**Free vibration of a pendulum system**

This demonstration shows *the definitions of natural frequency and period of a simple system.*

Fig. 15-8: Free vibration and natural frequency of a pendulum system

Fig. 15-8 shows a pendulum system consisting of five balls suspended by strings with equal length (known as Newton’s cradle). Push the balls in a direction parallel to the plane of the frames to apply an initial displacement and release the balls. The balls then move from side to side in free vibration at the natural frequency of the system.

Count the number of oscillations of the balls in 30 seconds and the number of oscillations divided by 30 is the natural frequency of the oscillations of the pendulum system in cycles per second or Hz. The inverse of the natural frequency is the period of the oscillations. For the studied system, 40 cycles of oscillation in 30 seconds were counted. Thus the natural frequency of the system is 40/30 = 1.33 Hz and the period is 1/1.33 = 0.75s.

The vertical distance between the supports of the strings and the connecting points on the balls is 140mm so the theoretical value of the natural frequency is: