Central-Force Motion Chapter 8

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Central-Force MotionChapter 8

• Prof. Claude A Pruneau
• Physics and Astronomy Department
• Wayne State University

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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

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8.2 Reduced Mass

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

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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
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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.

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Two-body CM Coordinates

• We have

Solving for r1 and r2
Substitute in the Lagrangian r1 and r2
2 to 1 reduction
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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.
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• Lagrangian

Lagrangian cyclic in q implies Angular momentum,
pq, conjugate to q, is a conserved quantity.
First integral of motion
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• 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

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Interpretation of l as Areal velocity

• The radius vector sweeps out an area dA in a time
  interval dt.

The areal velocity is thus
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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.

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• 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
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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
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• 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

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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.

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Solution using Lagrange equations

• Lagrange equation for r

Use variable change
Remember
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• Compute

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

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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!
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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
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Similarly for r(t), remember
And write
Answer (2)
Where l and C are determined by the initial
conditions
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Example 8.3 Total Energy

• What is the total energy of the orbit of the
  previous two examples?

Solution Need U
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Given the reference
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8.5 Orbits in a central field

• Radial velocity of a particle in central field

• Vanishes at the roots of the radical

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• 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.

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• 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

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• 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.

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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

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• 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

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• 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

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Energy
Turning point(s) (apsidal distances)
E1
unbound
r3
r4
r
r1
r2
E2
bound
E3
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• 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).

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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.

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

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• 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.

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Hyperbola, egt1
Parabola, e1
Ellipse, 0ltelt1
Directrix For parabola
Circle, e0
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• 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

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

Keplers Third Law
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• 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.

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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.

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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

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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.

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Inner Solar System

• The inner solar system contains
• Sun
• Mercury
• Venus
• Earth
• Mars

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Outer Solar System

• The planets of the outer solar system are
• Jupiter,
• Saturn,
• Uranus,
• Neptune,
• Pluto

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Some basic facts
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Nine Planets
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Nine Planets
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Principle Characteristics of the Planets
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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.