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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.