For a rigid slab in centroidal rotation, show that the

Chapter 16, Problem 16.24

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For a rigid slab in centroidal rotation, show that the system of the effective forces consists of vectors \(-\left(\Delta m_{i}\right) \mathrm{v}^{2} \mathbf{r}_{i}^{\prime} \text { and }\left(\Delta m_{i}\right)\left(A \times \mathbf{r}_{i}^{\prime}\right)\) attached to the various particles \(P_{i}\) of the slab, where V and A are the angular velocity and angular acceleration of the slab, and where \(\mathbf{r}_{i}^{\prime}\) denotes the position vector of the particle \(P_{i}\) relative to the mass center G of the slab. Further show, by computing their sum and the sum of their moments about G, that the effective forces reduce to a couple \(\overline{I}\mathrm{A}\).