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4. Conservation of linear momentum
One particle


p  mv


F  0  p  const
 
pF
System of particles


 ext

pn  Fn   Fnm  Fn
mn



Fnm   Fmn    Fnm  0
n mn


P   pn
  ext
PF
 ext  ext
 Fn  F
n
 ext

F  0  P  const
n
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Example (Completely inelastic collision):
The objects stick together after collision, so there is only one final velocity.






Pf  Pi  m1v  m2 v  m1v1  m1v1 


 m1v1  m1v1
v
m1  m2
Example (Rockets):
There are two objects: rocket and exhaust. The external force is zero.
This is a 1-D problem.
P  Pr  Pex


P0
dP  dPr  dPex  mdv  vdm   dmv  vex 
P  mv  m v  0
ex
m
v  vex
m
v  v0  vex lnmo m 
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Comments about Newton’s Third Law
Example (Two moving charges)
Electric and magnetic fields created by charge e at a point P,
where R is a vector from charge to the point P:


1 v2 c2 

E

e
R

R
v
c
Electric field:
 3
R  Rv c



 
Magnetic field: H  R  E  R

z
v2
y
v1
x
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Center of Mass (Center of inertia)
System of particles:

d  1

 d

P   pn   mn rn   mn rn  M 
dt n
dt  M
n
n
The system is moving as a particle
with mass M and radius vector R
Continuous mass distribution:
 
  ext
 

n mn rn  P  MR P  MR  F
 1

M   mn
R
m
r
 nn
M n
n
R
1 
1

r
dm


r
dV


M
M
Two inertial systems of reference:
  
vn  vn  v

 

 


P   mn vn   mn vn  v  mn  P  P  Mv
n
n



P  0  P  Mv
n
Always there is a system of reference
where the momentum is zero.
 
The velocity of this system is v  P m .
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5. Conservation of angular momentum
One particle
  
Angular momentum: l  r  p
  
Torque:   r  F
      
l  r  p  r  p  r  F
Newton’s second
Law for rotation:
 
l 
Compare with:
 
pF
Conservation of


angular momentum:   0  l  const
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