For a rigid body in plane motion, show that the system of the inertial terms consists of vectors $\left(\Delta m_{i}\right) \overline{\mathbf{a}},-\left(\Delta m_{i}\right) \omega^{2} \mathbf{r}_{i}^{\prime},$ and $\left(\Delta m_{i}\right)\left(\boldsymbol{\alpha} \times \mathbf{r}_{i}^{\prime}\right)$

attached to the various particles $P_{i}$ of the body, where $\overline{\mathrm{a}}$ is the acceleration of the mass center $G$ of the body, $\omega$ is the angular velocity of the body, $\alpha$ is its angular acceleration, and $\mathbf{r}_{i}^{\prime}$ denotes the position vector of the particle $P_{i}$ relative to $G$. Further show, by computing their sum and the sum of their moments about $G$, that the inertial terms reduce to a vector $m \overline{\mathbf{a}}$ attached at $G$ and a couple $\bar{I} \alpha$.