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Summary of oscillations

Oscillations are characterised by an amplitude $$(x_{0})$$ and a period $$(T)$$ (or frequency $$(f)$$).

Angular frequency: $$\omega=2\pi f=2\pi/T$$

Simple harmonic motion (SHM) : periodic and $$a\propto -x$$

Equations for SHM

  • in terms of position :

    $$$v=\pm\omega\sqrt{x_{0}^{2}-x^{2}} \quad \quad a=-\omega^{2}x $$$
  • in terms of energy:

    $$$E_{\text{T}}=\frac{1}{2}m\omega^{2}x_{0}^{2}$$$$$$E_{k}=\frac{1}{2}m\omega^{2}(x_{0}^{2}-x^{2}) \quad \quad E_{\text{p}}=E_{\text{T}}- E_{\text{k}}=\frac{1}{2}m\omega^{2}x^{2}$$$

Damped oscillations:

  • Light damping: amplitude of oscillation decreases gradually and period is constant.

  • Critical damping: no oscillation and shortest time to reach the state of rest.

  • Heavy damping: no oscillation and long time to reach the state of rest.

Forced oscillation: damped oscillation maintained by a driver force.

Resonance : driver frequency=natural frequency $$\Rightarrow$$ peak in the amplitude.