Title: Central-Force Motion Chapter 8
1Central-Force MotionChapter 8
- Prof. Claude A Pruneau
- Physics and Astronomy Department
- Wayne State University
28.1 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
38.2 Reduced Mass
- Description of a two-particle system
- Discussion restricted to frictionless
(conservative) systems.
48.2 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
5Lagrangian 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.
6Two-body CM Coordinates
Solving for r1 and r2
Substitute in the Lagrangian r1 and r2
2 to 1 reduction
78.3 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.
8Lagrangian 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
10Interpretation of l as Areal velocity
- The radius vector sweeps out an area dA in a time
interval dt.
The areal velocity is thus
11Keplers 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
138.4 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
15Remarks
- 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.
16Solution using Lagrange equations
Use variable change
Remember
17 18- Solving
- Substitute back into
- Which is useful if one wishes to find the force
law that produces a particular orbit rr(q).
19Example 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!
20Example 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
21Similarly for r(t), remember
And write
Answer (2)
Where l and C are determined by the initial
conditions
22Example 8.3 Total Energy
- What is the total energy of the orbit of the
previous two examples?
Solution Need U
23Given the reference
248.5 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.
288.6 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(No Transcript)
32Energy
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).
348.7 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.
37Hyperbola, 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
39a
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.
42Keplers Laws
- Planets move in elliptical orbits about the sun
with the sun at one focus. - The area per unit time swept out by a radius
vector from the sun to a planet is constant. - The square of a planets period is proportional
to the cube of the major axis of the planets
orbit.
43Example 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
44Basics 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.
45Inner Solar System
- The inner solar system contains
- Sun
- Mercury
- Venus
- Earth
- Mars
46Outer Solar System
- The planets of the outer solar system are
- Jupiter,
- Saturn,
- Uranus,
- Neptune,
- Pluto
47Some basic facts
48Nine Planets
49Nine Planets
50Principle Characteristics of the Planets
51Shoemaker-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.