A uniform thin rod $\mathrm{AB}$ of length $L$ and mass $m$ is undergoing rotation about a fixed axis passing through end $\mathrm{A}$ and perpendicular to the rod, such that end $\mathrm{A}$ remains stationary as shown. The kinetic energy of section $\mathrm{AP}$ of the rod is equal to the kinetic energy of section $\mathrm{BP}$ of the rod at an instant. Then what is the value of ${\left(\frac{\mathrm{A}\mathrm{B}}{\mathrm{A}\mathrm{p}}\right)}^3$?($\mathrm{AB}$ and $\mathrm{AP}$ are lengths of respective parts of the rod)