[QMSE] W01 - Oscillations

• 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

• from newton’s second law
  • $F=Ma=M\frac{d^{2}y}{d{t}^2}=-Ky$
• rearranging
  • $\frac{d^{2}y}{d{t}^2}=-\frac{K}{M}y=-{\omega }^{2}y$
• 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