1
Systems of particles
REFERENCE: Arya, Chapter 8.
The ideas of Newtonian mechanics and the conservation theorems can be straightfor
wardly extended to systems of
N
particles.
1.1
Center of Mass
Consider a system of
N
particles 1
,
2
,
3
...N
, with masses
m
1
,m
2
,...,m
N
located a distance
r
1
,
r
2
,...,
r
3
from the origin
O
. The center of mass is at a point
R
(
x,y,z
) from the origin,
de±ned by
N
±
k
=1
m
k
R
=
N
±
k
=1
m
k
r
k
(1)
Where we can de±ne the total mass
M
=
∑
N
k
=1
m
k
. The velocity and acceleration of the
center of mass is
V
=
˙
R
=
1
M
N
±
k
=1
m
k
˙
r
k
(2)
A
=
¨
R
=
1
M
N
±
k
=1
m
k
¨
r
k
(3)
For example, the center of mass of a pair of particles is
R
=
m
1
r
1
+
m
2
r
2
m
1
+
m
2
(4)
The center of mass lies on the line connecting the two particles, and divides the line in the
ratio
m
2
:
m
1
. The center of mass of three identical particles,
R
=
m
r
1
+
m
r
2
+
m
r
3
m
+
m
+
m
=
r
1
+
r
2
+
r
3
3
(5)
For a continuous distribution of mass, the summation is replaced by an integral:
R
=
1
M
²
r
dm
(6)
example:
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 Spring '10
 RogerMelko
 Center Of Mass, Force, Mass, Momentum, Fundamental physics concepts

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