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Central-Force Motion Chapter 8

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Title: Central-Force Motion Chapter 8


1
Central-Force MotionChapter 8
  • Prof. Claude A Pruneau
  • Physics and Astronomy Department
  • Wayne State University

2
8.1 Introduction
  • Consider motion of two particles affected by a
    force connecting the center of the two bodies.
  • One of few problems that can be solved
    completely.
  • Historically important I.e. motion of planets,
    alpha-particle scattering on nuclei

3
8.2 Reduced Mass
  • Description of a two-particle system
  • Discussion restricted to frictionless
    (conservative) systems.

4
8.2 Reduced Mass (contd)
  • Assume a force is acting between the two
    particles along a line joining them.
  • Particle positions described in some arbitrary
    reference frame as positions r1 and r2, or in
    terms of the C.M. frame position, R, and the
    relative position vector r r1 - r2 .

m1
m1
CM
CM
m2
m2
Arbitrary Frame
CM Frame
5
Lagrangian for a two-body system
  • Assume the potential energy is only a function of
    the distance between the particles, UU(r)
  • Lagrangian may be written
  • Translational motion of the system uninteresting
  • Use R0.

6
Two-body CM Coordinates
  • We have

Solving for r1 and r2
Substitute in the Lagrangian r1 and r2
2 to 1 reduction
7
8.3 Conservation Theorems
  • Particle of mass m in a central force field
    described by the potential function U(r).
  • Symmetry implies conservation of angular momentum.

Radius vector and momentum lie in a plane normal
to the angular momentum vector L. The problem
reduced to 2 dimensions I.e. along r and q.
8
  • Lagrangian

Lagrangian cyclic in q implies Angular momentum,
pq, conjugate to q, is a conserved quantity.
First integral of motion
9
  • The systems symmetry permits the integration of
    one equation of motion.
  • pq is the first integral of the motion.
  • Denote it
  • Note
  • l can be negative or positive

10
Interpretation of l as Areal velocity
  • The radius vector sweeps out an area dA in a time
    interval dt.

The areal velocity is thus
11
Keplers 2nd law of planetary motion
  • No particular assumption made about the form of
    U(r) implies
  • This result is NOT limited to an inverse-square
    law force but is valid for all central forces.

12
  • Since the motion of the CM is not interesting,
    only one degree of freedom remains to be
    considered.
  • Linear momentum conservation adds nothing new
    here
  • Energy conservation provides the only remaining
    equation of motion.

Total Energy
13
8.4 Equations of Motion
  • Assume U(r) is specified. Solve for dr/dt

Solving for dt, and integrate to get a solution t
t(r). Invert it to get r r(t) Alternatively
obtain q q(r), starting with
14
  • Inversion of the result (if possible) yields the
    standard form (general) solution r r(t).
  • Because l is constant, dq/dt is a monotonic
    function of time.
  • The above integral is in practice possible only
    for a limited number of cases

15
Remarks
  • with F(r) rn, solutions may be expressed in
    terms of elliptic integrals for certain integers
    and fractional values of n.
  • Solution may be expressed in terms of circular
    functions for n1, -2, and 3.
  • Case n 1 is the harmonic oscillator.
  • Case n -2 is the inverse square law.

16
Solution using Lagrange equations
  • Lagrange equation for r

Use variable change
Remember
17
  • Compute

18
  • Solving
  • Substitute back into
  • Which is useful if one wishes to find the force
    law that produces a particular orbit rr(q).

19
Example 8.1 Log-spiral
  • Find the force law for a central-force field that
    allows a particle to move in a logarithmic spiral
    orbit given by, where k and a
    are constants.
  • Solution
  • Calculate
  • Now use
  • To find

Force is Attractive and Inverse cube!
20
Example 8.2 r(t), q(t)
  • Determine the functions r(t) and q(t) for the
    problem in Ex 8.1.

Solution Start with
Rearrange, integrate
Answer
21
Similarly for r(t), remember
And write
Answer (2)
Where l and C are determined by the initial
conditions
22
Example 8.3 Total Energy
  • What is the total energy of the orbit of the
    previous two examples?

Solution Need U
23
Given the reference
24
8.5 Orbits in a central field
  • Radial velocity of a particle in central field
  • Vanishes at the roots of the radical

25
  • Vanishing of dr/dt implies turning points
  • Two roots in general rmin and rmax.
  • Motion confined to an annular region between rmin
    and rmax.
  • Certain combinations of E and l may lead to a
    single root one then has a circular motion, and
    dr/dt0 at all times.

26
  • Periodic motion in U(r) implies the orbit is
    closed I.e. loops on itself after a certain
    number of excursions about the center of force.
  • The change in ? while going from rmin to rmax is
    a function of the potential and need not be 180o.
  • It can be calculated!
  • Because the motion is symmetric in time

27
  • Path closed only if Dq is a rational fraction of
    2p.
  • Dq 2p(a/b) where a and b are integers.
  • In this case, after b periods the particle will
    have completed a revolutions and returned to its
    original position.
  • For a closed noncircular
    path exists only for n-2 or 1.

28
8.6 Centrifugal Energy and Effective Potential
  • In dr/dt, dq/dt, , we have
  • Where each term has the dimension of energy.
  • Remember that
  • Write

29
  • Interpret as a
    potential energy
  • The associated force is
  • Traditionally called a centrifugal force.
  • Although it is, STRICTLY SPEAKING, NOT A FORCE
  • but rather a pseudo-force.
  • We continue to use the term nonetheless

30
  • The term can then be interpreted as
    the centrifugal potential energy, and included
    with U(r) to define an effective potential
    energy.
  • V(r) is fictitious potential that combines the
    real or actual potential U(r) with the energy
    term associated with the angular motion about the
    center of force.
  • For an inverse-square law central-force motion,
    one gets

31
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32
Energy
Turning point(s) (apsidal distances)
E1
unbound
r3
r4
r
r1
r2
E2
bound
E3
33
  • Values of E less than do
    not result in physically real motion given
    velocity is imaginary.
  • Techniques illustrated here are used in modern
    atomic, molecular and nuclear physics (but in the
    context of QM).

34
8.7 Planetary Motion Keplers Problem
  • Consider the specific case of an inverse-square
    force law.
  • Integral soluble for with variable substitution
    u1/r.
  • Define the origin of q so r is a minimum.

35
  • Define constants
  • Then one can re-write
  • To get the equation of a conic section with one
    focus at the origin

36
  • The quantity, e, is called eccentricity, and
  • 2a is termed the latus rectum of the orbit.
  • Conic sections are formed by the intersection of
    a plane and a cone.
  • More specifically by the loci of points (formed
    by a plane) where the ratio of the distance from
    a fixed point (the focus) to a fixed line (called
    the directrix) is a constant.

37
Hyperbola, egt1
Parabola, e1
Ellipse, 0ltelt1
Directrix For parabola
Circle, e0
38
  • q0 corresponds to a pericenter, i.e. rmin
    whereas rmax corresponds to the apocenter.
  • The general term for turning points is apsides.
  • Planetary Motion
  • Major axis
  • Minor axis

39
a
b
ae
P
P
a
40
  • Period of elliptic motion
  • The area of an ellipse is
  • The period is then.
  • Noting
  • One also finds

Keplers Third Law
41
  • Given the gravitational force
  • The square of the period
  • Where the last approx is realized for m1 ltlt m2.
  • Keplers statement is correct only if the mass m1
    of a planet can be neglected with respect to the
    mass m2 of the sun.
  • Correction needed for Jupiter given that it is
    1/1000 of the mass of the Sun.

42
Keplers Laws
  1. Planets move in elliptical orbits about the sun
    with the sun at one focus.
  2. The area per unit time swept out by a radius
    vector from the sun to a planet is constant.
  3. The square of a planets period is proportional
    to the cube of the major axis of the planets
    orbit.

43
Example 8.4
  • Halleys comet, which passed around the sun early
    in 1986, moves in a highly elliptical orbit with
    an eccentricity of 0.967 and a period of 76
    years. Calculate its minimum and maximum
    distances from the sun.
  • Solution
  • We thus find

44
Basics Facts
  • The solar system consists of
  • Sun
  • Nine planets
  • Sixty eight (68) satellites of the planets
  • A large number of small bodies
  • comets
  • Asteroids
  • Interplanetary medium.

45
Inner Solar System
  • The inner solar system contains
  • Sun
  • Mercury
  • Venus
  • Earth
  • Mars

46
Outer Solar System
  • The planets of the outer solar system are
  • Jupiter,
  • Saturn,
  • Uranus,
  • Neptune,
  • Pluto

47
Some basic facts
48
Nine Planets
49
Nine Planets
50
Principle Characteristics of the Planets
51
Shoemaker-Levy 9
  • A dramatic example of impact is the collision of
    20 large pieces of Comet Shoemaker-Levy 9 with
    Jupiter in the summer of 1994.
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