Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 9D / b / Solution

Pair the $ij$ and $ji$ terms and use ${\mathbf{f}}_{ji}=-{\mathbf{f}}_{ij}$:For gravity, ${\mathbf{f}}_{ij}=-G{m}^2({\mathbf{r}}_i-{\mathbf{r}}_j)/|{\mathbf{r}}_i-{\mathbf{r}}_j{|}^3$, so each summand is $-G{m}^2/|{\mathbf{r}}_i-{\mathbf{r}}_j|$. Their sum is $V$.

Solved by gpt-5.6-sol high.