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Central Forces - Classical Mechanics - Lecture Slides, Slides of Classical Mechanics
These key points are discussed in these slides of classical mechanics : Central Forces, Two-Body System, Reduced Mass, Smaller Mass, Central Motion, Momentum, Orbital Angular, Velocity, Coplanar, Perpendicular
Typology: Slides
2012/2013
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Central
Forces
Two
‐Body
System
•^
Center
of
mass
R
•^
Equal
external
force
on
both
bodies.
•^
Add
to
get
the
CM
motion
•^
Subtract
for
relative
motion ext 1
int 1 (^11)
F
F
r m^
^
m^2
r^1
F^2
int r^2
R^
m^1
F^1
int
F^1
ext
F^2
ext^
r = r
- r 1
2
ext 2
int 2 (^22)
F
F
r m^
^
ext 2
ext 1
F
F
R
M
^
int 2 2
int 1 1 2 1
F m
F m r r
^
^
Central
Motion
•^
Central
motion
takes
place
in
a
plane.
-^
Force,
velocity,
and
radius
are
coplanar.
•^
Orbital
angular
momentum
is
constant.
•^
If
the
central
force
is
time
‐independent,
the
orbit
is
symmetrical
about
an
apse.
-^
Apse
is
where
velocity
is
perpendicular
to
radius
r r r r Jd dt
r r p r J
^
^
Use
J^
to avoid confusion
with Lagrangian
L
Central
Force
Equations
•^
Use
spherical
coordinates.
-^
Makes
r^ obvious
from
central
force.
-^
Generalized
forces
Q
= Q
=
-^
Central
force
need
not
be
from
a
potential.
-^
Kinetic
energy
expression
sin
(^
2 2 2 2 2 2
1 2
^
^
r
r r
T^
Q^ r T r T r d^ dt
^
T
T
d dt
T
T
d dt
Angle
Equation
•^
T^
doesn’t
depend
on
^ directly.
•^
Also
represents
constant
angular
momentum.^ –
A
constant
of
the
motion
•^
Change
the
time
derivative
to
an
angle
derivative.
T
T
d dt
T
d dt
J
r T^
^
2
constant^
d d J r
d d d dt d^ dt
2
Orbit
Equation
Let
u
= 1/
r
Qr T r T r d^ dt
^
Q^ r T r T r d d J^ r
^
^
2
Q^ r
r
r r
r
r r
d d J^ r
^
)]
[
)]
[^
2 2 2 1 2
2 2 2 1 2
2
Q^ r J r
dr d J r d d J r
J r r rd d J^ r
^
2 3
2
2 2 2
2
2
3
2
2
Q J
r
dr d r d d r
r
^
^
2 2
2 2
u Q J
u
d
u d^
r
^
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