Vibrations of the Mass Variable Systems
29
where r S1 , v S1 , L S1 are the position vector, velocity and angular momentum of the
mass center S, respectively, the total angular momentum after body separation is
L Oa = L O1 + L O2 = r S1 × (M − m)v S1 + L S1 + r S2 × mv S2 + L S2
(12)
The difference between the angular momentum before and after separation is
L O = L S1 + L S2 − L S + r S1 × (M − m)v S1 + r S2 × mv S2 − r S × Mv S (13)
Introducing the position vectors r S1 and r S2 (Fig. 1)
r S1 = r S + ρ S1 , r S2 = r S + ρ S2
(14)
the relation (10) transforms into
L O = L S1 + L S2 − L S + ρ S1 × (M − m)v S1 + ρ S2 × mv S2 − r S × Mv S
(15)
i.e., for (6) it is
L O = L S1 + L S2 − L S + ρ S1 × (M − m)v S1 + ρ S2 × mv S2 + r S × K (16)
The relation (13) gives the dependence of the angular momentum on the linear
momentum.
Otherwise, using (11) and the position of the mass center S
r S =
M + m
M
r S1 +
m
M
r S2
(17)
we obtain
(M − m)ρ S1 = −mρ S2
(18)
Substituting (15) into (13), it is
L O = r S × K + L S1 − L S + L S2 − ρ S2 × m(v S1 − v S2 )
(19)
According to the principle of the angular momentum, variation of the angular
momentum (16) in the time interval t is equal to the impulse J
M , which is the sum
of the impulse of the moment M
Fr
O of the resultant force for the point O and of the
impulse of the resultant torque M, caused by active and reaction torques, i.e.,
L O =
M
Fr
O + M
t = J
M
(20)
Substituting (7) and (2.16) for mass separation, the relation (3.3) gives