288 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



Exactly as in Article 65 we can see that, if the period is real, 

 the motion can be indefinitely small ; otherwise it soon becomes 

 so large that we cannot simplify our equations by neglecting B 2 . 

 In the former case the equilibrium is stable and in the latter 

 unstable. 



The process adopted shows that we might have reduced the 

 expression for T by substituting zero for 6 in J., and the expres 

 sion for V might have been taken to be simply the term of the 

 series which contains 2 . These simplifications might have been 

 made before differentiating the energy equation. Thus if we 

 express the kinetic energy correctly to the second order of small 

 quantities in the form JJ.# 2 ,and the potential energy also correctly 

 to the second order of small quantities in the form -^Ctf 2 , the 

 period of the small oscillations is 27r\/(A/C). In the case of a 

 simple pendulum of mass m and length I, A is ml 2 and C is mgl, 



so that 



A/G=l/g. 



In any other case we may compare the motion with that of a 

 simple pendulum and then the quantity gA/C is the length of a 

 simple pendulum which oscillates in the same time as the system. 

 It is called the length of the simple equivalent pendulum for the 

 small oscillations of the system. 



254. Examples. 



1. Two rings of masses m, m connected by a rigid rod of negligible mass 

 are free to slide on a smooth vertical circular wire of radius a, the rod sub 

 tending an angle a at the centre. Prove that the length of the simple equiva 

 lent pendulum for the small oscillations of the system is 



(m + m ) a/,J(m 2 + m 2 + 2mm cos a). 



2. One end of an inextensible thread is attached to a fixed point J., and 

 the thread passes over a small pulley B fixed at the same height as A and at 

 a distance 2a from it and supports a body of mass P. A ring of mass M can 

 slide on the thread and the system is in equilibrium with M between A and 

 B. Prove that the time of a small oscillation is 



47T ,J{aMP (M+ P}jg (4P 2 - M 2 ) f }. 



3. A particle is suspended from two fixed points at the same level by 

 equal elastic threads of natural length a, and hangs in equilibrium at a depth 

 h with each thread of length I. Prove that, if it is slightly displaced, parallel 

 to the line joining the fixed ends of the threads, the length of the simple 

 equivalent pendulum for the small oscillation is 



hl 2 (l-a~)/(l 3 -k 2 a}. 



