Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 10C / a / Solution

The torque about the origin isSince $\dot{L}=\tau$, the angular momentum $L=mr\times \dot{r}$ is conserved. The force is the negative gradient of the inverse-square potentialso the conserved energy isDirectly, $\dot{E}=\dot{r}\cdot (m\ddot{r}+kmr/r^3)=0$.