Elliptical orbit centered at the origin

[Natchanon]

Homework Statement

Under the influence of a central force F(r), a particle of mass m is observed to move in an elliptical orbit centered at the origin (the force center is not at one of the foci, as would be the case for a gravitational orbit)
a.) Show that the polar equation has the form 1/r = sqrt(A + B*sin^2(θ)) , where A and B are constants
b.) Show that the force f(r) comes from a simple spring (obeying Hooke's law) connected to the origin.

Homework Equations

Transformed radial equation u'' = -u - (mF)/(L^2u^2) where u = 1/r

The Attempt at a Solution

I know that in order to find r we must substitute F(r) in the equation. The problem is in this case the force comes from a spring or F = -kr, which when substituted into the eq, we will have u^3 in the denominator.

 

[mjc123]

F is not -kr, but -k(r - L), where L is the undistorted length of the spring.

 

[Orodruin]

F is not -kr, but -k(r - L), where L is the undistorted length of the spring.

If this was the case the orbit would not be elliptic. The potential needs to be a harmonic potential in two dimensions, which means that the force needs to vary as -kr and nothing else.