﻿Coupled 
  Vibrations 
  : 
  Unequal 
  Masses 
  or 
  Periods. 
  67 
  

  

  Initial 
  Conditions. 
  Case 
  I. 
  — 
  Suppose 
  the 
  heavj 
  bob 
  of 
  

   mass 
  Q 
  (which 
  = 
  />P) 
  is 
  pulled 
  aside 
  and 
  that 
  the 
  light 
  one 
  

   of 
  mass 
  P 
  is 
  allowed 
  to 
  hang 
  at 
  rest 
  in 
  its 
  more 
  or 
  less 
  

   displaced 
  position 
  according 
  to 
  the 
  coupling 
  in 
  use. 
  Then 
  

   we 
  may 
  write 
  : 
  

  

  For 
  £ 
  = 
  let 
  z=f, 
  

  

  Rn-f 
  

  

  then 
  it 
  follows 
  statically 
  that 
  y 
  = 
  - 
  — 
  ' 
  _ 
  : 
  

  

  > 
  u 
  i 
  + 
  P+/3p' 
  y 
  . 
  . 
  . 
  (31) 
  

  

  di=° 
  and 
  i 
  =0 
  - 
  

  

  Differentiating 
  with 
  respect 
  to 
  time 
  (27) 
  and 
  (28) 
  without 
  

   its 
  accents, 
  and 
  writing 
  in 
  the 
  latter 
  n 
  for 
  m/s/(l 
  + 
  /3), 
  we 
  

   find 
  

  

  dz 
  t 
  

  

  -j- 
  =Ee~ 
  ps 
  [in 
  cos 
  (mt 
  + 
  e)—ps 
  sin 
  (mt 
  -H 
  e)] 
  

  

  + 
  F<? 
  8t 
  [n 
  cos 
  (nt 
  + 
  <j>) 
  — 
  s 
  sin 
  (nt 
  + 
  <j>)], 
  . 
  (32) 
  

   dt 
  

  

  -J- 
  = 
  — 
  pEe 
  pa 
  '[mcos 
  (mt 
  + 
  e) 
  — 
  ps 
  sin 
  (mt 
  + 
  e)'] 
  

  

  + 
  Fe~ 
  st 
  [ncos(nt 
  + 
  <j>) 
  — 
  ssin(nt 
  + 
  (l>)]. 
  . 
  (33) 
  

  

  The 
  conditions 
  (31) 
  introduced 
  in 
  equations 
  (27), 
  (28), 
  

   (32), 
  and 
  (33) 
  give 
  

  

  /= 
  E 
  sin 
  e 
  + 
  F 
  sin 
  <£, 
  . 
  . 
  . 
  (34) 
  

  

  l 
  + 
  p 
  + 
  ^ 
  = 
  "" 
  />Esin6 
  + 
  Fsin 
  *'- 
  * 
  * 
  (35) 
  

  

  = 
  E(mcose 
  — 
  />s 
  sin 
  e) 
  + 
  F(n 
  cos 
  <£ 
  — 
  ssin<£), 
  . 
  (36) 
  

  

  0=— 
  pEi(m 
  cose 
  — 
  ps 
  sine) 
  +F(ncos 
  <j> 
  — 
  s 
  sin 
  <£). 
  (37) 
  

  

  But, 
  by 
  reason 
  of 
  the 
  smallness 
  of 
  ps 
  in 
  comparison 
  with 
  

   m 
  (of 
  the 
  order 
  O01) 
  and 
  of 
  s 
  in 
  comparison 
  with 
  n 
  (still 
  

   less), 
  we 
  may 
  write 
  instead 
  of 
  (36) 
  and 
  (37) 
  the 
  following 
  : 
  

  

  = 
  Emcose 
  + 
  Fncos<£, 
  .... 
  (38) 
  

  

  and 
  0=-pEmcose 
  + 
  Fncos</>. 
  . 
  . 
  . 
  (39) 
  

  

  These 
  are 
  satisfied 
  by 
  

  

  e=j 
  and 
  </>=| 
  (40) 
  

  

  These 
  values 
  inserted 
  in 
  (34) 
  and 
  (35) 
  give 
  

   /=E 
  + 
  F, 
  

  

  ^ 
  M 
  - 
  „F 
  + 
  F 
  

  

  WheDCe 
  E=— 
  /_ 
  and 
  F-ii±«¥- 
  • 
  W 
  

  

  l 
  + 
  p 
  + 
  £/> 
  l+P 
  + 
  /3p 
  v 
  ; 
  

  

  ¥2 
  

  

  