94 6. SYSTEMS OF PARTICLES
and
U
1!2
D
n
X
iD1
Z
s
2
s
1
.
F
t
/
i
ds
in which
.
F
t
/
i
represents the tangential component of the resultant force acting on particle P
i
by several forces. at is, explicitly, the work energy principle for a system of particles becomes
1
2
n
X
iD1
m
i
v
2
i
1
C
n
X
iD1
Z
s
2
s
1
.
F
t
/
i
ds D
1
2
n
X
iD1
m
i
v
2
i
2
or in more concise form
T
1
C U
1!2
D T
2
: (6.33)
6.8 CONSERVATION OF ENERGY FOR A SYSTEM OF
PARTICLES
Since the kinetic energy of a system of particles is given by
T D
1
2
n
X
iD1
m
i
v
2
i
:
is can be expressed in terms of velocity at the mass center G and velocities reference to this
mass center. To this end, one first considers the velocity
*
v
i
of particle P
i
;
*
v
i
D
*
v C
*
v
0
i
;
as illustrated in Figure 6.4 in which
*
v is the velocity of the mass center G.
Note that
*
v
i
*
v
i
D v
2
i
: erefore, the kinetic energy of a system of particles becomes
T D
1
2
n
X
iD1
m
i
v
2
i
D
1
2
n
X
iD1
m
i
*
v C
*
v
0
i
*
v C
*
v
0
i
) T D
1
2
n
X
iD1
m
i
!
Nv
2
C
*
v
n
X
iD1
m
i
*
v
0
i
C
1
2
n
X
iD1
m
i
v
0
i
2
:
According to Equation (6.25a), the second summation term on the rhs is zero. erefore,
the kinetic energy of the system of particles reduces to
T D
1
2
m Nv
2
C
1
2
n
X
iD1
m
i
v
0
i
2
: (6.34)
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