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

3. Is the linear combination useful? initial displacement, begin at rest x o, v o = 0 initial velocity, begin at origin v o, x o = 0

4. initial displacement x o and velocity v o Solve each for A 3, equate, solve for  Plug back into top equation to get A 3 : Solution: 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. x1x1 x2x2 x 1 +x 2

5. “real” x-axis “imaginary” y-axis Describe the position… j -> “rotate 90 degrees” …algebraically …geometrically OR j 2 b -> “go along original axis in opposite direction” … an imaginary number! r z

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

7. Describe SHM in the complex plane… Keepin’ it real ! Does it work? Yes, if: A a nd  take any value Don’t forget!!!

8. Same A and , but different  :

9. x 1 and x 2 in phase x 1 and x 2 out of phase x t x t x 1 x 2 x 1 x 2

10. Same A and , different  : Then: yields beats. Stick with trig…