Page 202 - Dynamics of Mechanical Systems

0593_C06_fm Page 183 Monday, May 6, 2002 2:28 PM

Forces and Force Systems 183

S
P(m)
P (m )
2 2
P (m )
i
i
p
P (m )
1 1 p
P (m ) i
N N
O O
FIGURE 6.8.1 FIGURE 6.8.2
A particle P and reference point O. A set S of particles and a reference point O.

S
P (m )
2 2 P (m )
i i
r
P (m ) i
1 1
p
i
G
FIGURE 6.8.3 p G
A set S of particles with mass center G. O
From Figure 6.8.3, we have:

p = p + or r = p − p (6.8.4)
r
i G i i i G
Hence, by substituting into Eq. (6.8.3), we obtain:

N
ii ∑
∑ m r = N m p − )
i(
p
i
G
i=1 i=1
N
i ∑
= ∑ m p − N m p G
i
i
i=1 i=1 (6.8.5)

N
i ∑
= ∑ m p −   N m p G
i
i
i=1  i=1 
= 0
By solving the last equation for p , we obtain:
G
N
p = (1 M) ∑ m p (6.8.6)
G i i
i=1
where M is the total mass of S. That is,
N
M = ∑ m i (6.8.7)
i=1