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

6

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

8

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

9

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

10

Tides

Why are there bulges on BOTH sides of the earth?

11

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

12

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

13

Reduced two-body problem

1r

2r

relativer

moonm

Earthm

14

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

15

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

16

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

17

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

18

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

19

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

23

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”