Theory  of  The  moment  of  this  round  the   axis   is   nr^drd^r  .  d/l.     Hence

ZirnQ/l .   /    rzdr  —  %7rna*d/l  is  the  total  moment  over  the  cross-

J  o

section.    But  this,  divided  by  the  twist  6/1,  is  the  torsional  rigidity.
Hence  (25)  above.
The  integral  just  found  is  the  total  moment  of  the  elastic  forces
in  each  cross-section  producing  angular  acceleration  of  the  whole
moving  system  towards  the  position  of  equilibrium.  Besides
these  forces,  the  system  is  acted  on  by  forces  of  viscous  resist-
ance (that  is  resisting  forces  depending  on  the  velocity),  partly  in
the  wire  and  partly  between  the  moving  system  and  the  air  ;  and
these  we  shall  assume  as  everywhere  proportional  to  the  velocity,
and  therefore  to  have  a  moment  round  the  axis  directly  propor-
tional to  the  angular  velocity


FIG.  41.
We  shall  denote  this   moment  by   Wcpddldt.     The   equation
of  motion  is  then

-0     ....     (30)


the  differential  equation  found  on  p.  221,  above,  which,  under  the
conditions  there  stated,  denotes  simple  harmonic  oscillations  with
amplitude  diminishing  at  logarithmic  rate  k.  From  the  solution
there  given  we  have

......  (31)


which  yields  at  once  (26)  and  (27).
MOMENT  OF  INERTIA  OF  VIBRATOR.  237

If  the  wire  be  not  of  truly  circular  section,  T,  its  torsional   Theory  of
rigidity,  must  be  used  instead  of  irna*/2  in  (30)  and  (31),  which    Method,
then  give  (29)  above.
Since  the  torsional  rigidity  of  a  cylindric  wire  is  as
the  fourth  power  of  its  radius,  or  the  square  of  its  area
of  cross-section,  it  is  frequently  preferable  to  use  a
number  of  wires  side  by  side  to  support  a  given  load,
rather  than  a  single  wire  of  equivalent  cross-section.
