of  Loaded  and  Unloaded  Bars.  371 
A  general  method  has  been  given  by  Lord  Kayleigh.     It 
is  applied  to  a  special  case  in  the  next  paragraph,  in  which 
I' =  Moment  of  Inertia  of  concentrated  mass  about  an 
axis  perpendicular  to  the  plane  of  bending  (  =  mk'2). 
T  and  Y  are  the  Kinetic  and  Potential  Energies  of  the 
system. 
§  25.  For    a    Fixed-Free    Massless    Bar    (cf.    Rayleigh's 
<  Sound/  Art.  183) .      Origin  at  fixed  end  and  mass  m  at  free 
end,  the  equation  of  the  bar  is 
=  (3yi-Z*)(£)S  +  (0-2yi)(*  )' 
where  yx  and  0  are  the  values  of  y  and  ^  at  the  point  where 
the  load  is  attached.     This  equation  is  deduced  f 
rom 
mpi=mi/\(i-w)+ife. 
dxA 
Integrating,  and  determining  yx  and  6  from 
me^my^+vm 
we  get  the  desired  result. 
Following  Rayleigh's  solution,  the  equations  of  motion  are 
2EI 
%'i+-^-(%i-3^)=0         | 
V     2EI  r\      U'2  /'      W2      W+  \ 
whence 
answering  to  the  two  different  periods. 
§  26.  The  solution  can  be  simplified  if  we  assume  to  start 
with  that  the  effect  of  I'  is  small.  This  is  usually  the  case 
in  practice,  and  the  method  has  been  fully  investigated  bv 
Dr.  Ghree. 
Adopting  this  simpler  method  (cf.  Chree,  I.e.  p.  511,  §  9) 
