Bukhovtsev-et-al-Problems-in-Elementary-Physics

MECHANICS
239
t-4----- 7j------......
i+T
?i+l------~
Fig. 865
sider an arbitrary section with the number i (Fig. 365). The acceleration of the
various points in this section will be different, since the distances from the
points to the axis of rotation are not the same. If the difference ';+1-'1 is
small, however, the acceleration of the i-th section may be assumed as equal
to ro2 'i+
2+'1
, and this will be the more accurate, the smaller is the
length of the section.
The i-th section is acted upon by the elastic force T;+l from the side of
the deformed section i+ 1 and the force Ti from the side of the section
i-I. Since the mass of the i-th section is ~ ('1+1 -rd, on the
Newton's second law we can write that
m
'·+1+,.
T;-Ti+l=T (ri+l- f ; ) 00 2 I 2 t
or
mID? ( 2 2)
Ti+1-T;=-y r'+l-r/
basis of
Let us write the equations of motion for the sections from k to n, inclusive,
assuming that 'n+i=l and ',,=x:
moo? a
-Tn=--(12_ r n)
. 21
moo? I I
Tn-T n-l = -Y ~rn-fn-l)
mID! t 2
Tk+I-Tk+ 1=-21 (rk+I-'k+J
mID'}, 2
Tk+l-T,,=-21(rk+l-X!)
;;
The first equation in this system takes into account the fact that the
elastic force does not act on the end of the rod, te., Tn+l =0. Upon
summing up the equations of the system, we find that the sought tension
• moo'/,
IS equal to T"=2(11_X 2 ) .