## Resonance

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.