## Resonance

Every object and physical system has a natural frequency - or frequencies - of vibration. For many simple systems there is an equation for this frequency.
For a pendulum,
$f=\frac{1}{2 \pi} \sqrt{\frac{g}{l}}$

For a vertically oscillating spring,
$f=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}$

Any oscillating system can be made to oscillate at any frequency
$f$
resonate by applying an external stimulus
$f_{EXTERNAL}$
times per second. The frequency
$f_{EXTERNAL}$
is called the driving frequency. If the driving frequency is equal to the natural frequency, then the amplitude of the oscillation reaches a maximum, and so does the energy of the system. Many systems dissipate this energy by some mechanism - friction or air resistance. If this is not possible the the system may oscillate to destruction. The best example is the Tacoma narrow bridge - nicknamed 'Galloping Gertie', which collapsed when the driving force was applied by wind in 1940.