
P
  i
 p i

Center of mass
 m i v
 i
  i i m i d
 r i dt

1 dt d


 i m i
 r i



M dt d


1
M
 i m i
 r i


M
  i m i v
 cm
 d
 r cm dt r


1 cm
M
 i m i
 r i

1
M
 i m i v
 i

P

M
 v cm
 a cm

 d v cm dt


F i

M
 a cm

Some basic properties of the CM
Equal Masses: in the middle
Unequal Masses: closer to heavy one m
1 d
1 d
2 m
2 d
1
:d
2
= m
2
: m
1
Note: inverted ratio!
A triangle of equal masses
Connect each corner to the center of the opposite side
Triangle of unequal masses
Connect each corner to the center of masses of the two other masses

 r cm
M


1
M
 i m i
 r i
 i m i
Continuous mass distribution
 r cm
M

 
1
M
 m ( r ) dr dm
Some basic properties of the CM
• Symmetry:
• The CM need not be inside the object!
• For a system composed of many parts, first condense each part to its CM and then treat each CM as a pointlike particle
1kg
C of M
2kg dm
1kg
C of M
2kg

Example: The disk shown in figure 1 is uniform and has its CM at the center. Suppose the disk is cut in half and its pieces arranged as shown in figure 2. Where is the CM of (2) compared to the CM of (1)?
Fig. 1
CM
Fig. 2
CM
A.
Higher
CM of top half
CM of bottom half
B.
Lower
C.
At the same level
Center of gravity
(the point where the gravitational force can be considered to act)
• It is the same as the center of mass as long as the gravitational force does not vary among different parts of the object.
• It can be found experimentally by suspending an object from different points.

In (a), the diver’s motion is pure translation; in (b) it is translation plus rotation .
There is one point that moves in the same path a particle would take if subjected to the same force as the diver.
This point is the center of mass.

High jumpers have developed a technique where their CM actually passes under the bar as they go over it. This allows them to clear higher bars.

Two people on frictionless roller blades, initially at rest, throw a ball back and forth. After a couple of throws, they are (ignore friction)
1. standing where they were initially.
2. standing farther away from each other.
3. standing closer together.
4. moving away from each other.
5. moving toward each other.

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:
K system, lab

 i

1
2 m i v i
2
, lab

1
2 i
 m i v i,
2
CM
 i

1
2 m i
( v i

, CM
1
2
2

 i
 m i v
 i, CM
 v

CM, lab
) 2

 v

CM, lab

1
2

 i
 m i

 v 2
CM, lab

1
2 i
 m i v i,
2
CM

1
2
Mv 2
CM, lab
 K system, CM
 m i v
 i, CM
 M v

CM, CM
 0 i
Velocity of the CM relative to the CM
K system, lab
 K
CM, lab
 K system, CM
 K
CM, lab