1 / 16

Summary From Last Lecture

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:. Why Care About SHO?.

rfrench
Télécharger la présentation

Summary From Last Lecture

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)

More Related