Re: A central force problem
From: George Jones (george_llew_jones_at_yahoo.com)
Date: 11/09/04
- Next message: jem: "Re: The Nature of Mathematics - a plea for help"
- Previous message: pete: "Re: PHYSICAL REVIEW FOCUS 8 November 2004"
- In reply to: Edward Green: "A central force problem"
- Next in thread: Mel Lep: "Re: A central force problem"
- Messages sorted by: [ date ] [ thread ]
Date: Tue, 9 Nov 2004 08:20:31 -0500
"Edward Green" <spamspamspam3@netzero.com> wrote in message
news:eca320d0.0411081723.1ed2f102@posting.google.com...
> A long time ago, a graduate student named Gregory said he was stumped
> by the following problem in preparing for his qualifying exams:
>
> Determine the central force law which gives a circular orbit with the
> force center on the circumferance.
>
> And a professor who had long since passed his qualifying exams
> suggested that such a force law was impossible, since angular momentum
> about the force center is invariant in an orbit under a central force
> law, but if the particle passes through the force center then the
> angular momentum must be zero.
>
> Years passed, and Gregory passed his qualifying exams, and the
> professor remained a professor, and eventually Ed, a wandering
> demi-savant who would philosophize for food, happened upon this
> question, bobbing to the oily surface of Usenet. And he thought it is
> true what was written, that in a central force, dL/dt = 0 about the
> force center; and that a particle passing through that center
> certainly seems to have zero angular momentum about it; but we may
> ask, when is zero not zero? Why, when it is multiplied by infinity!
>
> Suppose the velocity were infinite at r = 0, hence L indefinite, but
> that in the approach to the center v -> oo in such a way that that L
> remained constant.
>
> Now, would anybody like to complete the problem?
You hit the nail squarely on the head!
Take the origin as the centre of force. In polar coordinates, the
magnitude of the angular momentum about the origin is
L = m*r^2*(thetadot)^2.
So, L can be a finite, non-zero constant if thetadot -> oo as r -> 0.
Your post triggered a distant memory - years ago I had to do a similar
problem as an assignment question from Goldstein. Goldtein's question
gives more information, though:
------------------------------------------------------------------------
a) Show that if a particle describes a circular orbit under the
influence of an attractive central force directed toward a point on the
circle, then the force varies as the inverse fifth power of the
distance.
b) Show that for the orbit described the total energy is zero.
c) Find the period of the motion.
d) Find xdot, ydot, and v as a function of angle around the circle, and
show that all three quantities are infinite as the particle goes
through the centre of force.
------------------------------------------------------------------------
Goldstein gives
L^2*u^2/m*(d^2u/dtheta^2 + u) = -f(1/u), (1)
where u = 1/r, as the differential equation that describes orbits for
central forces.
r = d*cos(theta) is an equation of a diameter d circle that passes
through the orgin (centre of force). Take f = -k/r^n, with k > 0.
Then
d^2u/dtheta^2 = (1 + sin^2(theta))/(d*cos^3(theta)).
Substituting this, u, and f into (1), and simplifying gives
2*L^2/(m*d^3*cos^5(theta) = k/(d^n*cos^n(theta)).
L constant => n = 5.
Some more playing around gives v = d*thetadot, so
L = m*r^*(thetadot)^2 = m*r^2*v/d,
and
v = L*d/(m*r^2).
Regards,
George
- Next message: jem: "Re: The Nature of Mathematics - a plea for help"
- Previous message: pete: "Re: PHYSICAL REVIEW FOCUS 8 November 2004"
- In reply to: Edward Green: "A central force problem"
- Next in thread: Mel Lep: "Re: A central force problem"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
