Download System consisting of two objects

PHYS 172: Modern Mechanics
Lecture 6 – Momentum Conservation, Complex Systems
Summer 2012
Read 3.11-3.15
Electric force: the electric charges
Charges: property of an object
• Two types: positive (+) and negative (-)
• Like charges: repel.
• Opposite charges: attract
• Net charge of a system:
algebraic sum of all the charges
• Conservation of charge
• The force exerted by one point charge
on another acts along the line
joining the charges
Charge: measured in C (Coulomb)
Elementary charge: e = 1.602×10-19 C
Charge of electron is –e, of a proton +e
The electric force law (Coulomb’s law)
q1
Felec on1by 2
q2
r21
Felec on 2by1
Coulomb’s law
r̂21
Felec on 2by1 
1
q2 q1
4 0 r21
2
2
N×m
 9 109
4 0
C2
1
rˆ21
Predicting the future of a gravitational system
Massive star
And small planets
fixed
Two body: ellipse (or circle)
fixed
Determinism:
If we know the positions and momenta of all particles in
the Universe we can predict the future
Predicting the future of gravitational system
Solar system
Binary star
Sun, Earth and Moon
Problems:
Sensitivity Initial condition and t
Inability to account for all interactions
1025 molecules in glass of water !
Small particles: quantum mechanics
Probability and uncertainty
Example: a free neutron decays with ~15 minutes:
n  p   e  
Probability
t
Clicker:
Can we predict the motion of an electron near a free neutron?
A)Yes
B) No
Probability and uncertainty
Clicker poll:
The photon will:
A)
B)
Semitransparent mirror
photon
Reflect
Pass through
Where is the electron?
Classical
Quantum
The Heisenberg uncertainty principle
Werner Karl
Heisenberg
1901-1976
Position and momentum of a particle cannot be exactly measured simultaneously
1927
Nobel Prize: 1932
xpx  h
Planck’s constant
h = 6.6×10-34 kg.m2/s
E=h
1900
Nobel Prize: 1918
Max Planck
1858-1947
System consisting of two objects
F1,surr
system
p1
Momentum principle:
F1,int
(
)
Dp1 º p f ,1 - pi,1 = F1,surr + F1,int Dt
+
(
)
Dp2 º p f ,2 - pi,2 = F2,surr + F2,int Dt
F2,int
p2
F2,surr
(
)
p f ,1 + p f ,2 - pi,1 - pi,2 = F1,surr + F2,surr + F1,int + F2,int Dt
ptotal, f - ptotal,i
Fnet ,surr
Clicker question 2:
What is the last term in that equation?
A) Zero
B) Not zero, directed toward the larger object
C) Not zero, directed toward the smaller object
D) It is always positive
E) It is always negative
System consisting of two objects
F1,surr
system
p1
Momentum principle:
F1,int
(
)
Dp1 º p f ,1 - pi,1 = F1,surr + F1,int Dt
+
(
)
Dp2 º p f ,2 - pi,2 = F2,surr + F2,int Dt
F2,int
p2
F2,surr
(
)
p f ,1 + p f ,2 - pi,1 - pi,2 = F1,surr + F2,surr + F1,int + F2,int Dt
ptotal, f - ptotal,i
Fnet ,surr
0
Dptotal º ptotal, f - ptotal,i = Fnet Dt
Only from surrounding!
System consisting of many objects
Because all the forces inside the system come in pairs they cancel out.
The only forces left over are forces from the surroundings!
System consisting of several objects
The Momentum Principle for a system:
ptotal  ptotal , f  ptotal ,i  Fnet t
Total momentum of the system:
ptotal  p1  p2  p3  ...
The sum of all external forces
due to surrounding
Conservation of momentum
system
p1
F1,int
+
p1  p f ,1  pi ,1  F1,int t
p2  p f ,2  pi ,2  F2,int t


p f ,1  p f ,2  pi ,1  pi ,2  F1,int  F2,int t
F2,int
ptotal , f  ptotal ,i
p2
0
In the absence of external forces
system
p1  p2  0
p1
F1
Conservation of momentum
psystem  psurrounding  0
F2
p2
surrounding
ptotal  0
Reciprocity: Newton’s 3rd law
Fspring on mass
Fmass on spring
p  Fnet t
Force magnitudes are the same
Directions are opposite
They act on different objects
Fspring on mass   Fmass on spring
Reciprocity (Newton’s 3rd law):
The forces of two objects on each other are always equal
and are directed in opposite directions
NOTE: Velocity-dependent forces (e.g., magnetic forces) do not obey Newton’s 3rd
law!
Collisions: negligible external forces
p1 f
p2 f
p1i
1. Sticky ball
p1,i  p2,i  p1, f  p2, f
Assume =1:
m1
p2i
Momentum conservation:
m1v1i  m2v2i  m1v f  m2v f
m1v1i  m2v2i
vf 
m1  m2
m2
What if balls bounce?
m1v1i  m2v2i  m1v1 f  m2v2 f
Two unknowns, one equation