516 



BELL SYSTEM TECHNICAL JOURNAL 



of the energy being lost in internal friction at each oscillation. Some 

 resonant elements which have been used in timekeepers are illustrated in 

 Fig. 3. 



The simplest appearing of all these is that of a mass, M, supported by a 

 spring with stiffness, S. From the equation of motion 



Sx = M 



d X 

 If 





X.W 



ELECTRICAL RESONANT 

 CIRCUIT 



GRAVITY PENDULUM 



STEEL OR QUARTZ 

 TUNING FORK 



£ 



STEEL OR QUARTZ ROD 



IN FLEXURE OR 



LONGITUDINAL VIBRATION 



HAIRSPRING AND 

 BALANCE 



VERTICAL 

 PENDULUM 



Fig, 3t— Typical resonant elements used in timekeeping. 



the period of oscillation may be derived simply and is found to be 



= 2.^": 



M 



S 



Similarly for the simple electrical resonant circuit where current flowing 

 in an inductance, L, behaves like a mass, and current flowing in a condenser,. 

 C, behaves like the reciprocal of a stiffness, the period may be written. 



T = IttVLC 



Similar expressions are derivable for the periods of oscillation of all simple 

 oscillating systems, including the pendulum for which the period (for 

 small amplitudes) is given by 



-vl 



