Scientific context

Example 1: A swinging pendulum. The photograph in Fig. 3.1 shows the motion of a swinging pendulum. The multiple images of the pendulum, obtained using stroboscopic lighting, represent its position at the times when the stroboscope flashed which was every tenth of a second. In Fig. 3.2 we have drawn a graph of the horizontal displacement of the pendulum bob from its central position (i.e. when the string is vertical).

None of the functions we have considered so far provides a model for the way the displacement changes with time and this is obvious if we draw the graph of the displacement over a much longer time interval. This graph is shown in Fig. 3.3. As the bob swings backwards and forwards its displacement varies from – 0.076 m to + 0.076 m in a periodic way.

Polynomials and exponential functions are not periodic and so it is clear that a new class of functions is going to be needed if we are to describe the kind of motion shown. This wave-type behaviour is a very common feature of many periodic systems in science.