PROCEEDINGS OF SECTION A. 35 



To do this first bring the system to rest, then steady the beam with 

 one hand and with the other hold, the bob of the first penduhim slightly 

 deflected, and then let bob and beam go simultaneously. 



It will be seen that the second pendulum begins to swing, and 

 continues with increasing amplitude, while the amplitude of the first at 

 the same time diminishes. This goes on to a certain point when the 

 reverse takes place, and the transfer of energy forwards and backwards 

 many times between the two pendulums is strikingly demonstrated. 



From this motion the resultant frequencies wj and w^, can easily 

 be obtained by observation. For the motion of the second pendulum 

 is given by 



02 



E c \ 



= P2(l— < cos Wit - COS U>2tf 



E . u)2 - ioi . (H2 + <>>i 



= 2p2a — sm — - — t sin — ^r— t 

 c 2 2 



which shows that it is a vibration whose amplitude 



E . b)z ~ wi , 

 2«2a — sm — —= — t 

 c 2 



varies harmonically with a frequency of 



i (W2 - <"i) 

 while the oscillations of the pendulum have a frequency of 



Hence if we count the number of double swings the pendulum 

 makes ( = n say) between 11 points of rest, that is, during 5 periods 

 of amplitude change, then 



n W2 + wi 



5 (1)2 — OJi 



giving the ratio of w^ to w^. 



Now measure in the usual way the period of the individual 

 swings of the second pendulum, which is equal to 



W2 + Wj 



From these two results wi and W2 can be deduced, and the values 

 so obtained can be compared with those computed by means of the 

 triangle given in § 4. 



In this connexion it is interesting to note that if the pendulums are 

 of the same length /, and have bobs of equal mass m, then one of the 

 resultant frequencies is always the same as that of an ordinary simple 

 pendulum of length I with a rigid point of suspension, that is, it is 

 equal to >Jg/l no matter what the coupling may be. 



