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0593_C06_fm Page 196 Monday, May 6, 2002 2:28 PM

196 Dynamics of Mechanical Systems

Section 6.8 Mass Center
P6.8.1: Let a system of 10 particles P (i = 1,…, 10) have masses m and coordinates (x , y , z )
i
i
i
i
i
– – –
relative to a Cartesian coordinate system as in Table P6.8.1. Find the coordinates x, y, z
of the mass center of this set of particles if the m are expressed in kilograms and the x , i
i
y , z are in meters. How does the result change if the m are expressed in slug and the x , i
i
i
i
y , z in feet?
i
i
TABLE P6.8.1
Masses and Coordinates of a Set of Particles
P i m i x i y i z i
6 –1 0 2
P 1
4 3 –7 4
P 2
3 4 –5 –5
P 3
8 8 0 7
P 4
1 –2 –3 –9
P 5
9 1 –2 0
P 6
4 7 –8 3
P 7
5 –3 4 –2
P 8
5 1 –8 9
P 9
2 6 –5 1
P 10
P6.8.2: See Problem P6.8.1. From the definition of mass center as expressed in Eq. (6.8.3)
show that the coordinates of the mass center may be obtained by the simple expressions:
N
N
N
) ∑ ) ∑ ) ∑
x = (1 m m ii y = (1 m m ii z = (1 m m ii
y
x
z
i=1 i=1 i=1
∑
N
where m = m i .
i=1
P6.8.3: Use the definition of mass center as expressed in Eq. (6.8.3) to show that the mass
center of a system of bodies may be obtained by: (a) letting each body B be represented
i
by a particle G located at the mass center of the body and having the mass m of the body;
i
i
and (b) by then locating the mass center of this set of particles.
P6.8.4: See Problem P6.8.3. Consider a thin, uniform-density, sheet-metal panel with a
circular hole as in Figure P6.8.4. Let the center O of the hole be on the diagonal BC 13 in.
from corner B. Locate the mass center relative to corner A (that is, distance from A in the
AB direction and distance from A in the AC direction).
C D
3.7 in.
17 in.
O
OB = 13 in.
FIGURE P6.8.4
A thin, uniform-density panel with a A B
circular hole. 31 in.