Download Center of mass Equal Masses

Center of mass


dri


 M  1
1 
P   pi   mi vi   mi
 d   mi ri  
d
dt dt  i
i
i
i
 dt  M

1
rcm 
M
M   mi
i

vcm

acm

drcm
1


dt
M

dvcm

dt

 mi ri
i

 mi v i
i


 Fi Macm

i mi ri 


P  Mvcm
Some basic properties of the CM
Equal Masses: in the middle
Unequal Masses: closer to heavy one
m1
d1
A triangle of equal masses
Connect each corner to
the center of the opposite side
d2
m2
d1:d2= m2: m1
Note: inverted ratio!
Triangle of unequal masses
Connect each corner to the center
of masses of the two other masses
Continuous mass distribution

1
rcm 
M

 mi ri
i
M   mi
i

1
rcm 
m(r )dr

M
dm
M   dm
Some basic properties of the CM
•Symmetry:
•The CM need not be inside the object!
1kg
•For a system composed of many parts,
first condense each part to its CM and
then treat each CM as a pointlike particle
C of M
2kg
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
Fig. 2
CM
×
CM
×
×
×
CM of top half
CM of bottom half
A.
Higher
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.
Center of Mass
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.
CM for the Human Body
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.
Catch on roller blades
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


1
1
2
2

m
(
v

v
)
  mivi ,lab 
i
i ,CM
CM, lab
2
2
i
i
 
1
1 
1
 2
2
  mivi,CM  2  mivi,CM v CM,lab    mi v CM, lab
2 i
2  i
2 i


1
1
2
2
  mivi,CM  Mv CM,
lab
2 i
2


m
v

M
v
 i i,CM
CM, CM  0
i
Velocity of the CM relative to the CM
 K system, CM  K CM, lab
K system, lab  K system, CM  K CM, lab