﻿Coupled 
  Vibrations 
  : 
  UnequaUMasses 
  or 
  Periods. 
  11 
  

  

  from 
  general 
  considerations. 
  But 
  it 
  seems 
  at 
  first 
  sight 
  in 
  

   direct 
  contradiction 
  to 
  the 
  theory 
  which 
  shows 
  that 
  they 
  and 
  

   z 
  vibrations 
  for 
  the 
  light 
  and 
  heavy 
  bobs 
  respectively 
  involve 
  

   the 
  selfsame 
  damping 
  factors. 
  But 
  by 
  equations 
  (23) 
  and 
  

   (24) 
  we 
  see 
  that 
  one 
  damping 
  coefficient 
  is 
  p-times 
  the 
  other. 
  

   Again, 
  by 
  equation 
  (48) 
  the 
  amplitude 
  of 
  the 
  slow 
  vibrations 
  

   of 
  the 
  heavy 
  bob 
  is 
  p-times 
  that 
  of 
  its 
  quick 
  ones. 
  In 
  the 
  

   present 
  experimental 
  case 
  p 
  equals 
  20, 
  hence 
  almost 
  all 
  the 
  

   vibration 
  visible 
  is 
  the 
  slow 
  one 
  with 
  the 
  negligibly 
  small 
  

   damping 
  coefficient. 
  On 
  the 
  other 
  hand, 
  by 
  equation 
  (47) 
  

   we 
  see 
  that 
  the 
  amplitudes 
  of 
  the 
  slow 
  and 
  quick 
  vibrations 
  

   of 
  the 
  light 
  bob 
  are 
  numerically 
  equal. 
  Consequently 
  the 
  

   large 
  damping 
  coefficient, 
  which 
  is 
  20 
  times 
  the 
  small 
  one, 
  

   affects 
  at 
  least 
  half 
  of 
  the 
  amplitude 
  visible. 
  

  

  Logarithmic 
  decrements. 
  — 
  The 
  lower 
  trace 
  on 
  fig. 
  11 
  just 
  

   dealt 
  with, 
  led 
  to 
  the 
  theoretical 
  introduction 
  of 
  the 
  damping 
  

   of 
  the 
  light 
  bob 
  as 
  expressed 
  by 
  the 
  constant 
  k 
  in 
  equation 
  

   (4). 
  It 
  also 
  became 
  necessary 
  to 
  estimate 
  the 
  experimental 
  

   vnlue 
  of 
  k. 
  To 
  do 
  this 
  one 
  pendulum 
  with 
  a 
  light 
  bob 
  was 
  

   allowed 
  to 
  oscillate 
  alone, 
  the 
  other 
  being 
  meanwhile 
  dis- 
  

   connected. 
  The 
  traces 
  for 
  the 
  lighter 
  bobs 
  P 
  were 
  taken 
  

   when 
  their 
  masses 
  were 
  respectively 
  as 
  used 
  in 
  the 
  experi- 
  

   ments, 
  so 
  as 
  to 
  be 
  one-twentieth 
  and 
  one-fifth 
  of 
  those 
  of 
  the 
  

   corresponding 
  heavy 
  ones. 
  The 
  results 
  are 
  given 
  in 
  fig. 
  12. 
  

   From 
  the 
  upper 
  trace 
  with 
  the 
  very 
  light 
  bob 
  consisting 
  

   simply 
  of 
  a 
  cardboard 
  funnel, 
  a 
  few 
  weights 
  and 
  sand 
  (total 
  

   mass 
  about 
  50 
  gms.), 
  we 
  find 
  that 
  the 
  logarithmic 
  decrement 
  

   is 
  of 
  the 
  order 
  A 
  = 
  0'017. 
  

  

  Then 
  by 
  (57) 
  we 
  have 
  

  

  / 
  L 
  -=^ 
  = 
  (0-005)m 
  (76) 
  

  

  The 
  lower 
  trace 
  with 
  bob 
  about 
  J 
  20 
  gms. 
  shows 
  consider- 
  

   ably 
  less 
  damping 
  and 
  the 
  decrement 
  need 
  not 
  be 
  evaluated. 
  

  

  Masses 
  5 
  : 
  1 
  — 
  The 
  masses 
  of 
  the 
  bobs 
  used 
  in 
  these 
  

   experiments 
  were 
  of 
  the 
  order 
  600 
  gms. 
  and 
  120 
  cms. 
  

   respectively. 
  

  

  Figs. 
  13-19 
  in 
  Plate 
  II. 
  show 
  double 
  traces 
  obtained 
  with 
  

   this 
  arrangement. 
  In 
  figs. 
  13-16 
  we 
  see 
  very 
  plainly 
  the 
  

   beat 
  effects 
  on 
  the 
  lower 
  trace 
  which 
  is 
  left 
  by 
  the 
  lighter 
  

   bob. 
  The 
  traces 
  of 
  the 
  heavier 
  bob 
  also 
  show 
  distinct 
  but 
  

   much 
  slighter 
  fluctuations 
  of 
  amplitude. 
  In 
  this 
  respect 
  

   they 
  are 
  seen 
  to 
  present 
  an 
  intermediate 
  state 
  between 
  the 
  

   cases 
  of 
  equal 
  masses 
  and 
  masses 
  as 
  20 
  : 
  1. 
  And 
  this 
  is 
  just 
  

   what 
  we 
  should 
  naturally 
  expect. 
  Further, 
  the 
  beat 
  cycles 
  

   contain 
  fewer 
  and 
  fewer 
  vibrations 
  as 
  the 
  coupling 
  increases. 
  

  

  