f = (1/2π)(g/L)1/2
where g is the magnitude of the earth’s gravitational field (post 16.16).
This statement about frequency is equivalent to saying that the time, T, that the object (in the picture above) takes to move from A through B to A’ and back again to A, is given by
T = 2π(L/g)1/2.
The relationship between frequency and this time period is explained in post 16.14.
How can we keep our device oscillating? We have to supply energy to compensate for the energy dissipated. Supplying energy is equivalent to doing work (post 16.21). And doing work means applying a force to move an object (post 16.20).
Returning to the picture above, when the object at the end of the pendulum is at A, we need to push it (apply a force – see post 16.13) towards A’. To keep the pendulum moving, we need to push the object every time it arrives at A. So we need to drive it whenever a time interval T (defined in the second equation above) has elapsed. In other words, we need to drive it with a frequency f (defined in the first equation above). If we push it at a higher or lower frequency, we risk disturbing the natural motion of the pendulum.
It is easiest to drive the pendulum at the frequency at which it oscillates naturally – its natural frequency or resonant frequency. Anyone who has pushed a child on a swing will have learned this result intuitively. The response of the oscillating system to being driven at its natural frequency is called resonance.
Resonance is not confined to mechanical systems. The cavity magnetron was developed in 1940 by John Randall and Harry Boot. It is a device that enables electrons to oscillate at the natural frequency of microwaves. Originally it was used in radar for tracking the movement of aircraft and ships. Now most of us have one in our kitchens – in a microwave oven.
If you go to hospital for an MRI scan, the nuclei of the hydrogen atoms in your body (most of them in water molecules) will resonate at the natural frequency of radio waves. The radio waves emitted are used to form an image of the inside of your body.
So resonance can be very useful – but it can also cause problems. The London Millenium Footbridge was opened in the year 2000. The distinguished sculptor Sir Anthony Caro had a brilliant idea for what the bridge should look like (see above). Unfortunately, the equally distinguished architects (Foster and Partners) and civil engineers (Arup Group) were less brilliant. The bridge swayed with a natural frequency that was close to the frequency of people walking over it – it was known as the “wobbly bridge” and had to be closed for two years while its structure was modified.
At least the problem with the London Millenium Footbridge could be solved. The Tacoma Narrows bridge in the USA shook itself to pieces because its resonant frequency matched the frequency of gusts of wind (https://www.youtube.com/watch?v=j-zczJXSxnw).
So, like so many things, resonance is a very useful phenomenon but it can also be the source of serious problems.