Summary From Last Lecture

1. Summary From Last Lecture • Hooke’s Law for Springs: • Simple Harmonic Oscillator • Equation of Motion: • Solution: • Amplitude: • Angular Frequency: • Phase: • Period: • Frequency: • Pendulum • Like a SHO for small angles: • Physical Pendulum:

2. Why Care About SHO? • Because any potential energy function with a dip (a stable fixed point) acts like a SHO for small oscillations about that point • Thus the atoms in a lattice vibrate as if connected by springs (at low temperatures)

3. Example: Falling Through the Earth • If you were to drill a hole through the earth, what would be the motion of a falling particle • Remember, only that part of the earth inside the current position attracts the particle

4. Example: Falling Through the Earth • So falling object will undergo SHM. • What’s the period? • This is the same time for an orbit at the surface!

5. More on the General Solution • Use trigonometric angle sum formula to look at general solution of SHO • Look at behavior of x at small t • So B = Acosfis the initial position and Cw = wAsinf is the initial velocity

6. Solving the SHO with Exponentials • Look again at equation of motion • Try (formal) solution of the form: • Plug these into EOM • Have turned 2nd order differential equation into a quadratic equation in w • The complex exponential is a good solution, we just have to take real part

7. SHO with Periodic Forcing • Sometimes there is a periodic external force driving a SHO • Swinging on a swing, bowing a stringed instrument • How do we deal with this? • Model the driving force as a sine/cosine/exponential of a given size, with an arbitrary (i.e. we control it) frequency w • Assume solution has the same form (same frequency, different amplitude)

8. SHO with Periodic Forcing • Write down the equation of motion, and plug in expected form of solution • So amplitude “blows up” when driving frequency reaches natural frequency • Never really happens because of damping • This is Resonance

9. SHO with Damping • Include a non-conservative force proportional to velocity, a damping force (and, to begin, no periodic forcing) • Use same assumed form for solution

10. SHO with Damping • Now plug this solution for w into general solution for x • For b2 < 4k (underdamped case) get oscillations at nearly the natural frequency with exponentially diminishing size

11. SHO with Damping • The two other cases are • Critically damped (b): b2 = 4k • Fastest approach to zero • No oscillation (doesn’t cross zero) • Overdamped (c): b2 > 4k • Still no oscillation but slow approach to zero

12. SHO with Damping • Note that the total energy decreases with time (for underdamped case)

13. SHO with Forcing and Damping • Now we can try the full, general case of a SHO with both forcing and damping • Note that A is now complex. This means there is a phase between the driving force and the motion of the system

14. SHO with Forcing and Damping • What is the phase and (magnitude of the) amplitude of the system?

15. SHO with Forcing and Damping • What is the phase and (magnitude of the) amplitude of the system?

16. SHO with Forcing and Damping • Resonance gets larger as damping gets smaller • Phase goes from “in phase” to “out of phase” passing through 90 degrees at the resonance (maximizing the power Fv)