animated clip showing the dynamics for two unequal masses coupled by springs. Coupled equations of motion. Equations of motion in matrix form. Solution as an eigenvalue problem. Solution for the normal mode frequencies. The updated formula sheet below contains useful information about the analysis of coupled oscillators. List of useful formulae from lecture 11.12 More on coupled oscillators. Calculation of displacement amplitudes (eigenvectors) for the two modes of an asymmetric oscillator with different masses. Intepretation of the limit M/m >> 1. Normal modes of CO_2 (its a symmetric linear triatomic molecule.) Finding the normal mode frequencies from symmetry and mass ratios. Comparison of frequencies of the symmetric and antisymmetric stretching motions. Comparion of prediction with experimental data (predicted frequency ratio is too high by 8%). 11.15 One dimensional mass and spring lattice: equations of motion. Simplification for "slowly" varying displacement field. The wave equation in one dimension. Click here for a tutorial on various one dimensional wave equations and methods of their solution. Wave equation for transverse displacements of a stretched string. Solution by the method of "propagation of solutions" (D'Alembert's method). List of useful formulae from lecture Note: Problem 9.5 on this week's homework set is postponed until the first problem set after the midterm exam 11.17 Solution of the wave equation in one dimension by propagation of solutions. Separating left and right moving solutions. Solution for right moving Gaussian pulse, and fission of an initially stationary Gaussian pulse. Here's an animation of the fission of an initially stationary Gaussian pulse. 11.19 Pulse propagation on a string following a localized impulse. Click here for an animation of pulse propagation. Solution by separation of variables. Click here for an animation of a separable solution built from counterpropagating cosine waves. 11.29 Propagation of a triangular waveshape on a clamped string. Click here for an animation of the subsequent motion. Modes of oscillation of the clamped string. here for an animation of the oscillation of the bound modes. Boundary conditions on the solutions. Quantization of wavelengths. Fourier's trick for extracting the amplitudes. List of useful formulae from lecture` 12.1 Expansion of a triangular waveform in a Fourier series. Interpretation of the Fourier expansion coefficients. Energy consideration in wave motion. Here is an animation of wave generation at the center of a stretched string. Also check out the graphic of energy densities in a Gaussian pulse that fissions. 12.3 Kinetic and potential energy densities in one dimensional wave motion. Calculation of energy density and total energy for a Gaussian pulse. Why the kinetic and potential energy densities are equal for propagating pulses. 12.6 More on energy propagation. Kinetic and potential energies. Superposition rules for amplitudes and energy densities. Work, power and energy propagation for a sinusoidal propagating wave. 12.8 Reflection and transmission of waves at intefaces. Matching rules: continuity of string displacement and string slope. Reflection and transmission coefficients.
Here are graphics showing the reflection of Gaussian pulses at an interface to a denser medium at an inteface to an infinitely dense medium at an interface to a less dense medium at an inteface to a very less dense medium
The following images show the scattering of a harmonic wave at an interface to a denser medium at an inteface to a less dense medium List of useful formulae from lecture 12.10 Superposition of harmonic waves, wave packets. Click here for a frequency distribution of harmonic waves that gives the waveshapes shown here. Here is a localized packet made by integrating over all the harmonic waves in a finite interval of frequency. Phase velocity versus group velocity. Here is a propagation animation where the phase velocity exceeds the group velocity. Here is a propagation animation where the group velocity exceeds the phase velocity. Example: what is the "color" of the light from a femtosecond laser.
