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Announcements 10/3/12

Announcements 10/3/12. Prayer Exam 1 ends tomorrow night Lab 3: starts Sat 6 Oct Lab 4: starts Sat 13 Oct Lab 5: also starts Sat 13 Oct Lab 5 = computer simulations, see website Colton’s complex number review handout Taylor’s Series review: cos(x) = 1 – x 2 /2! + x 4 /4! – x 6 /6! + …

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Announcements 10/3/12

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  1. Announcements 10/3/12 • Prayer • Exam 1 ends tomorrow night • Lab 3: starts Sat 6 Oct • Lab 4: starts Sat 13 Oct • Lab 5: also starts Sat 13 Oct • Lab 5 = computer simulations, see website • Colton’s complex number review handout • Taylor’s Series review: • cos(x) = 1 – x2/2! + x4/4! – x6/6! + … • sin(x) = x – x3/3! + x5/5! – x7/7! + … • ex = 1 + x + x2/2! + x3/3! + x4/4! + … • (1 + x)n = 1 + nx + … Guy & Rodd

  2. From warmup Extra time on? (nothing in particular) Other comments? The book is really funny... If this has been my favorite section to read so far, what field of physics should I go into? Optics?

  3. Complex Number Basics What’s the square root of -1? What’s the difference between i and –i? The Complex Plane How to plot 3 + 4i? Or -10 + 10i? What does “take the complex conjugate” mean, graphically?

  4. Warmup question: • What’s the complex conjugate of: just switch all i’s to –i’s • or… much • longer method… (the same!)

  5. From warmup Consider the collection of all points in the complex plane that have the same magnitude--for the sake of discussion let's say the magnitude is 5. What's special about this group of points? They form a circle with radius 5.

  6. Complex Numbers – Polar Coordinates • Where is 10ei(p/6) located on complex plane? • Proof that it is really the same as 1030

  7. Complex Numbers, cont. • Adding • …on complex plane, graphically? • Multiplying • …on complex plane, graphically? • How many solutions are there to x2=1? x2=-1? • What are the solutions to x5=1? (xxxxx=1) • Subtracting and dividing • …on complex plane, graphically?

  8. Polar/rectangular conversion • Warning about rectangular-to-polar conversion: tan-1(-1/2) = ? • Do you mean to find the angle for (2,-1) or (-2,1)? Always draw a picture!!

  9. Using complex numbers to add sines/cosines • Fact: when you add two sines or cosines having the same frequency, you get a sine wave with the same frequency! • “Proof” with Mathematica • Worked problem: how do you find mathematically what the amplitude and phase are? • Summary of method: Just like adding vectors!!

  10. HW 16.5: Solving Newton’s 2nd Law • Simple Harmonic Oscillator (ex.: Newton 2nd Law for mass on spring) • Guess a solution like what it means, really: and take Re{ … } of each side (“Re” = “real part”)

  11. Complex numbers & traveling waves • Traveling wave: A cos(kx – wt + f) • Write as: • Often: • …or • where = “A-tilde” = a complex number • the amplitude of which represents the amplitude of the wave • the phase of which represents the phase of the wave • often the tilde is even left off

  12. Clicker questions: • Which of these are the same? (1) A cos(kx – wt) (2) A cos(kx + wt) (3) A cos(–kx – wt) • (1) and (2) • (1) and (3) • (2) and (3) • (1), (2), and (3) • Which should we use for a left-moving wave: (2) or (3)? • Convention: Usually use #3, Aei(-kx-wt) • Reasons: (1) All terms will have same e-iwt factor. (2) The sign of the number multiplying x then indicates the direction the wave is traveling.

  13. Reflection/transmission at boundaries: The setup x = 0 • Why are k and w the same for I and R? (both labeled k1 and w1) • “The Rules” (aka “boundary conditions”) • At boundary: f1 = f2 • At boundary: df1/dx = df2/dx Region 1: light string Region 2: heavier string transmitted wave in-going wave Goal: How much of wave is transmitted and reflected? (assume k’s and w’s are known) reflected wave

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