1 class #19 central force motion gravitational law properties of inverse-square forces center of...
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1
Class #19
Central Force Motion Gravitational law Properties of Inverse-square
forces Center of Mass motion Lagrangian for Central forces Reduced Mass and CM
reference frame
2
Reading
Read Ch. 6 – Answer 4 Questions
1. Why is it called the “Principle of Least Action”?
2. What does Chapter 6 have to do with the Lagrangian method?
3. What part of the chapter did you find most interesting or novel?
4. What question(s) did the reading leave you with?
3
Test #2Lagrangian Method
Setup Generating equations of motion Looking for equilibria or solving equations for limiting
conditionsEnergy
Work and Work-KE Theorem Line integrals and gradients Conditions for conservative forces
Energy methods and dissipation Rolling and KE of rotation 1-D potentials and small oscillations about stable
pointsOscillations
Types of solutions Damped oscillators Driven oscillators
4
11
2
2
1 2
1 22
6.673(75) 10
ˆmm
F
G
N
G r
m
kg
m mU G
r
r
Gravity and Electrostatics
Gravity Electrostatics
Universal Constant
Force Law
Potential
9
0
2
1 22
2
1 2
0
0
16.987551... 10
4
1
1ˆ
4
4
Nm
Coulomb
q qU
q
r
qF r
r
5
Gauss’s Law – I - Flux
cosG g dA gdA
2A
1A
dA
g
θ
Figures from Dave Raymond
1 2g A g A
Flux (Phi) may be thought of as the number of liters of fluid per second flowing through an area.The field-strength (g) may be thought of as # liters/second-m2
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Gauss’s Law - II4G Enclosedg dA GM
Figure from Dave Raymond
Figure from Dave Raymond
Total Flux through a surface depends ONLY on the amount of mass contained inside that surface.
7
Gauss’s Law - II
Unique property of inverse-square forces.For spherical shell and r>rshell, treat the entire shell mass as if concentrated at a point at the center of the shell
4G Enclosedg dA GM
2
2 3
(4 ) 4
ˆ
Enclosed
Enclosed Enclosed
G
g dA g R GM
GM GMg r r
R R
Finally F mg
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Potential of a sphere
;
;
shellshell
shellshell
shell
mMFor r r U G
rmM
For r r U Gr
Using above formulae – 1) What is potential at radius r
outside a solid sphere of radius R and mass M?
2) What is potential inside a solid sphere of radius R (at radius r<R)?
3) What is force? Give answers in terms of r, R, and M
shellr
r
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Vectors and Central forces
Vectors Many forces are of
form Remove
dependence of result on choice of origin
1 2r r
1r
2r
Origin 1Origin 2
1 2( )F r r
2r
1r
1 2r r
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Tides
Why are there bulges on BOTH sides of the earth?
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Two particles with central forces
1 1 2
1 2 1 21 2 1 2 1 23
1 2 1 2
1 21 1 1 2 2 2
1 2
2 11 2
2 21 1 2 2
2 21
21 2 1 2
2 1
( ) ( ) ( )( )
1 1
2 2
;
; ;
( )
mm mmF r r G r r U r r G
r r r r
m mm r
m r m
r m r r Gr r
m mr R r r R
rR M m
rM M
m r m r
m
m r r rM
m m R m rM
L
2 2
12
mm r
M
1 2r r
1r
2r
Origin
1m
2m
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Two particles with central forces
2 2 2
1 2
2 2 2 1 2
21 22
1 2 1( ) ( )
2
1( )
1
2
1( )
2
0:
:
;
2
CM CM
r r U r
mmr r G
r
m mr r G
r
m mr G r
r
M R R
m m
M
MR R
r
rel
CM rel
rel
L
LL L
L
+
r
CMR
1m
2m
1 2mm MG G
r r
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Reduced two-body problem
1r
2r
relativer
moonm
Earthm
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The power of
21 2
2 2 2 2 21 2
2 21 2 1 2
1 2 2 2
1 1 1 1
2 2 2 2
i
i
Two real masses One reduced mass
I I I r
T T I I T I r
m v m m mmvr r r G G
r r r r
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The power of - part 22 2
1 2 1 1 2 2
2 2
2 11 2
2 222 1
1 22 2
2 22 1 1 21 2 2
2 1
2
I I m r m r
m mm r m r
M M
m mm m r
M M
m m mmmm r r
M m m
r
1r
2r
moonm
Earthm
21 2
2 2 2 2 21 2
2 21 2 1 22 2
1 1 1 1
2 2 2 2
i
i
Two real masses One reduced mass
I I I r
T I I T I r
m v m m mmvG G
r r r r
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Equivalent 1-D problem
2 2 2 1 2
21 22
21 22 3
1( )
2
:
mmr r G
rm m
r r G rr
m mr G
r r
relL
Relative Lagrangian
Radial equation
Total Radial ForcetotalF
total pseudoF dr U
2r
2
1 222pseudo
mmU G
r r
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Pseudopotential and Energy
21 2
22pseudo
mmU G
r r
1 2mmG
r
2
22 r
pseudoU
2 2 2 1 2
2 2 2 1 2
22 1 2
2
2
2
1( )
21( )
2
1
2 2
1
21
2
pseudo
mmT U r r G
rm m
T U r r Gr
m mE r G
r r
E r U
E r
rel
rel
rel
L
H
H
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Pseudopotential and Energy2
1 222pseudo
mmU G
r r
2
2
1
21
2
pseudoE r U
E r
0E bounded orbit
0E unbounded orbit
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Tides
Why are there bulges on BOTH sides of the earth?
20
Earth and Moon
211
26.67 10
28
2,419,200sec
?
NmG
kg
Lunar month days
T
24
22
6.0 10
7.3 10
Earth
moon
m kg
m kg
1 2 1
11 2
2
1 11
2 2
1
1; (1 )
mm mmm mm
m mIf m
m m
1. What is the reduced mass for
the earth-moon system (in kg)?2. How many percent different is
it than the lunar mass3. What is theta-dot?4. What is the radius of a circular
orbit?5. How would this change if the
earth were fixed in space by the hand of God or a Borg tractor beam?
21
Earth and Moon
211
26.67 10
28
2,419,200sec
?
NmG
kg
Lunar month days
T
24
22
6.0 10
7.3 10
Earth
moon
m kg
m kg
1 2 1
11 2
2
1 11
2 2
1
1; (1 )
mm mmm mm
m mIf m
m m
1. What is the reduced mass for
the earth-moon system (in kg)?2. How many percent different is
it than the lunar mass3. What is theta-dot?4. What is the radius of a circular
orbit?5. How would this change if the
earth were fixed in space by the hand of God or a Borg tractor beam?
22
Potential and Force for solid spheres
1
2
1
3
ˆ;
ˆ;
spheresphere
spheresphere
sphere
mMFor r R F G r
rm M
For r Rr
F G rR
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Class #19 Windup
2 2 2
1 2
1( ) ( )
2
1
2
1( )2
r r U r
mm
Mr G
M
rM R R r
rel
gravity
L
L
1
2
1
3
ˆ;
ˆ;
spheresphere
spheresphere
sphere
mmFor r R F G r
rm m r
For r R F G rR
<- Gravitational Lagrangian
<- General Central Force “1-D eqn”