Particle under central force in circular motion

[steven09]

Homework Statement

A particle moves under the influence of a central force, F_r(r) = − α r^−3. At time t = 0, the radius = r0 and the velocity = v0 as shown in the figure.
(A) Determine the potential energy function U(r).
(B) Determine the velocity vc such that the particle will move on the circle.
(C) Now suppose v0 = vc /2. The particle will spiral into the origin. Calculate the time it will take the particle to reach r = 0.

Homework Equations

U(r) = - integral F(r) dr

The Attempt at a Solution

I got U(r).

For my Part B) I don't know how to get velocity and Part C) how to get time. I think I might have to set ma=− α r^−3 but that doesn't seem right

 

[AvroArrow]

. For Part B) I think I need to use the equation v^2/r = (dU/dr) but I'm not sure how to relate it with the given information. For Part C) I think I need to use the conservation of energy U + K = 0, but I don't know how to get the kinetic energy K. Any help is appreciated. Thanks!