[UPSC CSE 2012] Central Force Motion and Gravitation (Q97, 20M)

A particle is moving in a central force field on an orbit given by $r = ke^{\alpha\theta}$ , where $k$ and $\alpha$ are positive constants, $r$ is the radial distance and $\theta$ is the polar angle.

(i) Find the force law for the central force field.

(ii) Find $\theta(t)$ .