﻿Xongenerating 
  Force 
  of 
  High 
  Frequency. 
  

  

  619 
  

  

  3. 
  As 
  a 
  second 
  example 
  we 
  shall 
  determine 
  the 
  period 
  of 
  

   a 
  pendulum 
  the 
  point 
  of 
  suspension 
  of 
  which 
  is 
  moved 
  to 
  and 
  

   fro 
  vertically 
  in 
  a 
  small 
  period 
  2t, 
  with 
  constant 
  acceleration 
  

   a 
  in 
  one 
  half 
  of 
  the 
  path 
  and 
  equal 
  and 
  opposite 
  acceleration 
  

   in 
  the 
  other. 
  Let 
  I 
  be 
  the 
  length 
  of 
  the 
  simple 
  equivalent 
  

   pendulum 
  ; 
  then 
  the 
  spring 
  is 
  (g 
  + 
  aj/l 
  and 
  (g—a)/l 
  altern- 
  

   ately 
  for 
  equal 
  intervals 
  t. 
  The 
  spring 
  variation 
  about 
  the 
  

   mean, 
  the 
  required 
  time 
  integral, 
  and 
  its 
  square, 
  are 
  shown 
  

   graphically 
  : 
  — 
  

  

  The 
  graph 
  of 
  the 
  squared 
  integral 
  is 
  made 
  up 
  of 
  parabolic 
  

   arcs 
  OP', 
  P'Q, 
  . 
  . 
  . 
  

  

  Now 
  NP'=NP*=g) 
  2 
  . 
  

  

  Hence 
  the 
  mean 
  height 
  of 
  OP'Q 
  is 
  ~l 
  -=- 
  J 
  ; 
  therefore 
  

  

  , 
  If 
  1 
  (ar) 
  2 
  ) 
  

  

  and 
  the 
  period 
  of 
  the 
  pendulum 
  is 
  

  

  WWW}- 
  

  

  It 
  must 
  be 
  remembered 
  that 
  r 
  is 
  small 
  compared 
  with 
  \/l/g, 
  

   but 
  ar 
  may 
  be 
  comparable 
  with 
  x^lg. 
  

  

  4. 
  In 
  the 
  investigation 
  of 
  § 
  1 
  the 
  problem 
  is 
  limited 
  to 
  

   the 
  case 
  when 
  the 
  disturbance 
  is 
  made 
  up 
  of 
  a 
  series 
  of 
  

   simple 
  elements 
  which 
  are 
  all 
  in 
  zero 
  phase 
  at 
  the 
  same 
  

   instant. 
  Removing 
  this 
  restriction, 
  we 
  now 
  examine 
  the 
  

   motion 
  when 
  the 
  spring 
  has 
  any 
  rapid 
  periodic 
  variation 
  

   2/(0 
  about 
  its 
  mean 
  value 
  X. 
  

  

  