Center of Mass and
Momentum
Lecture 08
Monday: 9 February 2004
A special point…
•If the net external force on a system of particles is
zero, then (even if the velocity of individual objects
changes), there is a point associated with the
distribution of objects that moves with zero
acceleration (constant velocity).
•This point is called the “center of mass” of the system.
It is the balancing point for the mass distribution.
CENTER OF MASS
Three component equations :
1
1
1
xcm 
mn xn
ycm 
mn yn
zcm 


M
M
M
 M = m1 + m2 + ···
M = S mn
m z
n n
Position, Velocity and Acceleration of
CENTER OF MASS
xcm
1

M
m
xn
v cm
1

M
m
vn
acm
1

M
m
an
n
n
n
MOMENTUM
• For a system of particles, total momentum is:
P = p1 + p2 + p3 ··· or
P = S mnvn along with
v cm
1

M
m v
n
• P = Mvcm
n
MOMENTUM
dv cm
dP
M
 Ma cm
dt
dt
 Fext
dP

dt
CENTER OF MASS (CONT.)
•The overall translational motion of a system
of particles can be analyzed using Newton's
laws as if all the mass were concentrated at
the center of mass and the total external force
were applied at that point.
MOMENTUM
• For a particle, translational momentum is:


p  mv
• Momentum is a vector
• Its direction is the same as the object’s
velocity
• Momenta must be added as vectors
MOMENTUM (Review)
• For a particle, translational momentum is:


p  mv
• Momentum is a vector
• Its direction is the same as the object’s
velocity
• Momenta must be added as vectors
MOMENTUM of a SYSTEM
(Review)
• A system is any group of objects that we wish to
consider as a group.
• For a system of particles, momentum is:

 


ptotal  p1  p2  p3  ...  pn
i n

 i n 
ptotal   pi   mi vi
i 1
i 1
• This is a vector sum!!
We have
to take direction
into account.
dp
 F  dtimply?
What does
(Review)
1. Forces cause changes in an object’s momentum.
That is, forces cause the object’s velocity to change
over time.
2. We can determine the change in an object’s
velocity over time using this expression in the
form:
dp Pf  Pi
 F  dt  t  t
f
o
Internal and External Forces
•There are two classes of forces that act on and
within systems.
•INTERNAL FORCES are forces between an
object within the system and another object within
the system.
•EXTERNAL FORCES are forces between an
object within the system and an object outside the
system.
Only External Forces Change the
Total Momentum of the System
•
Recall Newton’s third law. For every force there is an equal
and opposite force. These “paired” forces are called Newton’s
third law pairs.
•
Two forces are a Newton’s third law pair if
1. The two different forces are between the same two objects
2. The object exerting the force and the object being acted on
switch roles in the two forces.
•
Third Law force pairs cancel out. So only forces external to a
system matter. ALL internal forces are third law pairs.
Conservation of Momentum
dp
Recall that  F 
dt
dp
For a system this is  FEXTERNAL 
dt
If the sum of external forces is zero, then
dp
0
dt
What does this imply?
Conservation of Momentum
•
dP
0
dt
states that the momentum
does not change over time. That is, it is
constant.
• If you know the momentum at any one time,
then you know it for all other times.
EXAMPLE: FExt = 0, FInt  0
Two masses, initially at rest.
PBefore = PAfter
0 = m1v1 + m2v2
m1v1 = – m2v2
If, m1 = m2
v1 = – v2
Relationship between Force and
Momentum
In one dimension (so dropping the vector symbol)
p  mv
if we take the derivative of both sides with respect to time
dp d (mv)

dt
dt
for a constant mass
dp
dv
m
 ma
dt
dt
and  F  ma so
F 
dp
dt
What does
dp
 F  dtimply?
1. Forces cause changes in an object’s momentum
over time. For constant mass objects, that means
forces cause the object’s velocity to change over
time.
2. We can determine the change in an object’s
velocity over time using this expression in the
form:
dp Pf  Pi
 F  dt  t  t
f
o
Reworking
dp
 F  dt
dp
dp
or Fnet 
dt
dt
we can rewrite this as Fnet (dt )  dp
F 
integratin g both sides of the equation

tf
ti

tf
ti
pf
Fnet dt 
 dp
pif
Fnet dt  p f  pi  p
IMPULSE
 tf 


J   Fdt  p f  p i
ti
In many cases, the integral is easily evaluated as
the area under a curve.
4
F (N)
0
5
8
Time (sec)
MOMENTUM of a SYSTEM
• A system is any group of objects that we wish to
consider as a group.
• For a system of particles, the total momentum is:

 


ptotal  p1  p2  p3  ...  pn
i n

 i n 
ptotal   pi   mi vi
i 1
1
• This is a vector sum!!
We ihave
to take direction
into account.
MOMENTUM (Review)
• For a particle, translational momentum is:


p  mv
• Momentum is a vector
• Its direction is the same as the object’s
velocity
• Momenta must be added as vectors
MOMENTUM of a SYSTEM
(Review)
• A system is any group of objects that we wish to
consider as a group.
• For a system of particles, momentum is:

 


ptotal  p1  p2  p3  ...  pn
i n

 i n 
ptotal   pi   mi vi
i 1
i 1
• This is a vector sum!!
We have
to take direction
into account.
dp
 F  dtimply?
What does
(Review)
1. Forces cause changes in an object’s momentum.
That is, forces cause the object’s velocity to change
over time.
2. We can determine the change in an object’s
velocity over time using this expression in the
form:
dp Pf  Pi
 F  dt  t  t
f
o
Internal and External Forces
•There are two classes of forces that act on and
within systems.
•INTERNAL FORCES are forces between an
object within the system and another object within
the system.
•EXTERNAL FORCES are forces between an
object within the system and an object outside the
system.
Only External Forces Change the
Total Momentum of the System
•
Recall Newton’s third law. For every force there is an equal
and opposite force. These “paired” forces are called Newton’s
third law pairs.
•
Two forces are a Newton’s third law pair if
1. The two different forces are between the same two objects
2. The object exerting the force and the object being acted on
switch roles in the two forces.
•
Third Law force pairs cancel out. So only forces external to a
system matter. ALL internal forces are third law pairs.
Conservation of Momentum
dp
Recall that  F 
dt
dp
For a system this is  FEXTERNAL 
dt
If the sum of external forces is zero, then
dp
0
dt
What does this imply?
Conservation of Momentum
•
dP
0
dt
states that the momentum
does not change over time. That is, it is
constant.
• If you know the momentum at any one time,
then you know it for all other times.
Conversation of Momentum
In one dimension,
m v  m v  ..  m dp
v  m v  ...
Recall that  F

dt
1
1i
2
2i
1
1f
2
2f
EXTERNAL
dp
If the sum of external forces is zero, then 0 
dt
(That is, the momentum is constant.)
This does not mean that the momentum of any one object in the system stays the same. It
means that if you add up all of the momenta for all of the objects in the system that this
total doesn’t change as time passes.
This is only true for systems on which
no net external force acts.
When motion occurs in two dimensions,
TWO conservation of momentum equations are
required.
mv m v
2 iy
mv m v
2 fy
mv m v
2 ix
mv m v
2 fx
1
1
1 iy
1 ix
2
2
1
1
1 fy
1 fx
2
2
Momentum is conserved or not conserved in each
direction SEPERATELY.
When motion occurs in two or more directions, we
need to consider each direction separately..
dp x
 FEXT , x  dt
dp y
 FEXT , y  dt
dp z
 FEXT , z  dt
If the sum of external forces is zero in any one
or all of the directions , then the momentum in
that (or those) direction( s) is constant.
Conversation of Momentum in Two
Dimensions
m v  m v  ...  m v  m v
2 fy
 ...
m v  m v  ...  m v  m v
2 fx
 ...
1
1
1 iy
1 ix
2
2
2 iy
2 ix
1
1
1 fy
1 fx
2
2
Remember, you know how to get the x and y
components of velocity…..
f
V
Vy= VCos f
Vx=VSinf
Momentum Transfers
In systems made up of several objects,
momentum can be transferred from one object
to another and still be conserved.
However,
momentum in the x direction can not be
transferred into the y direction and visa-versa.
Collisions
Momentum is conserved only if there is
no net external force acting on the system.
•Remember, if a system is composed of two or more objects
and these objects undergo a collision, then all forces between
the colliding objects are INTERNAL forces
and so sum to zero.
•There can be external forces acting on a system in which a
collision occurs. An example is friction between the objects
and the surface on which they move.
EXAMPLE: FExt = 0, FInt  0
Two masses, initially at rest.
PBefore = PAfter
0 = m1v1 + m2v2
m1v1 = – m2v2
If, m1 = m2
v1 = – v2