Re: [Phys-l] Sharing a problem for students

["Bob Sciamanda" <trebor@winbeam.com>]

Hi Ludwik,
A:
>  . . . I do not have Fowles' book; what does
> he mean by "circular orbits are stable?'  Does he refer to a
> potential-energy minimum (which implies a zero net force)  or does 
> he
> refer to something else? . . .

Here is a summary of Fowles' treatment of a small perturbation in r of 
aa circular orbit:
The radial component of N2:
1) mr" -mh^2/r^3=f(r)
where r" is the 2nd time derivative of r, and h =L/m is a constant of 
the motion.

For a circular orbit with r = a:
2) -mh^2/a^3 = f(a)

Now write 1) in terms of the perturbation x = r-a:

3) mx"-mh^2(x+a)^-3 = f(x+a)

Expand in powers of x:

4) mx"- m(h^2)(a^-3)(1-3x/a + . . .)= f (a) + f '(a)x    or (using 2):

5) mx" + [-3f(a)/a - f '(a)]x = 0

Now if the coefficient of x is positive, this is SHM, and the circular 
orbit is stable to first order in the perturbation x = r-a . . . . . 
.etc.

B:
> . . .Ueff(R) = -G*M*m/R + m*v^2 / 2
> In other words, the so-called U_effective is the sum of  two 
> energies,
> kinetic and potential.  But the m*v^2 / 2 term is treated as if it 
> was
> potential energy corresponding to a repulsive force of some kind. 
> Any
> comments on this? . . .

This visualization aid of an effective potential is very commom in 
intermediate mechanics texts.  Fowles relegates this and related items 
to problems.
Ex:
Prob 6.21:  Show that the radial differential equation of central 
force motion is the same as that of a particle undergoing rectilinear 
motion in an "effectrive potential" U(r) = V(r) + mh^2/(2r^2) . . .

Prob 6.22: Show that the stability condition for a circular orbit of 
radius a is equivalent to the condition that U"(r) >0 for r = a. 
[U"(r) is the second derivative of U(r) with respect to r.]


I still don't appreciate your sense of a "paradox" here.


Bob Sciamanda
Physics, Edinboro Univ of PA (Emeritus)
www.winbeam.com/~trebor
trebor@winbeam.com