CHAPTEE I 



THEORY OF VIBRATIONS 



4. The Pendulum. 



A vibrating body, such as a string or a bar or a plate, 

 cannot give rise to a sound except in so far as it acts on the 

 surrounding medium, which in turn exerts a certain reaction 

 on the body. The reaction is however in many cases so slight 

 that its effects only become sensible after a large number of 

 oscillations. Hence, to simplify matters, we begin by ignoring 

 it, and investigate the nature of the vibrations of a mechanical 

 system considered as completely isolated. 



The theory of vibrations begins, historically and naturally, 

 with the pendulum. With this simple apparatus 

 we are able to illustrate, in all essentials, many 

 important principles of acoustics, the mere differ- 

 ences of scale as regards amplitude and period, 

 enormous as they are, being unimportant from the 

 dynamical point of view. 



A particle of mass M, suspended from a fixed 

 point by a light string of length I, is supposed 

 to make small oscillations, in a vertical plane, 

 about its position of equilibrium. If the inclina- 

 tion of the string to the vertical never exceeds 

 a few degrees, the vertical displacement of the 

 particle may (to a first approximation) be neg- 

 lected, and the tension (P) of the string may be 

 equated to the gravity Mg of the particle. Since the horizontal 

 displacement (x) is affected only by the horizontal component 

 of the tension, we have 



M = - P - = - Ma - m 



Fig. 1. 



