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

On Dec 24, 2007, at 9:24 AM, Bob Sciamanda wrote:

Hi Ludwik,
. . . 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
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 . . . . .

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

Dear Bob,
1) What is the better word than paradox?
On one hand we know that the net force must be zero at a potential minimum, on the other hand we know that the net force, G*M*m/r^2, is not zero.

2) Thanks for telling me that what I found in "Mechanical Universe" is common in intermediate mechanics textbooks. I never had a chance to teach such course.
Ludwik Kowalski, a retired physicist
5 Horizon Road, apt. 2702, Fort Lee, NJ, 07024, USA
Also an amateur journalist at