9.1 Fundamental Conceptions in Mechanics
521
Fig. 9.2 The position
vectors of a particle system
at arbitrary reference point
O
z
y
c.m.
mi
m j
m k
P
r c
r P
ρ c
ρ i
Now find the kinetic energy associated with arbitrary reference point P, as shown
in Fig. 9.2. In this case there is
r i = r P + ρ i
(9.1.24)
Substituting the expression (9.1.24) into expression (9.1.23), we obtain
T =
1
2
n
i=1
m i (˙ r P + ˙
ρ i ) · (˙ r P + ˙
ρ i ) =
1
2
m ˙
r
2
P +
1
2
n
i=1
m i ˙
ρ
2
i + ˙
r P ·
n
i=1
m i ˙
ρ i
(9.1.25)
where, m the total mass of the particle system. Relative to the reference point P, the
position vector ρ c of the mass center is
ρ c =
1
m
n
i=1
m i ρ i
(9.1.26)
Thus
T =
1
2
m ˙
r
2
P +
1
2
n
i=1
m i ˙
ρ
2
i + ˙
r P · m ˙
ρ c
(9.1.27)
In later examples and exercises of this chapter, the expression (9.1.27) will be
often used.
The expression (9.1.27) shows that the total kinetic energy of particle system is
the sum of three parts: (1) The kinetic energy of motion of the particle system with