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

On Dec 23, 2007, at 2:30 PM, Bob Sciamanda wrote:
Fowles defines and investigates the stability of circular, central
force orbits. (Paragraph 6.13 of the 2nd edition). He concludes that
for power law central forces, f(r)= -c*r^n, circular orbits are
stable if n > -3 . Thus the inverse square law (n = -2) and the
harmonic force law (n = 1) lead to stability. Nowhere does he speak
of "dynamic" equilibrium/stability.

1) OK, perhaps I was wrong in saying that the term "dynamic equilibrium" is used by some physicists to describe orbits. In any case, labels are not important. 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?

2) The conflict (between the durability of a circular orbit of a planet (m) around a sun (M>>m) and the minimum of potential energy) disappears when a positive term is added to the negative potential energy, U=-G*M*m/r. The magnitude of the positive term is L^2/(2*m*r^2), where L is the magnitude of the orbiting angular momentum. That approach is taken in "The Mechanical Universe: Mechanics and Heat," by S.C. Frautschi et al. (Advanced Edition, 1986). They refer to

Ueff(r) = - G*M*m/r + L^2/(2*m*r^2)

as an"effective potential energy." The plot of the Ueff versus the radius r does indeed have a minimum at the expected r=R. For a circular orbit of radius R, L=m*v*R and

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?

P.S.
Most of our textbooks explain why the planet speed v, on a circular orbit of radius R, must satisfy this equation:

v^2=G*M / R

where M is the mass of the sun. But multiplying each side by m/2 one has

m*v^2 /2 = G*M*m / (2*R)

Thus the kinetic energy, for a stationary circular orbit, is only one half of the absolute value of potential energy. Note that the negative term in the Ueff is proportional to the 1/r while the positive term is proportional to to 1/r^2.
______________________________________________________
Ludwik Kowalski, a retired physicist
5 Horizon Road, apt. 2702, Fort Lee, NJ, 07024, USA
Also an amateur journalist at http://csam.montclair.edu/~kowalski/cf/