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Free SHM

Superposition. Free SHM. EOM:. Solutions:. Superposition: the sum of solutions to an EOM is also a solution . . . . . . if the EOM is linear . . …or a linear combination: c 1 x 1 + c 2 x 2. x and its derivatives appear only to first power. Is the linear combination useful?.

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Free SHM

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  1. Superposition Free SHM EOM: Solutions: Superposition: the sum of solutions to an EOM is also a solution . . . . . . if the EOM is linear. …or a linear combination: c1x1 + c2x2 x and its derivatives appear only to first power.

  2. Is the linear combination useful? initial velocity, begin at origin vo ,xo = 0 initial displacement, begin at rest xo , vo = 0

  3. x1+x2 x3 x1 x2 initial displacement xoand velocity vo Solve each for A3, equate, solve for f: Superposition: The motion resulting from two simultaneous initial conditions is equal to the sum of the motions resulting from each initial condition. . . . . . if the EOM is linear. Plug back into top equation to get A3: Solution:

  4. The trig is getting complicated, let’s try something else… Imagine SHM in 1D… …as a projection of 2D motion. +x

  5. The trig is getting complicated, let’s try something else… Imagine SHM in 1D… …as a projection of 2D motion. +x “imaginary” y-axis “real” x-axis

  6. “imaginary” y-axis “real” x-axis Describe the position… z r OR …geometrically …algebraically j-> “rotate 90 degrees” j2b-> “go along original axis in opposite direction” … an imaginary number!

  7. “imaginary” y-axis “real” x-axis “complex number” z r “real” “imaginary” “complex plane”

  8. Circular motion in complex plane: …expansion of exp(jt)! “Euler’s Formula” - imaginary exponentials oscillate !

  9. Euler’s Identity “..we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth.” -Benjamin Peirce “if this formula is not immediately apparent to a student being told it, the student would never be a first-class mathematician.” -Gauss “this amazing jewel … the most remarkable formula in mathematics.” -Feynman “The most famous equation in all mathematics.” -Constance Reid “Greatest equation ever.” -Physics World

  10. Physical Graffiti

  11. Describe SHM in the complex plane… Does it work? Yes, if: A andftake any value Keepin’ it real ! Don’t forget!!!

  12. Same A and w, but different f:

  13. x1 and x2 in phase x1 and x2 out of phase x x x1 x1 x2 x2 t t

  14. Same A and f, different w: Stick with trig… Then: yields beats.

  15. Systems with linear EOMs obey the Principle of Superposition: solutions to the EOM can be summed to make more complicated solutions. Re Im

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