Motion of the CM
Motion of the CM
cm
a
M
F
G
G
=
ext
net
v
P
G
G
=
total
The CM is a good “representation” of the extended object.
Internal forces among the parts may change the velocities
and accelerations of the parts, but
the velocity of the CM
of a system remains constant unless it is acted on by an
external force.
Conservation of
linear momentum
Constant velocity of
center of mass
=
Trajectory of the CM
When a shell explodes, the CM keeps moving along
the parabolic trajectory the shell had before the
explosion.
Reference frame of the CM
Motion of a system of particles can be broken down to:
1. Motion of each particles relative to the CM
2. Motion of the CM relative to the lab
In particular, for the kinetic energy:
∑
=
i
m
K
2
lab
,
lab
system,
2
1
∑
+
=
2
lab
CM,
CM
,
)
(
2
1
G
G
2
lab
CM,
lab
CM,
CM
i,
2
CM
i,
2
1
2
2
1
2
1
ii
+
+
=
∑
∑∑
G
G
2
lab
CM,
2
CM
i,
2
1
2
1
Mv
∑
+
=
0
CM
CM,
CM
i,
=
=
∑
G
G
Velocity of the CM relative to the CM
lab
CM,
CM
system,
lab
system,
+
=
lab
CM,
CM
system,
+
=
System of particles
System of particles
• The momentum of the system is the
momentum of the center of mass
• The kinetic energy of a system is the kinetic
energy of the center of mass
plus
the kinetic
energy of the parts of the system relative to
the center of mass.
Phy 221 2007S Lecture 21
N’s First Laws of C of M motion
The velocity of the C of M of a system remains constant unless
it is acted on by an external force.
– Note: Internal interactions within the system do not alter
this assertion.
• By conservation of momentum
does not change if there
are no external forces.
P
v
M
cm
G
G
=
Phy 221 2007S Lecture 21
N’s Second Laws of C of M motion
• The total external force on a system is
equal to (the total mass)(acceleration
of CofM)
• Note however that how that net force is
applied makes a difference in terms of what
happens within the system.
external
tot
cm
F
F
a
M
G
G
G
=
=
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View Full DocumentPhy 221 2007S Lecture 21
Lecture 21
Rotation of a Rigid body
Phy 221 2007S Lecture 21
Today’s Lecture
Today’s Lecture
• Review circular motion
• Energy of rotational motion
• Moment of inertia
Phy 221 2007S Lecture 21
Review of circular motion
;
;
dd
dt
θ
ω
θω
α
==
vR
=
y
x
s
R
Relation to
linear quantities:
s = R
Description in terms
of angular quantities
(in radians!):
tan
aR
=
Centripetal acceleration
2
2
c
v
a
tan
c
total
Phy 221 2007S Lecture 21
The righthand rule
•In 3D
ω
is a vector defined by the righthand rule
•Curl fingers of right hand in the direction of motion.
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 Spring '06
 Johnson
 Force, Kinetic Energy, Mass, Moment Of Inertia, Rigid Body, Rotation

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