Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 3C / a / Solution

The parallel axis theorem says that for an axis through $x$, parallel to the centre-of-mass axis in direction $\hat{d}$,LetMoments about the axis through $x_C$ add, soUsing $M_A=M_C-M_B$ and solving giveswhen the denominator is nonzero. In the degenerate equal-distance case, this inertia equation alone does not determine $M_B$; the centre-of-mass equation $M_C{x}_C=M_A{x}_A+M_B{x}_B$ supplies the remaining information.