Paper 4, Section I, A

(a) Explain what is meant by a central force acting on a particle moving in three dimensions.

(b) Show that the orbit of a particle experiencing a central force lies in a plane.

(c) Show that, in the approximation in which the Sun is regarded as fixed and only its gravitational field is considered, a straight line joining the Sun and an orbiting planet sweeps out equal areas in equal times (Kepler's second law).

[With respect to the basis vectors $\left(\mathbf{e}_{r}, \mathbf{e}_{\theta}\right)$ of plane polar coordinates, the velocity $\dot{\mathbf{x}}$ and acceleration $\ddot{\mathbf{x}}$ of a particle are given by $\dot{\mathbf{x}}=(\dot{r}, r \dot{\theta})$ and $\ddot{\mathbf{x}}=\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right)$.]

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