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Description and Reading |
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I. Vectors: Velocity and Acceleration.
Mathematical Aside. Definition of vectors in Cartesian coordinates. Boas 3.4; Thornton & Marion, 1.9. |
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Properties of vector addition
and multiplication -- scalar and vector products. Boas 3.4, 6.2, 6.3; Thornton & Marion, 1.10-1.12. |
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Permutation symbol and the curl. Concatenating vector and scalar products. Vector identities. |
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Holiday. Martin Luther King Day. |
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Homework 1 due. Curvilinear coordinates. Velocity and acceleration in polar coordinates. Marion & Thornton App. F, 1.14; Boas 6.2, 10.6-10.7. |
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Velocity and acceleration in
cylindrical coordinates. Thornton and Marion 1.14; Boas 6.2. |
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Velocity and acceleration in
spherical coordinates. Applications of velocity and acceleration
in curvilinear coordinates. Thornton and Marion 1.14. |
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Homework 2 due. II. Newton's Laws. Two formalisms of mechanics. The three laws. Motion under constant acceleration. Free body diagrams. Block sliding down a frictionless inclined plane. Thornton and Marion 2.1-2.4. |
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Block sliding down an inclined plane
with friction. Static versus kinetic friction. Mathematical
Aside: 1st order ODEs. Terminology. Separable equations. Thornton and Marion 2.4.; Boas 8.1-8.2. |
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Examples of separable 1st order
ODEs. Newton's second law: F0, F(t), F(x). Radioactive Decay.
Application of initial conditions. Linear first order ODEs. Thornton and Marion 2.4; Boas 8.2-8.3. |
Feb |
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Homework 3 due. Linear first order ODEs (cont). Taylor Series. Derivation of the Taylor Series formula. Applications. Boas 1.12-1.13, 1.15. |
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Retarding forces: 1D motion
and Stokes drag: F(v). Short time and long time response. Thornton and Marion 2.4. |
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1D motion and Newtonian drag: F(v^2).
2D motion without drag. Thornton and Marion 2.4. |
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Homework 4 due. 2D motion with Stokes drag. Perturbation expansions. Begin discussion of the Lorentz force. Thornton and Marion 2.4. |
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Lorentz force. Motion of a
charged particle in electric and magnetic fields.
Thornton and Marion 2.4. |
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Lorentz force (cont.) |
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Homework 5 due.
Mathematical Aside: Functions of a complex variable. Boas Ch. 2.9-2.14. |
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Mid-term exam 1. |
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Class cancelled (sick). |
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Mathematical Aside: Functions of a complex variable (cont.). Plots in the complex plane. Euler's formula and its use. |
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Mathematical Aside: Functions of a complex variable (cont.). Powers and roots. Hyperbolic functions. |
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Mathematical Aside: Functions of a complex variable (cont.). Hyperbolic functions. Logarithms. Complex powers and roots. Mathematical Aside: grad, div, curl, and line integrals. Grad and the directional derivative. | Mar |
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Homework 6 due. Mathematical Aside: grad, div, curl, and line integrals (cont.). Physical interpretation of grad. Divergence, curl, laplacian. Line integrals. |
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Mathematical Aside: grad, div, curl, and line integrals (cont.). Line integrals. Conservative forces: work, existence of a potential, curl of F, exact differentials. |
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Conservative forces: work, existence of a potential, curl of F, exact differentials (cont.). Finding potentials. Work - energy theorem. Examples. |
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Homework 7 due. Energy and physical problems. Examples: falling ball, pendulum, simple harmonic oscillator. Friction, loss of mechanical energy, conservation of eneryg in this case. |
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Application of energy conservation with frictional losses. Interpreting potential functions: equilibrium, turning points, `forbidden regions. |
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Homework 8 due. Mathematical Aside: 2nd order ODEs. Example: forced, damped SHO equation. |
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Mid-term exam 2. |
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No class. |
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Mathematical Aside: homogeneous 2nd order ODEs. Terminology and some theory. Linear operators, linear independence, the Wronskian, principle of superpostion. Linear, homogeneous, 2nd order ODEs with constant coefficients. Auxiliary equation. |
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Mathematical Aside: homogeneous 2nd order ODEs (cont.). Solutions with real roots, complex roots, repeated roots to the auxiliary equation. General solution. Initial conditions. | Apr |
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Mathematical Aside: inhomogeneous 2nd order ODEs (cont.). Method of undetermined coefficients and superposition. |
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Mathematical Aside: inhomogeneous 2nd order ODEs (cont.). Method of undetermined coefficients and superposition (cont.). Examples. Undamped, unforced simple harmonic oscillator equation. Initial conditions: drag & drop or kick. |
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Homework 9 due. Damped, unforced simple harmonic oscillator equation. Over, under, and critical damping. Decrement of motion, quality factor Q, and attenuation q. |
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Damped, forced simple harmonic oscillator equation: simple forcing (Fourier series not needed). Mathematical Aside: Fourier Series. The Fourier theorem, Dirichlet conditions. |
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Mathematical Aside: Fourier Series (cont). Orthogonality of the Fourier basis functions, finding Fourier coefficients. Example. |
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Homework 10 due. Mathematical Aside: Fourier Series (cont). Even and odd functions. More examples. Damped, forced simple harmonic oscillator equation: general forcing (Fourier series needed). (SHO with generalized forcing.) |
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Physics of forced oscillators. Transient and steady-state solutions, amplitude response, phase lag. |
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Strings. Derivation of the string equation. (Derivation of the 1D wave equation.) (Physics and solution of the wave equation.) (Application of initial conditions to the solution of the 1D wave equation.) |
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Homework 11 due. Solving the string equation. Separation of variables, applying boundary conditions. |
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Applying initial conditions to the 1-D wave equation. Traveling (D'Alembert's) solution to the wave equation. 2-D wave equation. (2-D wave equation.) |
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2-D wave equation: separation of variables, application of boundary conditions, modal indices, patterns and frequencies of oscillation, general solution. |
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No class. |
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Homework 12 due. Finish 2-D oscillations: application of initial conditions. Review for final exam. |