Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 2 / 12 / a

Solution

\frac{d E}{d t} = m \dot{r} \cdot \ddot{r} + k r \cdot \dot{r} = \dot{r} \cdot (m \ddot{r} + k r) = 0.

The equations of motion make the bracket zero. Hence the mechanical energy is constant in time. The central restoring force can exchange kinetic and potential energy but supplies no change in their sum.

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