For a rigid slab in plane motion, show that the system of

Chapter 16, Problem 16.47

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For a rigid slab in plane motion, show that the system of the effective forces consists of vectors \(\left(\Delta m_{i}\right) \overline{\mathbf{a}}\), \(-\left(\Delta m_{i}\right) \mathrm{v}^{2} \mathbf{r}_{i}^{\prime}\), 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 \(\bar{\mathbf{a}}\) is the acceleration of the mass center G of the slab, V is the angular velocity of the slab, A 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 effective forces reduce to a vector \(m \bar{\mathbf{a}}\) attached at G and a couple \(\bar{I}\mathrm{A}\).