472.   CeMrifligAl  TcMlMi  la  Whedi  aadi  Piril«7*— -The  rim  of  a
wheel,  moving  with  the  velocity  v,  is  subjected  to  a  centrifngal

tension  whose  amount  is  equal  to  the  weight  of  a  length  —  of  that

rim  (including  teeth,  if  it  is  a  toothed  wheel).  This  is  resisted
by  the  tenacity  of  the  mm  at  its  smallest  cross-section  (or  by
the  fiEistenings  of  the  rim,  if  it  is  made  in  s^rmeuts),  partly
assisted  by  Uie  tenacity  of  the  arms.  Each  of  the  arms  nas  to
bear  its  own  centrifugal  tension,  which,  at  a  point  close  to  the  boas,
is  equal  to  the  weight  of  a  length  of  the  arm  itself  expressed

ijo'y^  being  the  radius  of  the  wheel,  and  f  that  of  the

boss ;  and  on  the  whole,  it  is  an  error  on  the  safe  side  to  make  the
rim  strong  enough  to  bear  its  own  centrifugal  tension  without  aid
from  the  arms.  This  fixes  a  limit  of  safety  as  to  speed,  for  a  rim
of  a  given  material  and  construction.  Let  m  be  the  ratio  in  which
the  mean  sectional  area  is  greater  than  the  effective  sectional  area,
/the  greatest  working  tensile  stress,  and  let  u^  be  the  heaviness  of
the  material ;  then  the  greatest  proper  velocity,  being  that  which,
produces  the  stress/)  ia given  by  the  formula : —


^('-?;


VdQ-


.(1.)


The  modulus,  ^,  for  cast  iron  may  be  taken  at  800  feet^  or  \2ii

metres ;  so  that  when  m  =  1,  as  in  a  pulley,  or  a  fly-wheel  without
teeth,  we  have  v  =  160  feet,  or  49  metres,  per  second  nearly.  Let
a  cast-iron  spur  fly-wheel  be  so  designed  that  m  =  2;  then  t;  =  113
feet,  or  34  metres,  per  second  nearly.
