Center of Mass - Center of Mass The Center of Mass:Chap...

Center of Mass

The Center of Mass:Chap
13.2
n
There is a special point in a system or
object, called the center of mass , that
moves as if all of the mass of the
system is concentrated at that point.
n
The system will move as if an external
force were applied to a single particle of
mass M located at the center of mass
n
M is the total mass of the system

Center of Mass, Coordinates
n
Consider the system is made up of
many pieces labeled with the subscript i
n
The coordinates of the center of mass
are
n
where M is the total mass of the system
M = m
i
i

Center of Mass, position
n
The center of mass can be located by its
position vector, r
CM
n
r
i
is the position of the i th particle, defined by

C of M, Example
x
cm
=
(1.0 m )1.0 kg + (2.0 m )1.0 kg + (0 m )2.0 kg
4.0 kg
x
cm
= 0.75 m
y
cm
=
(0 m )1.0 kg + (0 m )1.0 kg + (2.0 m )2.0 kg
4.0 kg
y
cm
= 1.0 m
r
r
cx
= x
cn
ˆ
i + y
cm
ˆ
j = 0.75 m
ˆ
i + 1.0 m
ˆ
j
M
1
= M
2
= 1.0kg, M
3
= 2.0kg
x
cm
=
m
i
x
i
i
M

Center of Mass, Example 2
n
Both masses are on
the x -axis
n
The center of mass
is on the x -axis
n
The center of mass
is closer to the
particle with the
larger mass

C of M Example 2
r
r
cm
=
m
1
x
1
+ m
2
x
2
m
1
+ m
2
ˆ
i
e . g . take m
1
= 10. kg , x
1
= 1.0 m & x
2
= 10. m , m
2
= 1.0 kg
r
r
cm
=
10. + 10.
11
ˆ
i = 1.8 m
ˆ
i

Center of Mass, Extended
Object
n
Think of the extended object as a
system containing a large number of
particles
n
The particle distribution is small, so the
mass can be considered a continuous
mass distribution

Center of Mass, Example
n
An extended object
can be considered a
distribution of small
mass elements, D m
n
The center of mass
is located at position
r
CM

Center of Mass, Extended
Object, Coordinates
n
The coordinates of the center of mass
of the object are