• a lot of things oscillate in the CM world
    • musical instruments
    • loudspeakers
    • electronics devices such as
      • microwaves
      • radio waves for radio transmissions
      • wireless remote controls

mass-spring oscillator

  • a mass on a spring system is a simple harmonic oscillator
  • a simple spring is attached to a mass \(M\)
  • it’s restoring force \(F\) is proportional to \(y\)
    • \(y\) is the amount of stretch applied to the spring
  • \( F = -Ky \)
    • where \(K \) is a spring constant that characterizes the spring
    • sign is negative because it is “restoring”, pulling the mass back to equilibrium

simple harmonic oscillator

Simple-Harmonic-Motion

  • from newton’s second law
    • \( F = Ma = M\frac{d^2 y}{d t^2} = - Ky \)
  • rearranging
    • \( \frac{d^2 y}{d t^2} = - \frac{K}{M}y = -\omega^2y \)
  • where \( \omega \) is the angular frequency
    • \( \omega = \sqrt{\frac{K}{M}} \)
    • one possible solution is \( y \propto sin \omega t \)
  • angular frequency \( \omega = 2 \pi f\)
    • units: \(\frac{rad}{s}\)
    • \(f\): frequency

mass-spring system

  • a simple harmonic oscillator equation is given by
    • \( \frac{d^2 y}{d t^2} = -\omega^2 y \)

examples

  • mass on a spring
  • electrical resonant circuits
  • “helmholtz” resonators in acoustics
    • wine bottle resonator
    • the air inside the bottle behaves like a spring
    • the air at the neck acts like a mass
    • a given bottle produces only one note allegedly
  • linear oscillators in general
  • calculating the oscillation frequency in a mass-potential field